Okay, let's break this down step-by-step:
* River flows southeast at 10 km/hr
* Let's define southeast as 45° from the east
* So the river's velocity is 10 cos(45°)ax + 10 sin(45°)ay = 7.07ax + 7.07ay
* Boat moves in the direction of the river at vB
* Man walks on deck at 2 km/hr perpendicular to the boat
* So the man's velocity is 2ay
* To find the total velocity, we add the velocities:
Total velocity = River velocity + Boat velocity + Man's velocity
= 7.07ax + 7.07ay + vB + 2
Maxwell's equations describe the relationship between electric and magnetic fields. Gauss' law states that the divergence of the electric flux density equals the electric charge density. Gauss' magnetism law states that the divergence of the magnetic flux density is always zero. Faraday's law describes how a changing magnetic field generates an electric field. Ampere's law shows the relationship between electric current and the surrounding magnetic field. Maxwell unified electricity, magnetism, and light through his equations, which can be written in differential or integral form and describe fields in free space or harmonically varying fields.
The document discusses Kirchhoff's laws of electrical circuits and their applications. Kirchhoff's first law, also known as the junction law, states that the algebraic sum of all currents meeting at a junction is zero. Kirchhoff's second law states that the algebraic sum of the potential differences (voltage drops) around any closed network plus the emfs in the circuit is zero. The document also explains Wheatstone bridge circuit, meter bridge method for determining unknown resistances, Kelvin's method for measuring galvanometer resistance using meter bridge, sources of errors and their minimization in these experiments, and the principle and applications of potentiometer for measuring emf and internal resistance of a cell.
- Electromagnetic induction occurs when a changing magnetic field induces an electromotive force (emf) in a conductor. This induced emf can drive an electric current.
- Faraday's law of induction states that the induced emf in a coil is proportional to the rate of change of the magnetic flux through the coil. A changing magnetic field is necessary to induce an emf and current.
- Lenz's law describes the direction of the induced current: the current will flow in the direction that opposes the change producing it. This ensures the law of conservation of energy is obeyed.
- Transformers take advantage of electromagnetic induction to change the voltage of an alternating current (AC) while transmitting power efficiently over
The document discusses motional electromotive force (emf) generated when a conductor moves through a magnetic field. It explains that as the conductor moves, a potential difference is created between its ends due to the separation of positive and negative charges. This potential difference, known as motional emf, is equal to the product of the magnetic field strength, length of the conductor, and its velocity perpendicular to the field. The document also provides examples of how motional emf causes induced currents in circuits involving moving conductors in magnetic fields.
Maxwell's equation and it's correction in Ampere's circuital lawKamran Ansari
This document discusses Maxwell's correction to Ampere's circuital law. It notes that Ampere's law was incomplete as it did not account for changing electric fields. Maxwell added a "displacement current" term to account for this. His full corrected law states that the curl of the magnetic field equals the permeability times the sum of the conduction current and the displacement current. This resolved inconsistencies in Ampere's law and completed the description of classical electromagnetism.
thevenin theorem.
SLIDE NUMBER 3 EXPLANATION OF THEOREM: it is possible to simplify any electrical circuit, no matter how complex, to an equivalent two-terminal circuit with just a single constant voltage source in series with a resistance (or impedance) connected to a load. SLIDE NUMBER 4 INVENTION STORY THE THEOREM WAS INDEPENDENTLY DERIVED IN 1853 BY THE GERMAN SCIENTIST HERMANN VON HELMHOLTZ. SLIDE NUMBER 5 EXPLANATION OF Thevenin’s equivalent circuit As far as the load resistor RL is concerned, any complex “one-port” network consisting of multiple resistive circuit elements and energy sources can be replaced by one single equivalent resistance Rs and one single equivalent voltage Vs. Rs is the source resistance value looking back into the circuit and Vs is the open circuit voltage at the terminals. SLIDE NUMBER 6 EXPLANATION OF DIAGRAM 1
Let us consider a simple DC circuit as shown in the figure above, where we have to find the load current IL by the Thevenin’s theorem. In order to find the equivalent voltage source, rL is removed from the circuit as shown in the figure below and Voc or VTH is calculated. SLIDE NUMBER 7 EXPLANATION OF DIAGRAM 2
Now, to find the internal resistance of the network (Thevenin’s resistance or equivalent resistance) in series with the open circuit voltage VOC , also known as Thevenin’s voltage VTH, the voltage source is removed or we can say it is deactivated by a short circuit (as the source does not have any internal resistance) SLIDE NUMBER 9 As per Thevenin’s Statement, the load current is determined by the circuit shown above and the equivalent Thevenin’s circuit is obtained. Where, VTH is the Thevenin’s equivalent voltage. It is an open circuit voltage across the terminal AB known as load terminal RTH is the Thevenin’s equivalent resistance, as seen from the load terminals where all the sources are replaced by their internal impedance rL is the load resistance Steps for Solving Thevenin’s Theorem Step 1 – First of all remove the load resistance rL of the given circuit. Step 2 – Replace all the impedance source by their internal resistance. Step 3 – If sources are ideal then short circuit the voltage source and open the current source. Step 4 – Now find the equivalent resistance at the load terminals know as Thevenin’s Resistance (RTH). Step 5 – Draw the Thevenin’s equivalent circuit by connecting the load resistance and after that determine the desired response. Slide number-10 Thevenin Voltage The Thevenin voltage e used in Thevenin's Theorem is an ideal voltage source equal to the open circuit voltage at the terminals. In the example below, the resistance R2 does not affect this voltage and the resistances R1 and R3 form a voltage divider
Slide number-11 Thevinin resistance The Thevenin resistance r used in Thevenin's Theorem is the resistance measured at terminals AB with all voltage sources replaced by short circuits and all current sources replaced by open circuits.
The document discusses alternating current (AC) circuits and concepts. It covers:
1. The definitions and characteristics of alternating emf and current, including their sinusoidal waveform and the relationships between peak, average, and root mean square (RMS) values.
2. The behavior of AC circuits containing a resistor, inductor, or capacitor, including the phase relationships between current and voltage.
3. Resonance in an AC LCR circuit, including definitions of resonant frequency and quality factor.
4. Calculations of power in AC circuits with various components and the concept of power factor.
5. Oscillations in an LC circuit and the damping of oscillations over time.
1. Electric charges and fields deals with forces, fields, and potentials arising from static electric charges. An electric charge is a fundamental property of matter that experiences an attractive or repulsive force. There are two types of charges: positive and negative.
2. Objects can be charged through friction, contact, or induction. Conductors allow electric charges to move through them, while insulators do not.
3. An electric dipole consists of two equal and opposite charges separated by a distance. It has a net electric field even though its total charge is zero. The electric field due to a dipole depends on distance and orientation relative to the dipole.
Maxwell's equations describe the relationship between electric and magnetic fields. Gauss' law states that the divergence of the electric flux density equals the electric charge density. Gauss' magnetism law states that the divergence of the magnetic flux density is always zero. Faraday's law describes how a changing magnetic field generates an electric field. Ampere's law shows the relationship between electric current and the surrounding magnetic field. Maxwell unified electricity, magnetism, and light through his equations, which can be written in differential or integral form and describe fields in free space or harmonically varying fields.
The document discusses Kirchhoff's laws of electrical circuits and their applications. Kirchhoff's first law, also known as the junction law, states that the algebraic sum of all currents meeting at a junction is zero. Kirchhoff's second law states that the algebraic sum of the potential differences (voltage drops) around any closed network plus the emfs in the circuit is zero. The document also explains Wheatstone bridge circuit, meter bridge method for determining unknown resistances, Kelvin's method for measuring galvanometer resistance using meter bridge, sources of errors and their minimization in these experiments, and the principle and applications of potentiometer for measuring emf and internal resistance of a cell.
- Electromagnetic induction occurs when a changing magnetic field induces an electromotive force (emf) in a conductor. This induced emf can drive an electric current.
- Faraday's law of induction states that the induced emf in a coil is proportional to the rate of change of the magnetic flux through the coil. A changing magnetic field is necessary to induce an emf and current.
- Lenz's law describes the direction of the induced current: the current will flow in the direction that opposes the change producing it. This ensures the law of conservation of energy is obeyed.
- Transformers take advantage of electromagnetic induction to change the voltage of an alternating current (AC) while transmitting power efficiently over
The document discusses motional electromotive force (emf) generated when a conductor moves through a magnetic field. It explains that as the conductor moves, a potential difference is created between its ends due to the separation of positive and negative charges. This potential difference, known as motional emf, is equal to the product of the magnetic field strength, length of the conductor, and its velocity perpendicular to the field. The document also provides examples of how motional emf causes induced currents in circuits involving moving conductors in magnetic fields.
