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Fundamental Concepts on Electromagnetic Theory
The document summarizes key concepts from a presentation on electromagnetic theory. It discusses different types of fields, including scalar and vector fields. It also covers gradient, divergence, curl, coordinate systems, static electric and magnetic fields, Maxwell's equations, and other fundamental electromagnetic concepts. Multiple students contributed sections on topics including Coulomb's law, Biot-Savart law, Lorentz force, and Maxwell's equations in differential, integral and harmonic forms.
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Fundamental Concepts on Electromagnetic Theory
- 1. Department of Electrical& Electronic Engineering Presentation on: Fundamental Concepts on Electromagnetic Theory. Group: A Name ID Md:Sefat Ahmmad 17102931 Md:Razib Hossan 17102935 Md:Abdullah Rifat Khan 17102936 Md:Sadi Khan(Group leader) 17102940 AL-AMIN 16102904 Date: Sunday, September 23, 2018
- 2. Field: If at everypoint of region, there is a corresponding value of some physical function, the region is called field. There are two types of field: (i)Scalar field (ii)Vector field Scalar field: If the value of physical function at every point is scalar quantity then the field is scalar field. For example: temperature, density etc. Vector field: If the value of physical function at every point is vector quantity then the field is vector field. For vector field a number and a direction are required. For example: velocity, gravitational force etc.
- 3. Gradient: The first typeof multiplication is the multiplication of a vector by a scalar and the result is a vector. Physical interpretation of gradient: let scalar function ∅ represents temperature, then 𝛻 ∙ ∅ or grad ∙ ∅ represent temperature gradient or rate of change of temperature with distance. Thus although temperature is a scalar quantity having magnitude only the temperature gradient 𝛻 ∙ ∅ is a vector quantity having magnitude and direction both in the maximum rate of change of temperature. This gradient vector can be expressed: 𝛻 ∙ ∅ = 𝑖 𝜕∅ 𝜕𝑥 + 𝑗 𝜕∅ 𝜕𝑦 + 𝑘 𝜕∅ 𝜕𝑧
- 4. Similarly if scalarfunction represents electric potential or gravitational potential, then 𝛻 ∙ ∅ would represent electric potential gradient or gradient of gravitational potential respectively.
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- 6. What is Divergence? Invector calculus ,divergence is a operator that produces a scalar field ,given the quantity of vector field’s source at each point. More technically the divergence represents the volume density of the outward flux of vector field.
- 7. What is curl? •In vector calculus ,the curl is a vector operator that describes the infinitesimal rotation of a vector field in three dimension Euclidean space . • A vector field whose curl is zero is called irrational curl .
- 8. What is co-ordinatesystem? • A co-ordinate system is a system designed to establish position with respect to given reference points .The co-ordinate system consists of one or more reference points.
- 9. • There arethree kinds of co-ordinate system . That’s are : • Cartesian co-ordinate system(x, y ,z):That’s system species each point in a plane by the pair of numerical co-ordinate . • Cylindrical co-ordinate system(r, θ,z):That’s a three dimension co- ordinates ,where each point is specified by the two polar co- ordinate of its perpendicular projection onto same fixed point. • Spherical co-ordinate system(r , , Ф ,):It is a co-ordinate system for three dimensional space where position of the point is specified by three numbers.
- 11. Md. Abdullah RifatKhan ID: 17102936
- 12. Static Field Static ElectricField Static Magnetic Field
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- 14. Coulomb’s Law The Force(F) between two charges (𝑄1 and 𝑄2) varies directly as the product of the charges and inversely as the square of the distance (r) between them. 𝐹 ∝ 𝑄1 𝑄2 𝑟2 Or, 𝐹 = 𝑘 𝑄1 𝑄2 𝑟2 , k is proportionality constant. 𝑘 = 1 4𝜋𝜀 , 𝜀 is permittivity or dielectric constant of medium. So the equation becomes, 𝐹 = 1 4𝜋𝜀 . 𝑄1 𝑄2 𝑟2 In vector form, 𝑭 = 1 4𝜋𝜀 . 𝑄1 𝑄2 𝑟2 𝒂 𝒓 , where 𝒂 𝒓 is unit vector.