Maxwell's equation and it's correction in Ampere's circuital lawKamran Ansari
This document discusses Maxwell's correction to Ampere's circuital law. It notes that Ampere's law was incomplete as it did not account for changing electric fields. Maxwell added a "displacement current" term to account for this. His full corrected law states that the curl of the magnetic field equals the permeability times the sum of the conduction current and the displacement current. This resolved inconsistencies in Ampere's law and completed the description of classical electromagnetism.
thevenin theorem.
SLIDE NUMBER 3 EXPLANATION OF THEOREM: it is possible to simplify any electrical circuit, no matter how complex, to an equivalent two-terminal circuit with just a single constant voltage source in series with a resistance (or impedance) connected to a load. SLIDE NUMBER 4 INVENTION STORY THE THEOREM WAS INDEPENDENTLY DERIVED IN 1853 BY THE GERMAN SCIENTIST HERMANN VON HELMHOLTZ. SLIDE NUMBER 5 EXPLANATION OF Thevenin’s equivalent circuit As far as the load resistor RL is concerned, any complex “one-port” network consisting of multiple resistive circuit elements and energy sources can be replaced by one single equivalent resistance Rs and one single equivalent voltage Vs. Rs is the source resistance value looking back into the circuit and Vs is the open circuit voltage at the terminals. SLIDE NUMBER 6 EXPLANATION OF DIAGRAM 1
Let us consider a simple DC circuit as shown in the figure above, where we have to find the load current IL by the Thevenin’s theorem. In order to find the equivalent voltage source, rL is removed from the circuit as shown in the figure below and Voc or VTH is calculated. SLIDE NUMBER 7 EXPLANATION OF DIAGRAM 2
Now, to find the internal resistance of the network (Thevenin’s resistance or equivalent resistance) in series with the open circuit voltage VOC , also known as Thevenin’s voltage VTH, the voltage source is removed or we can say it is deactivated by a short circuit (as the source does not have any internal resistance) SLIDE NUMBER 9 As per Thevenin’s Statement, the load current is determined by the circuit shown above and the equivalent Thevenin’s circuit is obtained. Where, VTH is the Thevenin’s equivalent voltage. It is an open circuit voltage across the terminal AB known as load terminal RTH is the Thevenin’s equivalent resistance, as seen from the load terminals where all the sources are replaced by their internal impedance rL is the load resistance Steps for Solving Thevenin’s Theorem Step 1 – First of all remove the load resistance rL of the given circuit. Step 2 – Replace all the impedance source by their internal resistance. Step 3 – If sources are ideal then short circuit the voltage source and open the current source. Step 4 – Now find the equivalent resistance at the load terminals know as Thevenin’s Resistance (RTH). Step 5 – Draw the Thevenin’s equivalent circuit by connecting the load resistance and after that determine the desired response. Slide number-10 Thevenin Voltage The Thevenin voltage e used in Thevenin's Theorem is an ideal voltage source equal to the open circuit voltage at the terminals. In the example below, the resistance R2 does not affect this voltage and the resistances R1 and R3 form a voltage divider
Slide number-11 Thevinin resistance The Thevenin resistance r used in Thevenin's Theorem is the resistance measured at terminals AB with all voltage sources replaced by short circuits and all current sources replaced by open circuits.
The document discusses alternating current (AC) circuits and concepts. It covers:
1. The definitions and characteristics of alternating emf and current, including their sinusoidal waveform and the relationships between peak, average, and root mean square (RMS) values.
2. The behavior of AC circuits containing a resistor, inductor, or capacitor, including the phase relationships between current and voltage.
3. Resonance in an AC LCR circuit, including definitions of resonant frequency and quality factor.
4. Calculations of power in AC circuits with various components and the concept of power factor.
5. Oscillations in an LC circuit and the damping of oscillations over time.
1. Electric charges and fields deals with forces, fields, and potentials arising from static electric charges. An electric charge is a fundamental property of matter that experiences an attractive or repulsive force. There are two types of charges: positive and negative.
2. Objects can be charged through friction, contact, or induction. Conductors allow electric charges to move through them, while insulators do not.
3. An electric dipole consists of two equal and opposite charges separated by a distance. It has a net electric field even though its total charge is zero. The electric field due to a dipole depends on distance and orientation relative to the dipole.
This document discusses harmonic waves and travelling harmonic waves. It provides relevant equations including that the wave number k is equal to 2pi divided by the wavelength lambda. It also defines the angular frequency w as equal to 2pi divided by the period T. The displacement of a travelling harmonic wave D(x,t) is described by the equation D(x,t) = A sin(kx - wt) where A is the amplitude, k is the wave number, x is position, t is time, and w is the angular frequency. An example problem is worked through to find the displacement, wavelength, and frequency of a travelling harmonic wave given its displacement equation.
This document provides instructions for viewing a presentation as a slideshow and navigating between its slides and sections. It can be viewed as a slideshow by selecting "View" and "Slide Show" from the menu bar. Clicking the right arrow or space bar advances the slides. Clicking on resources from the resources slide or lessons from the Chapter menu screen goes directly to those sections. The Esc key exits the slideshow.
This document discusses sinusoidal steady state analysis and phasors. It introduces representing sine waves with phase using phasors, which characterize a sinusoidal voltage or current using just amplitude and phase angle. Circuits containing resistors, inductors and capacitors are examined, showing their phasor relationships: voltage and current are in phase for resistors, voltage lags current by 90 degrees for inductors and leads current by 90 degrees for capacitors. Impedance is defined as the total opposition to AC current, consisting of resistance and reactance. Phasor diagrams provide a graphical method to solve circuit problems by showing the relationship between phasor voltages and currents.
1. Electromagnetic induction occurs when a magnetic flux through a circuit changes over time, inducing an emf and current.
2. Faraday's experiments demonstrated this effect and led to his laws of electromagnetic induction.
3. Lenz's law states that the direction of induced current will be such that it creates magnetic fields opposing the change producing it.
The document discusses the magnetic effects of electric current. It begins by describing Oersted's experiment which showed that electric current produces a magnetic field. It then provides several rules for determining the direction of magnetic fields, including Ampere's swimming rule, Maxwell's corkscrew rule, and the right hand thumb rule. The document goes on to define Biot-Savart's law and describes how to use it to calculate magnetic fields produced by straight wires, circular loops of wire, and solenoids. It also discusses the magnetic force on a current-carrying conductor and the torque experienced by a current-carrying coil in a uniform magnetic field.
Maxwell's equations describe the fundamental interactions between electricity and magnetism. They include:
1) Gauss's law for electric fields, which relates the electric flux through a closed surface to the electric charge enclosed.
2) Gauss's law for magnetic fields, which states that the magnetic flux through a closed surface is always zero, since there are no magnetic monopoles.
3) Faraday's law, which describes how a changing magnetic field induces an electric field. It relates the circulating electric field to the rate of change of the magnetic field.
4) The Ampere-Maxwell law, which describes how electric currents and changing electric fields generate magnetic fields. It relates the magnetic field to the electric current
SUBJECT: PHYSICS - Chapter 6 : Superposition of waves (CLASS XII - MAHARASH...Pooja M
1. The document discusses the physics concept of superposition of waves. It defines superposition as when two or more waves pass through a common point, the resulting displacement is the vector sum of the individual displacements.
2. Examples of superposition include two pulses of equal amplitude and same phase combining to produce a pulse with double the amplitude, and two pulses of equal amplitude and opposite phases combining to produce no net displacement.
3. Stationary waves occur when two identical waves travel in opposite directions through a medium, resulting in points of no displacement called nodes and points of maximum displacement called antinodes.
1. Cells connected in series have their emfs add up but their currents remain equal. Cells in parallel have the same emf but their currents divide.
2. The internal resistance of cells in series adds up while the reciprocal of the internal resistance adds up for parallel cells.
3. A mixed grouping of cells has some cells in series forming rows, and the rows in parallel. The total resistance is minimized when the rows' resistance equals the series resistance within rows.
This document provides a summary of key concepts related to electromagnetic induction and Maxwell's equations:
1) Faraday's law describes how a changing magnetic flux induces an electromotive force (emf). A changing magnetic field can also induce an electric field.
2) Maxwell proposed adding a "displacement current" term to Ampere's law to account for time-varying electric fields. This completes the theory to show that changing electric fields generate magnetic fields.
3) Maxwell's full set of equations symmetrically relate the electric and magnetic fields and show they are interdependent. In the absence of charges, the equations imply a relationship between electromagnetic phenomena and the speed of light.
Class 12th physics current electricity ppt Arpit Meena
1. The document discusses key concepts related to electric current including definitions of current and conventional current, drift velocity, current density, Ohm's law, resistance, resistivity, conductance, conductivity, and temperature dependence of resistance.
2. It also covers color codes for carbon resistors, series and parallel combinations of resistors, definitions of emf and internal resistance of cells, and series and parallel combinations of cells.
3. The document provides formulas and explanations for many important electrical concepts in a comprehensive yet concise manner.
This document discusses numerical methods and errors. It introduces that numerical methods provide approximate solutions rather than exact analytical solutions due to errors from measurements, algorithms, and output. Accuracy refers to how close an approximation is to the true value, while precision refers to the reproducibility of results. Significant figures indicate the precision of a number. True error, relative error, and percent error are defined to quantify the error between approximations and true values. Round-off errors from floating point representation on computers are also discussed.
Electrostatic potential is a scalar quantity measured in joules (J) or electron volts (eV). Equipotential surfaces represent regions in space where the electric potential is constant around a charged body. They are always perpendicular to electric field lines and can be represented by concentric spheres or lines. The relationship between electric field and potential is such that the electric field is defined as the negative gradient of the potential.
The document provides information on electric current, including definitions of conventional current, drift velocity, current density, and Ohm's law. It discusses resistance, resistivity, conductance, and conductivity and how they relate to temperature, length, and other factors. The document also covers color codes for carbon resistors, and series and parallel combinations of resistors and cells. It defines emf and potential difference, and discusses the internal resistance of cells and how series and parallel connections of cells affect total emf, internal resistance, and current.
This document provides an overview of key concepts in electrostatics. It defines important physics terms like charge, electron, nucleus, ion, and polarization. It explains phenomena such as charging by contact and induction. Formulas are given for Coulomb's law, electric potential, electric potential difference, and the relationship between force, charge, and electric field. Example problems and diagrams are included to illustrate electrostatics concepts and problem solving approaches. Problem solving tips are also outlined.
Resonance in electrical circuits – series resonancemrunalinithanaraj
This document discusses electrical resonance in series RLC circuits. It explains that series resonance occurs when the inductive and capacitive reactances cancel each other out, resulting in a minimum impedance. This is useful for applications that require a stable oscillating frequency, like radio transmission. The document defines key terms like resonant frequency, bandwidth, and quality factor (Q factor). It describes how the Q factor relates the peak stored energy to energy lost, and how a higher Q factor results in a narrower bandwidth.
Electric potential is defined as the electric potential energy per unit charge. It is measured in volts and represents the work required to move a charge between two points. The electric potential difference between two points is equal to the work needed to move a positive test charge between those points. Equipotential surfaces represent points in space where the electric potential is the same. Electric field lines are always perpendicular to equipotential surfaces.
Photoelectric Effect And Dual Nature Of Matter And Radiation Class 12Self-employed
This document discusses the photoelectric effect and the dual wave-particle nature of matter and light. It covers:
1) An overview of the photoelectric effect and how it demonstrated the particle nature of light via Einstein's photoelectric equation.
2) De Broglie's hypothesis that matter has wave-like properties described by the de Broglie wavelength.
3) Daviesson and Germer's experiment demonstrating the wave-like diffraction of electrons from a crystal lattice, verifying matter waves.
Vectors are quantities that have both magnitude and direction. They can be represented by capital letters with an arrow or lowercase letters with a bar. A vector has components in different dimensions - for two dimensions it has x and y components, and for three dimensions it has x, y, and z components. Some key vector concepts are parallel vectors (same direction), equal vectors (same magnitude and direction), negative/opposite vectors, free vectors (originate from different points), and position vectors (originate from the same point). The dot product of two vectors is a scalar quantity that depends on the angle between the vectors, and can be used to determine properties like whether vectors are parallel or orthogonal.
1. Electromagnetic induction is the phenomenon by which a changing magnetic field induces an electromotive force (emf) in a conductor. Experiments by Michael Faraday and Joseph Henry in the 1830s demonstrated this effect and established its laws.
2. Faraday's experiments showed that a changing magnetic flux induces a current in a coil. He placed coils inside changing magnetic fields from moving magnets and observed induced currents.
3. Lenz's law defines the direction of induced current: the current flows such that its magnetic field opposes the change that caused it. This ensures the conservation of energy.
Class 12th Physics Electrostatics part 2Arpit Meena
This document discusses the concepts of electrostatics including electric field, electric field intensity, and electric field lines. It defines electric field as a region around charged particles where other charges will experience a force. Electric field intensity is the force per unit charge on a small test charge. The electric field due to a point charge is defined by the equation E=kq/r^2 and has spherical symmetry. The superposition principle states that the total electric field is the vector sum of the fields due to individual charges. Electric field lines are imaginary lines showing the direction of the electric field. Key properties of electric field lines are also discussed. The document further explains the electric field due to an electric dipole, and the torque and work
Dec ''too simple'' digital electronics demystified (mc grawChethan Nt
This document is the table of contents for the book "Digital Electronics Demystified" by Myke Predko. It lists 12 chapters that make up the book, divided into two parts - an introduction to digital electronics and digital electronics applications. The chapters cover topics such as boolean logic, number systems, combinational and sequential logic circuits, oscillators, counters, interfaces, and computer processors.
This document discusses harmonic waves and travelling harmonic waves. It provides relevant equations including that the wave number k is equal to 2pi divided by the wavelength lambda. It also defines the angular frequency w as equal to 2pi divided by the period T. The displacement of a travelling harmonic wave D(x,t) is described by the equation D(x,t) = A sin(kx - wt) where A is the amplitude, k is the wave number, x is position, t is time, and w is the angular frequency. An example problem is worked through to find the displacement, wavelength, and frequency of a travelling harmonic wave given its displacement equation.
This document provides instructions for viewing a presentation as a slideshow and navigating between its slides and sections. It can be viewed as a slideshow by selecting "View" and "Slide Show" from the menu bar. Clicking the right arrow or space bar advances the slides. Clicking on resources from the resources slide or lessons from the Chapter menu screen goes directly to those sections. The Esc key exits the slideshow.
This document discusses sinusoidal steady state analysis and phasors. It introduces representing sine waves with phase using phasors, which characterize a sinusoidal voltage or current using just amplitude and phase angle. Circuits containing resistors, inductors and capacitors are examined, showing their phasor relationships: voltage and current are in phase for resistors, voltage lags current by 90 degrees for inductors and leads current by 90 degrees for capacitors. Impedance is defined as the total opposition to AC current, consisting of resistance and reactance. Phasor diagrams provide a graphical method to solve circuit problems by showing the relationship between phasor voltages and currents.
1. Electromagnetic induction occurs when a magnetic flux through a circuit changes over time, inducing an emf and current.
2. Faraday's experiments demonstrated this effect and led to his laws of electromagnetic induction.
3. Lenz's law states that the direction of induced current will be such that it creates magnetic fields opposing the change producing it.
The document discusses the magnetic effects of electric current. It begins by describing Oersted's experiment which showed that electric current produces a magnetic field. It then provides several rules for determining the direction of magnetic fields, including Ampere's swimming rule, Maxwell's corkscrew rule, and the right hand thumb rule. The document goes on to define Biot-Savart's law and describes how to use it to calculate magnetic fields produced by straight wires, circular loops of wire, and solenoids. It also discusses the magnetic force on a current-carrying conductor and the torque experienced by a current-carrying coil in a uniform magnetic field.
Maxwell's equations describe the fundamental interactions between electricity and magnetism. They include:
1) Gauss's law for electric fields, which relates the electric flux through a closed surface to the electric charge enclosed.
2) Gauss's law for magnetic fields, which states that the magnetic flux through a closed surface is always zero, since there are no magnetic monopoles.
3) Faraday's law, which describes how a changing magnetic field induces an electric field. It relates the circulating electric field to the rate of change of the magnetic field.
4) The Ampere-Maxwell law, which describes how electric currents and changing electric fields generate magnetic fields. It relates the magnetic field to the electric current
SUBJECT: PHYSICS - Chapter 6 : Superposition of waves (CLASS XII - MAHARASH...Pooja M
1. The document discusses the physics concept of superposition of waves. It defines superposition as when two or more waves pass through a common point, the resulting displacement is the vector sum of the individual displacements.