- 15. Electric Field Intensity Electricfield intensity or electric field is denoted by E. If a unit charge (q) is placed at any point near a fix charge (Q), the unit charge experience a force. This force is know as Electric Field Intensity. Applying coulomb’s law, 𝐹 = 1 4𝜋𝜀 . 𝑄 𝑞 𝑟2 Now 𝐸 = 𝐹 𝑞 = 𝑄 𝑞 𝑞.4𝜋𝜀𝑟2 = 𝑄 4𝜋𝜀𝑟2 In vector form, 𝑬 = 𝑄 4𝜋𝜀𝑟2 𝒂 𝒓
- 16. Electric Potential If asmall body having charge Q and a test charge q is moved from infinity along radius line to point P at a distance R from the charge Q, then work done (W) on the system, 𝑊 = − ∞ 𝑅 𝐹. 𝑑𝑟 ∞ 𝑅 𝑄 𝑞 4𝜋𝜀𝑟2 Or, 𝑊 = − ∞ 𝑅 𝑄 𝑞 4𝜋𝜀𝑟2 𝑑𝑟 = − 𝑄 𝑞 4𝜋𝜀 1 𝑟 ∞ 𝑅 = 𝑄 𝑞 4𝜋𝜀𝑅 If the test charge is unit charge, then work done is potential V, 𝑉 = 𝑄 × 1 4𝜋𝜀𝑅 = 𝑄 4𝜋𝜀𝑅
- 17. Electric Charge Density𝜌 The electric charge density (𝜌) is the ratio of total charge Q in a volume V, to volume V. So, 𝜌 = 𝑄 𝑉 Volume charge density, 𝜌 𝑣 = lim ∆𝑣→0 ∆𝑄 ∆𝑣 , unit 𝐶𝑚−3 Surface charge density, 𝜌𝑠 = lim ∆𝑠→0 ∆𝑄 ∆𝑠 , unit 𝐶𝑚−2 Linear charge density, 𝜌 𝐿 = lim ∆𝐿→0 ∆𝑄 ∆𝐿 , unit 𝐶𝑚−1
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- 19. Faraday’s Law ofInduction If a varying magnetic field is placed near a conductor then an electromotive force (emf) is induced across the conductor. This emf is equal to the rate of change of the flux linkage of magnetic field. So, 𝑉 = − 𝑑𝜑 𝑑𝑡 Here, V = induced emf, in Volts 𝜑 = total magnetic flux, in weber
- 20. Biot-Savart’s Law It dealswith the magnetic field of current carrying element. The magnetic flux density (dB), is directly proportional to the length (dl) of the element, the current (I), the sine of the angle (𝜃) between direction of the current and the vector joining a given point of the field and the current element, and is inversely proportional to the square of the distance (r) of the given point. So, 𝑑𝐵 ∞ 𝐼 𝑑𝑙 𝑠𝑖𝑛𝜃 𝑟2 Or, 𝑑𝐵 = 𝑘 𝐼 𝑑𝑙 𝑠𝑖𝑛𝜃 𝑟2 , 𝑘 is proportional constant Here, 𝑘 = 𝜇 4𝜋 , 𝜇 is permeability In vacuum, 𝜇 = 4𝜋 × 10−7 H/m
- 21. Biot-Savart’s Law Now, 𝑑𝐵= 𝜇 4𝜋 𝐼 𝑑𝑙 𝑠𝑖𝑛𝜃 𝑟2 Integrating the equation, 𝑑𝐵 = 𝜇 4𝜋 𝐼 𝑑𝑙 𝑠𝑖𝑛𝜃 𝑟2 So, 𝐵 = 𝜇𝐼 4𝜋 𝑠𝑖𝑛𝜃 𝑑𝑙 𝑟2 The unit of magnetic flux density is T.
- 22. Lenz’s Law Lenz’s lawstates that when an emf is generated by a change in magnetic flux according to Faraday’s Law, the polarity of the induced emf is such, that it produces an current that’s magnetic field opposes the change which produces it. Case-1 Case-2
- 23. Lorentz Force Lorentz forceis the combination of electric force and magnetic force on a point charge due to electromagnetic fields. It is expressed mathematically in vector algebra as: 𝑭 = 𝑞𝑬 + 𝑞(𝒗 × 𝑩) Here, F = force experience by the particle q = charge of the particle E = electric field v = velocity of the particle B = magnetic flux
- 24. Md : SadiKhan ID : 17102940
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- 26. Current continuity equation: Acontinuity equation is the mathematical way to express this kind of statement. For example: The continuity equation for electric charge states that the amount of electric charge in any volume of space can only change by the amount of electric current flowing into or out of that volume through its boundaries
- 27. Let us considera small volume element dv as shown in figure placed inside a conducting medium. The current density 𝑗 a vector having the same direction as the current flow. In general the current density 𝑗 have three rectangular components which vary with position. ∇ ∙ 𝑗=0 This equation is the differential equation involving 𝑗 at a point. Continuous nature of 𝑗 and statement of Kirchhoff’s current law. Total flow of current out of the volume must be equal to the negative rate of charge with time. Total charge in the volume dv & 𝜌 is average charge density.