2. Examples of superposition include two pulses of equal amplitude and same phase combining to produce a pulse with double the amplitude, and two pulses of equal amplitude and opposite phases combining to produce no net displacement.
3. Stationary waves occur when two identical waves travel in opposite directions through a medium, resulting in points of no displacement called nodes and points of maximum displacement called antinodes.
1. Cells connected in series have their emfs add up but their currents remain equal. Cells in parallel have the same emf but their currents divide.
2. The internal resistance of cells in series adds up while the reciprocal of the internal resistance adds up for parallel cells.
3. A mixed grouping of cells has some cells in series forming rows, and the rows in parallel. The total resistance is minimized when the rows' resistance equals the series resistance within rows.
This document provides a summary of key concepts related to electromagnetic induction and Maxwell's equations:
1) Faraday's law describes how a changing magnetic flux induces an electromotive force (emf). A changing magnetic field can also induce an electric field.
2) Maxwell proposed adding a "displacement current" term to Ampere's law to account for time-varying electric fields. This completes the theory to show that changing electric fields generate magnetic fields.
3) Maxwell's full set of equations symmetrically relate the electric and magnetic fields and show they are interdependent. In the absence of charges, the equations imply a relationship between electromagnetic phenomena and the speed of light.
Class 12th physics current electricity ppt Arpit Meena
1. The document discusses key concepts related to electric current including definitions of current and conventional current, drift velocity, current density, Ohm's law, resistance, resistivity, conductance, conductivity, and temperature dependence of resistance.
2. It also covers color codes for carbon resistors, series and parallel combinations of resistors, definitions of emf and internal resistance of cells, and series and parallel combinations of cells.
3. The document provides formulas and explanations for many important electrical concepts in a comprehensive yet concise manner.
This document discusses numerical methods and errors. It introduces that numerical methods provide approximate solutions rather than exact analytical solutions due to errors from measurements, algorithms, and output. Accuracy refers to how close an approximation is to the true value, while precision refers to the reproducibility of results. Significant figures indicate the precision of a number. True error, relative error, and percent error are defined to quantify the error between approximations and true values. Round-off errors from floating point representation on computers are also discussed.
Electrostatic potential is a scalar quantity measured in joules (J) or electron volts (eV). Equipotential surfaces represent regions in space where the electric potential is constant around a charged body. They are always perpendicular to electric field lines and can be represented by concentric spheres or lines. The relationship between electric field and potential is such that the electric field is defined as the negative gradient of the potential.
The document provides information on electric current, including definitions of conventional current, drift velocity, current density, and Ohm's law. It discusses resistance, resistivity, conductance, and conductivity and how they relate to temperature, length, and other factors. The document also covers color codes for carbon resistors, and series and parallel combinations of resistors and cells. It defines emf and potential difference, and discusses the internal resistance of cells and how series and parallel connections of cells affect total emf, internal resistance, and current.
This document provides an overview of key concepts in electrostatics. It defines important physics terms like charge, electron, nucleus, ion, and polarization. It explains phenomena such as charging by contact and induction. Formulas are given for Coulomb's law, electric potential, electric potential difference, and the relationship between force, charge, and electric field. Example problems and diagrams are included to illustrate electrostatics concepts and problem solving approaches. Problem solving tips are also outlined.
Resonance in electrical circuits – series resonancemrunalinithanaraj
This document discusses electrical resonance in series RLC circuits. It explains that series resonance occurs when the inductive and capacitive reactances cancel each other out, resulting in a minimum impedance. This is useful for applications that require a stable oscillating frequency, like radio transmission. The document defines key terms like resonant frequency, bandwidth, and quality factor (Q factor). It describes how the Q factor relates the peak stored energy to energy lost, and how a higher Q factor results in a narrower bandwidth.
Electric potential is defined as the electric potential energy per unit charge. It is measured in volts and represents the work required to move a charge between two points. The electric potential difference between two points is equal to the work needed to move a positive test charge between those points. Equipotential surfaces represent points in space where the electric potential is the same. Electric field lines are always perpendicular to equipotential surfaces.
Photoelectric Effect And Dual Nature Of Matter And Radiation Class 12Self-employed
This document discusses the photoelectric effect and the dual wave-particle nature of matter and light. It covers:
1) An overview of the photoelectric effect and how it demonstrated the particle nature of light via Einstein's photoelectric equation.
2) De Broglie's hypothesis that matter has wave-like properties described by the de Broglie wavelength.
3) Daviesson and Germer's experiment demonstrating the wave-like diffraction of electrons from a crystal lattice, verifying matter waves.
Vectors are quantities that have both magnitude and direction. They can be represented by capital letters with an arrow or lowercase letters with a bar. A vector has components in different dimensions - for two dimensions it has x and y components, and for three dimensions it has x, y, and z components. Some key vector concepts are parallel vectors (same direction), equal vectors (same magnitude and direction), negative/opposite vectors, free vectors (originate from different points), and position vectors (originate from the same point). The dot product of two vectors is a scalar quantity that depends on the angle between the vectors, and can be used to determine properties like whether vectors are parallel or orthogonal.
1. Electromagnetic induction is the phenomenon by which a changing magnetic field induces an electromotive force (emf) in a conductor. Experiments by Michael Faraday and Joseph Henry in the 1830s demonstrated this effect and established its laws.
2. Faraday's experiments showed that a changing magnetic flux induces a current in a coil. He placed coils inside changing magnetic fields from moving magnets and observed induced currents.
3. Lenz's law defines the direction of induced current: the current flows such that its magnetic field opposes the change that caused it. This ensures the conservation of energy.
Class 12th Physics Electrostatics part 2Arpit Meena
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Okay, let's break this down step-by-step:
* River flows southeast at 10 km/hr
* Let's define southeast as 45° from the east direction
* So the river's velocity is 10 cos(45°)ax + 10 sin(45°)ay = 7.07ax + 7.07ay
* Boat moves in the direction of the river at some velocity v
* Man walks on the deck at 2 km/hr perpendicular to the boat's direction
* So the man's velocity is 2ay
* To find the boat's velocity v, we add the river velocity and man's velocity:
v = 7.07ax + 7.07ay + 2ay = 7
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1. This chapter discusses electrostatic fields, beginning with Coulomb's law and electric field intensity.
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Electromagnetic theory deals with the study of charges at rest and in motion and is fundamental to electrical engineering and physics. It is applicable to areas like communications, electrical machines, and quantum electronics. Electromagnetic theory can be considered a generalization of circuit theory and is necessary for understanding situations that cannot be analyzed solely through circuit theory. It involves the study of electric and magnetic fields as well as electric charges, which act as sources of electromagnetic fields that can redistribute other charges. Vector analysis is a useful mathematical tool for expressing and understanding electromagnetic concepts involving three spatial dimensions and time.
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2. Chapter 7
T•,—••• ' ' V ' '.S-f »
VECTOR ALGEBRA
One thing I have learned in a long life: that all our science, measured against
reality, is primitive and childlike—and yet is the most precious thing we have.
—ALBERT EINSTEIN
1.1 INTRODUCTION
Electromagnetics (EM) may be regarded as the study of the interactions between electric
charges at rest and in motion. It entails the analysis, synthesis, physical interpretation, and
application of electric and magnetic fields.
Kkctioniiiniutics (k.Yli is a branch of physics or electrical engineering in which
electric and magnetic phenomena are studied.
EM principles find applications in various allied disciplines such as microwaves, an-
tennas, electric machines, satellite communications, bioelectromagnetics, plasmas, nuclear
research, fiber optics, electromagnetic interference and compatibility, electromechanical
energy conversion, radar meteorology," and remote sensing.1'2 In physical medicine, for
example, EM power, either in the form of shortwaves or microwaves, is used to heat deep
tissues and to stimulate certain physiological responses in order to relieve certain patho-
logical conditions. EM fields are used in induction heaters for melting, forging, annealing,
surface hardening, and soldering operations. Dielectric heating equipment uses shortwaves
to join or seal thin sheets of plastic materials. EM energy offers many new and exciting
possibilities in agriculture. It is used, for example, to change vegetable taste by reducing
acidity.
EM devices include transformers, electric relays, radio/TV, telephone, electric motors,
transmission lines, waveguides, antennas, optical fibers, radars, and lasers. The design of
these devices requires thorough knowledge of the laws and principles of EM.
For numerous applications of electrostatics, see J. M. Crowley, Fundamentals of Applied Electro-
statics. New York: John Wiley & Sons, 1986.
2
For other areas of applications of EM, see, for example, D. Teplitz, ed., Electromagnetism: Paths to
Research. New York: Plenum Press, 1982.
3. 4 • Vector Algebra
+
1.2 A PREVIEW OF THE BOOK
The subject of electromagnetic phenomena in this book can be summarized in Maxwell's
equations:
V - D = pv (1.1)
V• B = 0 (1.2)
••*••• *-
• V X E = - — (1.3)
dt
VXH = J + — (1.4)
dt
where V = the vector differential operator
D= the electric flux density
B= the magnetic flux density
E = the electric field intensity
H = the magnetic field intensity
pv = the volume charge density
and J = the current density.
Maxwell based these equations on previously known results, both experimental and theo-
retical. A quick look at these equations shows that we shall be dealing with vector quanti-
ties. It is consequently logical that we spend some time in Part I examining the mathemat-
ical tools required for this course. The derivation of eqs. (1.1) to (1.4) for time-invariant
conditions and the physical significance of the quantities D, B, E, H, J and pv will be our
aim in Parts II and III. In Part IV, we shall reexamine the equations for time-varying situa-
tions and apply them in our study of practical EM devices.
1.3 SCALARS AND VECTORS
Vector analysis is a mathematical tool with which electromagnetic (EM) concepts are most
conveniently expressed and best comprehended. We must first learn its rules and tech-
niques before we can confidently apply it. Since most students taking this course have little
exposure to vector analysis, considerable attention is given to it in this and the next two
chapters.3 This chapter introduces the basic concepts of vector algebra in Cartesian coordi-
nates only. The next chapter builds on this and extends to other coordinate systems.
A quantity can be either a scalar or a vector.
Indicates sections that may be skipped, explained briefly, or assigned as homework if the text is
covered in one semester.
3
The reader who feels no need for review of vector algebra can skip to the next chapter.
I
4. 1.4 U N I T VECTOR
A scalar is a quantity that has only magnitude.
Quantities such as time, mass, distance, temperature, entropy, electric potential, and popu-
lation are scalars.
A vector is a quantity that has both magnitude and direction.
Vector quantities include velocity, force, displacement, and electric field intensity. Another
class of physical quantities is called tensors, of which scalars and vectors are special cases.
For most of the time, we shall be concerned with scalars and vectors.4
To distinguish between a scalar and a vector it is customary to represent a vector by a
letter with an arrow on top of it, such as A and B, or by a letter in boldface type such as A
and B. A scalar is represented simply by a letter—e.g., A, B, U, and V.
EM theory is essentially a study of some particular fields.
A field is a function that specifies a particular quantity everywhere in a region.
If the quantity is scalar (or vector), the field is said to be a scalar (or vector) field. Exam-
ples of scalar fields are temperature distribution in a building, sound intensity in a theater,
electric potential in a region, and refractive index of a stratified medium. The gravitational
force on a body in space and the velocity of raindrops in the atmosphere are examples of
vector fields.
1.4 UNIT VECTOR
A vector A has both magnitude and direction. The magnitude of A is a scalar written as A
or |A|. A unit vector aA along A is defined as a vector whose magnitude is unity (i.e., 1) and
its direction is along A, that is,
(1-5)
Note that |aA| = 1. Thus we may write A as
A = AaA (1.6)
which completely specifies A in terms of its magnitude A and its direction aA.
A vector A in Cartesian (or rectangular) coordinates may be represented as
(Ax, Ay, Az) or Ayay + Azaz (1.7)
4
For an elementary treatment of tensors, see, for example, A. I. Borisenko and I. E. Tarapor, Vector
and Tensor Analysis with Application. Englewood Cliffs, NJ: Prentice-Hall, 1968.
5. Vector Algebra
H 1 —-y
(a) (b)
Figure 1.1 (a) Unit vectors ax, ay, and az, (b) components of A along
ax, a^,, and az.
where Ax, A r and Az are called the components of A in the x, y, and z directions respec-
tively; ax, aT and az are unit vectors in the x, y, and z directions, respectively. For example,
ax is a dimensionless vector of magnitude one in the direction of the increase of the x-axis.
The unit vectors ax, a,,, and az are illustrated in Figure 1.1 (a), and the components of A along
the coordinate axes are shown in Figure 1.1 (b). The magnitude of vector A is given by
A = VA2X + Al + A (1-8)
and the unit vector along A is given by
Axax Azaz
(1.9)
VAT+AT+AI
1.5 VECTOR ADDITION AND SUBTRACTION
Two vectors A and B can be added together to give another vector C; that is,
C = A + B (1.10)
The vector addition is carried out component by component. Thus, if A = (Ax, Ay, Az) and
B = (Bx,By,Bz).
C = (Ax + Bx)ax + {Ay + By)ay + (Az + Bz)az (l.H)
Vector subtraction is similarly carried out as
D = A - B = A + (-B)
(1.12)
= (Ax - Bx)ax + (Ay - By)ay + (Az - Bz)az
6. 1.6 POSITION AND DISTANCE VECTORS
B
(a) (b)
Figure 1.2 Vector addition C = A + B: (a) parallelogram rule,
(b) head-to-tail rule.
Figure 1.3 Vector subtraction D = A -
B: (a) parallelogram rule, (b) head-to-tail
fA rule.
(a) (b)
Graphically, vector addition and subtraction are obtained by either the parallelogram rule
or the head-to-tail rule as portrayed in Figures 1.2 and 1.3, respectively.
The three basic laws of algebra obeyed by any giveny vectors A, B, and C, are sum-
marized as follows:
Law Addition Multiplication
Commutative A + B = B + A kA = Ak
Associative A + (B + C) = (A + B) + C k(( A) = (k()A
Distributive k(A + B) = kA + Zc
fB
where k and € are scalars. Multiplication of a vector with another vector will be discussed
in Section 1.7.
1.6 POSITION AND DISTANCE VECTORS
A point P in Cartesian coordinates may be represented by (x, y, z).
The position vector r,. (or radius vector) of point P is as (he directed silancc from
the origin () lo P: i.e..
r P = OP = xax + yay (1.13)
7. 8 • Vector Algebra
Figure 1.4 Illustration of position vector rP
3a, + 4a., + 5az.
4,5)
/I
--/ I
111 I A I
Figure 1.5 Distance vector rPG.
The position vector of point P is useful in defining its position in space. Point (3, 4, 5), for
example, and its position vector 3ax + 4a>( + 5az are shown in Figure 1.4.
The distance vector is ihc displacement from one point to another.
If two points P and Q are given by (xP, yP, zp) and (xe, yQ, ZQ), the distance vector (or
separation vector) is the displacement from P to Q as shown in Figure 1.5; that is,
r r r
PQ ~ Q P
= (xQ - xP)ax + (yQ - yP)&y + (zQ - zP)az (1.14)
The difference between a point P and a vector A should be noted. Though both P and
A may be represented in the same manner as (x, y, z) and (Ax, Ay, Az), respectively, the point
P is not a vector; only its position vector i> is a vector. Vector A may depend on point P,
however. For example, if A = 2xya,t + y2ay - xz2az and P is (2, - 1 , 4 ) , then A at P
would be — 4a^ + ay — 32a;,. A vector field is said to be constant or uniform if it does not
depend on space variables x, y, and z. For example, vector B = 3a^ — 2a^, + 10az is a
uniform vector while vector A = 2xyax + y2ay — xz2az is not uniform because B is the
same everywhere whereas A varies from point to point.
If A = 10ax - 4ay + 6a z andB = 2&x + av, find: (a) the component of A along ay, (b) the
EXAMPLE 1.1
magnitude of 3A - B, (c) a unit vector along A + 2B.
8. 1.6 POSITION AND DISTANCE VECTORS
Solution:
(a) The component of A along ay is Ay = - 4 .
(b) 3A - B = 3(10, - 4 , 6) - (2, 1, 0)
= (30,-12,18) - (2, 1,0)
= (28,-13,18)
Hence
|3A - B| = V28 2 + (-13) 2 + (18)2 = VT277
= 35.74
(c) Let C = A + 2B = (10, - 4 , 6) + (4, 2, 0) = (14, - 2 , 6).
A unit vector along C is
(14,-2,6)
V l 4 + (-2) 2 + 62
2
or
ac = 0.91 3ax - 0.1302a,, + 0.3906az
Note that |ac| = 1 as expected.
PRACTICE EXERCISE 1.1
Given vectors A = ax + 3a. and B = 5ax + 2av - 6a,, determine
(a) |A + B
(b) 5A - B
(c) The component of A along av
(d) A unit vector parallel to 3A 4- B
Answer: (a) 7, (b) (0, - 2 , 21), (c) 0, (d) ± (0.9117, 0.2279, 0.3419).