- 28. 𝑠 𝑗 ∙ 𝑑𝑠= − 𝜕 𝜕𝑡 𝑣 𝜌 dv 𝑠 𝑗 ∙ 𝑑𝑠= 𝑣 𝜕𝜌 𝜕𝑡 dv Applying divergence theorem, 𝑣 ∇ ∙ 𝑗 dv = 𝑣 𝜕𝜌 𝜕𝑡 ∙ 𝑑𝑣 since ∇ ∙ 𝑗 = 𝜕𝜌 𝜕𝑡 This equation is valid for all volume v & is called as the general equation of continuity between current density 𝑗 & charge density 𝜌 at a point. This also called equation of conservation of charge.
- 29. Displacement current: In electromagnetism,displacement current density is the quantity 𝜕𝐷 𝜕𝑡 appearing in Maxwell’s equations that is defined in terms of the rate of change of D, the electric displacement field. Displacement current density has the same units as electric current density, and it is a source of the magnetic field just as actual current is. However it is not an electric current of moving charges, but a time varying electric field.
- 30. Equation for displacementcurrent: The equation governing static electric field due to charge at rest and static magnetic field due to steady currents are already derived and are due to Gauss, Ampere & Faraday. They are reproduced here. From Gauss’s Law, From Faraday’s Law,
- 31. From Ampere’s Law, Equationof continuity for steady current is given by, Let us take divergence of both sides of (4), Now, ∇ ∙ 𝑗 = - 𝜕𝜌 𝜕𝑡 ⋯ ⋯ ⋯ ⋯(7)
- 32. From (7), Thus although∇ ∙ 𝑗 is not zero but the divergence of ( 𝑗 + 𝜕𝐷 𝜕𝑡 ) is always zero . Here , Maxwell made the assumption that the term 𝑗 of equation (4) must replaced by ( 𝑗 + 𝜕𝐷 𝜕𝑡 ) i.e. ∇x𝐻 = 𝑗 + 𝜕𝐷 𝜕𝑡
- 34. Department of Electrical& Electronic Engineering Name: AL-AMIN ID:16102904 Date: Sunday, September 23, 2018
- 35. 1.Definition & Historyof Maxwell’s equations: Gauss Law Gauss’ Magnetism Law Faraday’s Law Ampere’s Law Maxwell’s equations in- differential form integral form free space Harmonically Varying fields
- 36. Maxwell’s equations • Maxwell'sequations describe how electric charges and electric currents create electric and magnetic fields. Further, they describe how an electric field can generate a magnetic field, and vice versa.
- 37. Gauss Law • Gausslaw describes the nature of electric field around electric charges. The law is expressed in terms of electric charge density and electric charge density. • The inverted triangle is called as the divergence operator. • The equations hold good at any point in space. When the electric charge exists any somewhere, the divergence of D at that particular point is nonzero, else it is zero.
- 38. Gauss’ Magnetism Law •You can see that both the equations indicate the divergence of the field. The top equation states that the divergence of the Electric flux density D equals the volume of electric charge density. • The second equation states the divergence of the Magnetic Flux Density (B) is null.
- 39. Faraday’s Law Faraday wasa scientist whose experiment setup led to Faraday’s Law which is shown in the figure below. • The experiment is not very complex. When a battery is disconnected, no electricity flows through the wire. Hence, no magnetic flux is induced in the iron (Magnetic Core). The iron acts like a magnetic field that flows easily in a magnetic material. The purpose of the core is to form a path for the flow of magnetic flux.
- 40. Ampere’s Law • Thelaw shows the relationship between the flow of electric current and the magnetic field around it. Suppose the wire carries a current I, the current produces a magnetic field that surrounds the wire.
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- 43. Where Q isthe total charge within the volume v. Or, simply Or, simply Integral form:
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