Points P and Q are located at (0, 2, 4) and ( - 3 , 1, 5). Calculate
(a) The position vector P
(b) The distance vector from P to Q
(c) The distance between P and Q
(d) A vector parallel to PQ with magntude of 10
9. 10 Vector Algebra
Solution:
(a) i> = 0ax + 2av + 4az = 2a, + 4az
(b) rPQ = rQ - i> = ( - 3 , 1, 5) - (0, 2, 4) = ( - 3 , - 1 , 1)
or = - 3 a x - ay + az
(c) Since rPQ is the distance vector from P to Q, the distance between P and Q is the mag-
nitude of this vector; that is,
d = |i> e | = V 9 + 1 + 1 = 3.317
Alternatively:
d= V(xQ - xPf + (yQ- yPf + (zQ - zPf
= V 9 + T + T = 3.317
(d) Let the required vector be A, then
A = AaA
where A = 10 is the magnitude of A. Since A is parallel to PQ, it must have the same unit
vector as rPQ or rQP. Hence,
r
PQ (-3,-1,1)
3.317
and
I0( 3
A = ± ' * ' — - = ±(-9.045a^ - 3.015a, + 3.015az)
PRACTICE EXERCISE 1.2
Given points P(l, - 3 , 5), Q(2, 4, 6), and R(0, 3, 8), find: (a) the position vectors of
P and R, (b) the distance vector rQR, (c) the distance between Q and R,
Answer: (a) ax — 3a y + 5a z , 3a* + 33,, (b) —2a* - ay + 2a z .
A river flows southeast at 10 km/hr and a boat flows upon it with its bow pointed in the di-
EXAMPLE 1.3
rection of travel. A man walks upon the deck at 2 km/hr in a direction to the right and per-
pendicular to the direction of the boat's movement. Find the velocity of the man with
respect to the earth.
Solution:
Consider Figure 1.6 as illustrating the problem. The velocity of the boat is
ub = 10(cos 45° ax - sin 45° a,)
= 7.071a^ - 7.071a, km/hr
10. 1.7 VECTOR MULTIPLICATION • 11
Figure 1.6 For Example 1.3.
w-
The velocity of the man with respect to the boat (relative velocity) is
um = 2(-cos 45° ax - sin 45° a,,)
= -1.414a, - 1.414a,, km/hr
Thus the absolute velocity of the man is
uab = um + uh = 5.657a., - 8.485ay
| u j = 10.2/-56.3"
that is, 10.2 km/hr at 56.3° south of east.
PRACTICE EXERCISE 1.3
An airplane has a ground speed of 350 km/hr in the direction due west. If there is a
wind blowing northwest at 40 km/hr, calculate the true air speed and heading of the
airplane.
Answer: 379.3 km/hr, 4.275° north of west.
1.7 VECTOR MULTIPLICATION
When two vectors A and B are multiplied, the result is either a scalar or a vector depend-
ing on how they are multiplied. Thus there are two types of vector multiplication:
1. Scalar (or dot) product: A • B
2. Vector (or cross) product: A X B
11. 12 HI Vector Algebra
Multiplication of three vectors A, B, and C can result in either:
3. Scalar triple product: A • (B X C)
or
4. Vector triple product: A X (B X C)
A. Dot Product
The dot product of two vectors A and B, wrilten as A • B. is defined geometrically
as the product of the magnitudes of A and B and the cosine of the angle between
them.
Thus:
A • B = AB cos I (1.15)
where 6AB is the smaller angle between A and B. The result of A • B is called either the
scalar product because it is scalar, or the dot product due to the dot sign. If A =
(Ax, Ay, Az) and B = (Bx, By, Bz), then
A • B = AXBX + AyBy + AZBZ (1.16)
which is obtained by multiplying A and B component by component. Two vectors A and B
are said to be orthogonal (or perpendicular) with each other if A • B = 0.
Note that dot product obeys the following:
(i) Commutative law:
A-B = B-A (1.17)
(ii) Distributive law:
A (B + C) = A B + A C (1.18)
A - A = |A| 2 = A2 (1.19)
(iii)
Also note that
ax • ay = ay • az = az • ax = 0 (1.20a)
ax • ax = ay • ay = a z • a z = 1 (1.20b)
It is easy to prove the identities in eqs. (1.17) to (1.20) by applying eq. (1.15) or (1.16).
12. 1.7 VECTOR MULTIPLICATION H 13
B. Cross Product
The cross product of two vectors A ;ind B. written as A X B. is a vector quantity
whose magnitude is ihe area of the parallclopiped formed by A and It (see Figure
1.7) and is in the direction of advance of a right-handed screw as A is turned into B.
Thus
A X B = AB sin 6ABan (1.21)
where an is a unit vector normal to the plane containing A and B. The direction of an is
taken as the direction of the right thumb when the fingers of the right hand rotate from A to
B as shown in Figure 1.8(a). Alternatively, the direction of an is taken as that of the
advance of a right-handed screw as A is turned into B as shown in Figure 1.8(b).
The vector multiplication of eq. (1.21) is called cross product due to the cross sign; it
is also called vector product because the result is a vector. If A = (Ax Ay, Az) and
B = (Bx, By, Bz) then
ax av a
z
A X B = Ax Ay K (1.22a)
Bx By Bz
- AzBy)ax + (AZBX - AxBz)ay + (AxBy - AyBx)az (1.22b)
which is obtained by "crossing" terms in cyclic permutation, hence the name cross
product.
Figure 1.7 The cross product of A and B is a vector with magnitude equal to the
area of the parallelogram and direction as indicated.
13. 14 H Vector Algebra
AXB AX B
*- A
(a) (b)
Figure 1.8 Direction of A X B and an using (a) right-hand rule, (b) right-handed
screw rule.
Note that the cross product has the following basic properties:
(i) It is not commutative:
A X B ^ B X A (1.23a)
It is anticommutative:
A X B = -B X A (1.23b)
(ii) It is not associative:
A X (B X C) =h (A X B) X C (1.24)
(iii) It is distributive:
A X ( B + C) = A X B + A X C (1.25)
(iv)
A XA = 0 (1.26)
Also note that
ax X ay = az
a, X a z = ax (1.27)
az X ax = ay
which are obtained in cyclic permutation and illustrated in Figure 1.9. The identities in eqs.
(1.25) to (1.27) are easily verified using eq. (1.21) or (1.22). It should be noted that in ob-
taining an, we have used the right-hand or right-handed screw rule because we want to be
consistent with our coordinate system illustrated in Figure 1.1, which is right-handed. A
right-handed coordinate system is one in which the right-hand rule is satisfied: that is,
ax X ay = az is obeyed. In a left-handed system, we follow the left-hand or left-handed
14. 1.7 VECTOR MULTIPLICATION 15
(a) (b)
Figure 1.9 Cross product using cyclic permutation: (a) moving
clockwise leads to positive results: (b) moving counterclockwise
leads to negative results.
screw rule and ax X ay = -az is satisfied. Throughout this book, we shall stick to right-
handed coordinate systems.
Just as multiplication of two vectors gives a scalar or vector result, multiplication of
three vectors A, B, and C gives a scalar or vector result depending on how the vectors are
multiplied. Thus we have scalar or vector triple product.
C. Scalar Triple Product
Given three vectors A, B, and C, we define the scalar triple product as
A • (B X C) = B • (C X A) = C • (A X B) (1.28)
obtained in cyclic permutation. If A = (Ax, Ay, Az), B = (Bx, By, Bz), and C = (Cx, Cy, Cz),
then A • (B X C) is the volume of a parallelepiped having A, B, and C as edges and is
easily obtained by finding the determinant of the 3 X 3 matrix formed by A, B, and C;
that is,
A • (B X C) = Bx By Bz (1.29)
Cy C,
Since the result of this vector multiplication is scalar, eq. (1.28) or (1.29) is called the
scalar triple product.
D. Vector Triple Product
For vectors A, B, and C, we define the vector tiple product as
A X (B X C) = B(A • C) - C(A • B) (1.30)
15. 16 • Vector Algebra
obtained using the "bac-cab" rule. It should be noted that
(A • B)C # A(B • C) (1.31)
but
(A • B)C = C(A • B). (1.32)
1.8 COMPONENTS OF A VECTOR
A direct application of vector product is its use in determining the projection (or compo-
nent) of a vector in a given direction. The projection can be scalar or vector. Given a vector
A, we define the scalar component AB of A along vector B as [see Figure 1.10(a)]
AB = A cos 6AB = |A| |aB| cos 6AB
or
AR = A • afl (1.33)
The vector component AB of A along B is simply the scalar component in eq. (1.33) multi-
plied by a unit vector along B; that is,
AB = ABaB = (A (1-34)
Both the scalar and vector components of A are illustrated in Figure 1.10. Notice from
Figure 1.10(b) that the vector can be resolved into two orthogonal components: one com-
ponent AB parallel to B, another (A - A s ) perpendicular to B. In fact, our Cartesian repre-
sentation of a vector is essentially resolving the vector into three mutually orthogonal com-
ponents as in Figure l.l(b).
We have considered addition, subtraction, and multiplication of vectors. However, di-
vision of vectors A/B has not been considered because it is undefined except when A and
B are parallel so that A = kB, where k is a constant. Differentiation and integration of
vectors will be considered in Chapter 3.
-»- B •- B
(a)
Figure 1.10 Components of A along B: (a) scalar component AB, (b) vector
component AB.
16. 1.8 COMPONENTS OF A VECTOR • 17
Given vectors A = 3ax + 4ay + az and B = 2ay - 5az, find the angle between A and B.
EXAMPLE 1.4
Solution:
The angle dAB can be found by using either dot product or cross product.
A • B = (3, 4, 1) • (0, 2, - 5 )
= 0 + 8 - 5 = 3
A| = V 3 2 + 42 + I2 = V26
B| = VO2 + 22 + (-5) 2 = V29
A B 3
COS BAR =
A B = 0.1092
I II I V(26)(29)
9AR = cos" 1 0.1092 = 83.73°
Alternatively:
az
A XB = 3 4 1
0 2 - 5
= ( - 2 0 - 2)ax + (0 + 15)ay + (6 - 0)az
= (-22,15,6)
|A X B + 152 + 62 = V745
A X Bj V745
sin 6AB = = 0.994
/
(26X29)
dAB = cos" 1 0.994 = 83.73°
PRACTICE EXERCISE 1.4
If A = ax + 3az and B = 5a* + 2ay - 6a., find 6AB.
Answer: 120.6°.
Three field quantities are given by
EXAMPLE 1.5
P = 2ax - a,
Q = 2a^ - ay + 2az
R = 2ax - 33^, + az
Determine
(a) (P + Q) X (P - Q)
(b) Q R X P
17. 18 Vector Algebra
(c) P • Q X R
(d) sin0 eR
(e) P X (Q X R)
(f) A unit vector perpendicular to both Q and R
(g) The component of P along Q
Solution:
(a) (P + Q) X (P - Q) = P X (P - Q) + Q X (P - Q)
=PXP-PXQ+QXP-QXQ
=O+QXP+QXP-O
= 2Q X P
ay a,
= 22 - 1 2
2 0 - 1
= 2(1 - 0) ax + 2(4 + 2) ay + 2(0 + 2) az
= 2a r + 12av 4a,
(b) The only way Q • R X P makes sense is
ay a
Q (RX P) = ( 2 , - 1 , 2 ) 2 -3 1
2 0 -1
= (2, - 1 , 2 ) -(3, 4, 6)
= 6 - 4 + 12 = 14.
Alternatively:
2 - 1 2
Q (R X P) = 2 - 3 1
2 0 - 1
To find the determinant of a 3 X 3 matrix, we repeat the first two rows and cross multiply;
when the cross multiplication is from right to left, the result should be negated as shown
below. This technique of finding a determinant applies only to a 3 X 3 matrix. Hence
Q (RXP)= _
•+
= +6+0-2+12-0-2
= 14
as obtained before.
18. 1.8 COMPONENTS OF A VECTOR 19
(c) From eq. (1.28)
P (Q X R) = Q (R X P) = 14
or
P (Q X R) = (2, 0, - 1 ) • (5, 2, - 4 )
= 10 + 0 + 4
= 14
(d) |QXR
IQIIRI 1(2,
/45 V5
= 0.5976
3V14 V14
(e) P X (Q X R) = (2, 0, - 1 ) X (5, 2, - 4 )
= (2, 3, 4)
Alternatively, using the bac-cab rule,
P X (Q X R) = Q(P R) - R(P Q)
= (2, - 1 , 2)(4 + 0 - 1) - (2, - 3 , 1)(4 + 0 - 2 )
= (2, 3, 4)
(f) A unit vector perpendicular to both Q and R is given by
±Q X R ±(5,2, - 4 )
3
|QXR|
= ± (0.745, 0.298, - 0 . 5 9 6 )
Note that |a| = l , a - Q = 0 = a - R . Any of these can be used to check a.
(g) The component of P along Q is
PQ = cos 6PQaQ
(P Q)Q
= (P • a G )a e =
IQI 2
(4 2 =
(4+1+4) 9
= 0.4444ar - 0.2222av + 0.4444a7.
PRACTICE EXERCISE 1.5
Let E = 3av + 4a, and F = 4a^ - 10av + 5a r
(a) Find the component of E along F.
(b) Determine a unit vector perpendicular to both E and F.
Answer: (a) (-0.2837, 0.7092, -0.3546), (b) ± (0.9398, 0.2734, -0.205).
19. 20 • Vector Algebra
FXAMPIF 1 f. Derive the cosine formula
a2 = b2 + c2 - 2bc cos A
and the sine formula
sin A sin B sin C
a b c
using dot product and cross product, respectively.
Solution:
Consider a triangle as shown in Figure 1.11. From the figure, we notice that
a + b + c = 0
that is,
b + c = -a
Hence,
a2 = a • a = (b + c) • (b + c)
= b b + c c + 2bc
a2 = b2 + c2 - 2bc cos A
where A is the angle between b and c.
The area of a triangle is half of the product of its height and base. Hence,
l-a X b| = l-b X c| = l-c X al
ab sin C = be sin A = ca sin B
Dividing through by abc gives
sin A sin B sin C
Figure 1.11 For Example 1.6.
20. 1.8 COMPONENTS OF A VECTOR 21
PRACTICE EXERCISE 1.6
Show that vectors a = (4, 0, - 1 ) , b = (1,3, 4), and c = ( - 5 , - 3 , - 3 ) form the
sides of a triangle. Is this a right angle triangle? Calculate the area of the triangle.
Answer: Yes, 10.5.
Show that points Ptf, 2, - 4 ) , P2{, 1, 2), and P 3 ( - 3 , 0, 8) all lie on a straight line. Deter-
EXAMPLE 1.7
mine the shortest distance between the line and point P4(3, - 1 , 0).
Solution:
The distance vector fptp2 is given by
r
r
PJP2 — r
p2 P,= (1,1 ,2) - ( 5 , 2, -4)
= (-4, - 1 6)
Similarly,
Tp,P3 = Tp3-1>, = ("3, 0,8) - (5, 2, -4)
= (-8, - 2 , 12)
r
PtP4 = r
P4 - • >, = (3, - 1,0) - (5, 2, -4)
= (-2, - 3 , 4)
a* a, az
rP P X rP p = -4 -1 6
-8 2 12
= (0,0, 0)
showing that the angle between r>iP2 and rPiPi is zero (sin 6 = 0). This implies that Ph P2,
and P3 lie on a straight line.
Alternatively, the vector equation of the straight line is easily determined from Figure
1.12(a). For any point P on the line joining P, and P2
where X is a constant. Hence the position vector r> of the point P must satisfy
i> - i>, = M*p2 ~ rP)
that is,
i> = i>, + (i> 2 - i>,)
= (5, 2, - 4 ) - X(4, 1, - 6 )
i> = (5 - 4X, 2 - X, - 4 + 6X)
This is the vector equation of the straight line joining Px and P2- If P 3 is on this line, the po-
sition vector of F 3 must satisfy the equation; r 3 does satisfy the equation when X = 2.
21. 22 Vector Algebra
(a)
Figure 1.12 For Example 1.7.
The shortest distance between the line and point P 4 (3, - 1 , 0) is the perpendicular dis-
tance from the point to the line. From Figure 1.12(b), it is clear that
d = rPiPt sin 6 = |r P| p 4 X aP]p2
312
= 2.426
53
Any point on the line may be used as a reference point. Thus, instead of using P as a ref-
erence point, we could use P3 so that
d= sin
PRACTICE EXERCISE 1.7
If P, is (1,2, - 3 ) and P 2 is ( - 4 , 0,5), find
(a) The distance P]P2
(b) The vector equation of the line P]P2
(c) The shortest distance between the line PP2 and point P3(7, - 1 , 2 )
Answer: (a) 9.644, (b) (1 - 5X)ax + 2(1 - X) av + (8X - 3) a^, (c) 8.2.
SUMMARY 1. A field is a function that specifies a quantity in space. For example, A(x, y, z) is a vector
field whereas V(x, y, z) is a scalar field.
2. A vector A is uniquely specified by its magnitude and a unit vector along it, that is,
A = AaA.
22. REVIEW QUESTIONS M 23
3. Multiplying two vectors A and B results in either a scalar A • B = AB cos 6AB or a
vector A X B = AB sin 9ABan. Multiplying three vectors A, B, and C yields a scalar
A • (B X C) or a vector A X (B X C).
4. The scalar projection (or component) of vector A onto B is AB = A • aB whereas vector
projection of A onto B is AB = ABaB.
1.1 Identify which of the following quantities is not a vector: (a) force, (b) momentum, (c) ac-
celeration, (d) work, (e) weight.
1.2 Which of the following is not a scalar field?
(a) Displacement of a mosquito in space
(b) Light intensity in a drawing room
(c) Temperature distribution in your classroom
(d) Atmospheric pressure in a given region
(e) Humidity of a city
1.3 The rectangular coordinate systems shown in Figure 1.13 are right-handed except:
1.4 Which of these is correct?
(a) A X A = |A| 2
(b)AXB + BXA = 0
(c) A • B • C = B • C • A
(d) axay = az
(e) ak = ax - ay
where ak is a unit vector.
(a) (c)
-*• y
y
(d) (e) (f)
Figure 1.13 For Review Question 1.3.
23. 24 H Vector Algebra
1.5 Which of the following identities is not valid?
(a) a(b + c) = ab + be
(b) a X (b + c) = a X b + a X c
(c) a•b = b•a
(d) c • (a X b) = - b • (a X c)
(e) aA • aB = cos dAB
1.6 Which of the following statements are meaningless?
(a) A • B + 2A = 0
(b) A • B + 5 = 2A
(c) A(A + B) + 2 = 0
(d) A•A+ B•B= 0
1.7 Let F = 2ax - 63^ + 10a2 and G = ax + Gyay + 5az. If F and G have the same unit
vector, Gy is
(a) 6 (d) 0
(b) - 3 (e) 6
1.8 Given that A = ax + aay + az and B = <xax + ay + az, if A and B are normal to each
other, a is
(a) - 2 (d) 1
(b) -1/2 (e) 2
(c) 0
1.9 The component of 6ax + 2a}, — 3az along 3ax — 4a>( is
(a) -12a x - 9ay - 3az
(b) 30a, - 40a^
(c) 10/7
(d) 2
(e) 10
1.10 Given A = — 6ax + 3ay + 2az, the projection of A along ay is
(a) - 1 2
(b) - 4
(c) 3
(d) 7
(e) 12
Answers: Lid, 1.2a, 1.3b,e, 1.4b, 1.5a, 1.6b,c, 1.7b, 1.8b, 1.9d, 1.10c.
24. PROBLEMS 25
1.1 Find the unit vector along the line joining point (2, 4, 4) to point ( - 3 , 2, 2).
PROBLEMS
1.2 Let A = 2a^ + 53^, - 3a z , B = 3a^ - 4a y , and C = ax + ay + az. (a) Determine
A + 2B. (b) Calculate |A - 5 C | . (c) For what values of k is |kB| = 2? (d) Find
(A X B)/(A • B).
1.3 If
A = ay - 3a z
C = 3ax 5a v 7a z
determine:
(a) A - 2B + C
(b) C - 4(A + B)
2A - 3B
(d) A • C - |B| 2
(e) | B X (|A + | C )
1.4 If the position vectors of points T and S are 3a^ — 23^, + az and Aax 4- 6ay + 2a x , re-
spectively, find: (a) the coordinates of T and S, (b) the distance vector from T to S, (c) the
distance between T and S.
1.5 If
A = 5a x
B = -*x Aay + 6a z
C = 8a x + 2a,
find the values of a and /3 such that a A + 0B + C is parallel to the y-axis.
1.6 Given vectors
A = aax + ay + Aaz
o — ^ax ~~
T p3y O3Z
C = 5a x - 2a y + 7a,
determine a, /3, and 7 such that the vectors are mutually orthogonal.
1.7 (a) Show that
(A • B) 2 + (A X B) 2 = (AB)2
a z X ax a^ X ay
a, • aY X a.' a, =
a* • ay X a z
25. 26 • Vector Algebra
1.8 Given that
P = 2ax - Ay - 2a z
Q = 4a_, + 3ay + 2a2
C = ~ax + ay + 2az
find: (a) |P + Q - R|, (b) P • Q X R, (c) Q X P • R, (d) (P X Q) • (Q X R),
(e) (P X Q) X (Q X R), (f) cos 6PR, (g) sin 6PQ.
1.9 Given vectors T = 2ax — 6ay + 3az and 8 = 3^-4- 2ay + az, find: (a) the scalar projec-
tion of T on S, (b) the vector projection of S on T, (c) the smaller angle between T and S.
1.10 If A = — ax + 6ay + 5az andB = ax + 2ay + 3ax, find: (a) the scalar projections of A
on B, (b) the vector projection of B on A, (c) the unit vector perpendicular to the plane
containing A and B.
1.11 Calculate the angles that vector H = 3ax + 5ay - 8az makes with the x-,y-, and z-axes.
1.12 Find the triple scalar product of P, Q, and R given that
P = 2ax - ay + az
Q = a^ + ay + az
and
R = 2a, + 3az
1.13 Simplify the following expressions:
(a) A X (A X B)
(b) A X [A X (A X B)]
1.14 Show that the dot and cross in the triple scalar product may be interchanged, i.e.,
A • (B X C) = (A X B) • C.
1.15 Points Pi(l, 2, 3), P2(~5, 2, 0), and P3(2, 7, - 3 ) form a triangle in space. Calculate the
area of the triangle.
1.16 The vertices of a triangle are located at (4, 1, - 3 ) , ( - 2 , 5, 4), and (0,1,6). Find the three
angles of the triangle.
1.17 Points P, Q, and R are located at ( - 1 , 4, 8), ( 2 , - 1 , 3), and ( - 1 , 2, 3), respectively.
Determine: (a) the distance between P and Q, (b) the distance vector from P to R, (c) the
angle between QP and QR, (d) the area of triangle PQR, (e) the perimeter of triangle PQR.
*1.18 If r is the position vector of the point (x, y, z) and A is a constant vector, show that:
(a) (r - A) • A = 0 is the equation of a constant plane
(b) (r — A) • r = 0 is the equation of a sphere
*Single asterisks indicate problems of intermediate difficulty.
26. PROBLEMS 27
Figure 1.14 For Problem 1.20.
(c) Also show that the result of part (a) is of the form Ax + By + Cz + D = 0 where
D = -(A2 + B2 + C2), and that of part (b) is of the form x2 + y2 + z2 = r2.
*1.19 (a) Prove that P = cos 0i&x + sin 6xay and Q = cos 82ax + sin 02ay are unit vectors in
the xy-plane respectively making angles &i and 82 with the x-axis.
(b) By means of dot product, obtain the formula for cos(0 2 — #i)- By similarly formulat-
ing P and Q, obtain the formula for cos(0 2 + #i).
(c) If 6 is the angle between P and Q, find —|P — Q | in terms of 6.
1.20 Consider a rigid body rotating with a constant angular velocity w radians per second about
a fixed axis through O as in Figure 1.14. Let r be the distance vector from O to P, the
position of a particle in the body. The velocity u of the body at P is |u| = dw=
r sin 6 |co or u = <o X r. If the rigid body is rotating with 3 radians per second about
an axis parallel to ax — 2ay + 2az and passing through point (2, —3, 1), determine the
velocity of the body at (1, 3,4).
1.21 Given A = x2yax — yzay + yz2az, determine:
(a) The magnitude of A at point T(2, —1,3)
(b) The distance vector from T to 5 if S is 5.6 units away from T and in the same direction
as A at T
(c) The position vector of S
1.22 E and F are vector fields given by E = 2xa_,. + ay + yzaz and F = xya x — y2ay+
xyzaz. Determine:
(a) | E | a t ( l , 2 , 3)
(b) The component of E along F at (1, 2, 3)
(c) A vector perpendicular to both E and F at (0, 1 , - 3 ) whose magnitude is unity