Electromagnetic theory


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Electromagnetic theory

  1. 1. ElectromagneticTheory
  2. 2. 221RQkQF =It states that the force F between two point charges Q1 and Q2 isCoulomb’s LawIn Vector formOrIf we have more than two point charges
  3. 3. QFE =Electric Field Intensity is the force per unit charge when placed in theelectric fieldElectric Field IntensityIn Vector formIf we have more than two point chargesE
  4. 4. If there is a continuous charge distribution say along a line, on asurface, or in a volumeElectric Field due to Continuous ChargeDistributionThe charge element dQ and the total charge Q due to these chargedistributions can be obtained by
  5. 5. The electric field intensity due to each charge distribution ρL, ρS andρV may be given by the summation of the field contributed by thenumerous point charges making up the charge distribution.
  6. 6. The electric field intensity depends on the medium in which thecharges are placed.Electric Flux DensityThe electric flux ψ in terms of D can be defined asSuppose a vector field D independent of the medium is defined byED oε=The vector field D is called the electric flux density and is measured incoulombs per square meter.
  7. 7. Electric Flux DensityFor an infinite sheet the electric flux density D is given byFor a volume charge distribution the electric flux density D is givenbyIn both the above equations D is a function of charge and positiononly (independent of medium)
  8. 8. Gauss LawIt states that the total electric flux ψ through any closed surface isequal to the total charge enclosed by that surface.encQ=ψ(i)
  9. 9. Using Divergence Theorem(ii)Comparing the two volume integrals in (i) and (ii)This is the first Maxwell’s equation.It states that the volume charge density is the same as the divergenceof the electric flux density.
  10. 10. Electric PotentialElectric Field intensity, E due to a charge distribution can be obtainedfrom Coulomb’s Law.or using Gauss Law when the charge distribution is symmetric.We can obtain E without involving vectors by using the electric scalarpotential V.From Coulomb’s Law the force on pointcharge Q isEQF =The work done in displacing the chargeby length dl isdlFdW .−= dlEQ .−=The negative sign indicates that the work is being done by an external agent.
  11. 11. The total work done or the potential energy required in moving thepoint charge Q from A to B isdlEQWBA.∫−=Dividing the above equation by Q gives the potential energy per unitcharge.dlEQWBA.∫−= ABV=ABV is known as the potential difference between points A and B.1. If is negative, there is loss in potential energy in moving Qfrom A to B (work is being done by the field), if is positive, thereis a gain in potential energy in the movement (an external agent doesthe work).ABVABV2. It is independent of the path taken. It is measured in Joules perCoulomb referred as Volt.
  12. 12. The potential at any point due to a point charge Q located at the origin isThe potential at any point is the potential difference between thatpoint and a chosen point at which the potential is zero.Assuming zero potential at infinity, the potential at a distance r fromthe point charge is the work done per unit charge by an external agentin transferring a test charge from infinity to that point.ldEVr.∫∞−=If the point charge Q is not at origin but at a point whose positionvector is , the potential at becomesr r)( rV||4)(rrQrVo −=πεrQVoπε4=
  13. 13. For n point charges Q1, Q2, Q3…..Qn located at points with positionvectors the potential at isIf there is continuous charge distribution instead of point charges thenthe potential at becomesnrrrr .....,, 321 r∑= −=nk kko rrQrV1 ||41)(πεr
  14. 14. Relationship between E and VThe potential difference between points A and B is independent of thepath takenldEVBAAB .∫−=BAAB VV −=and ldEVABBA .∫=0. ==+ ∫ ldEVV BAAB0. =∫ ldEIt means that the line integral of along a closed path must be zero.E(i)
  15. 15. Physically it means that no net work is done in moving a charge alonga closed path in an electrostatic field.0).(. =×∇= ∫∫ SdEldEEquation (i) and (ii) are known as Maxwell’s equation for staticelectric fields.(ii)Applying Stokes’s theorem to equation (i)0=×∇ EEquation (i) is in integral form while equation (ii) is in differentialform, both depicting conservative nature of an electrostatic field.
  16. 16. AlsoIt means Electric Field Intensity is the gradient of V.VE −∇=The negative sign shows that the direction of is opposite to thedirection in which V increases.E
  17. 17. Consider an atom of the dielectric consisting of an electron cloud (-Q)and a positive nucleus (+Q).Polarization in DielectricsWhen an electric field is applied, the positive charge is displacedfrom its equilibrium position in the direction of by whilethe negative charge is displaced by in the oppositedirection.EE EQF =+A dipole results from the displacement of charges and the dielectric ispolarized. In polarized the electron cloud is distorted by the appliedelectric field.EQF =−
  18. 18. where is the distance vector between -Q to +Q.dQp =dIf there are N dipoles in a volume Δv of the dielectric, the total dipolemoment due to the electric fieldPThis distorted charge distribution is equivalent to the originaldistribution plus the dipole whose moment isFor the measurement of intensity of polarization, we definepolarization (coulomb per square meter) as dipole moment per unitvolume
  19. 19. The major effect of the electric field on the dielectric is the creation ofdipole moments that align themselves in the direction of electric field.This type of dielectrics are said to be non-polar. eg: H2, N2, O2Other types of molecules that have in-built permanent dipole momentsare called polar. eg: H2O, HClWhen electric field is applied to a polar material then its permanentdipole experiences a torque that tends to align its dipole moment in thedirection of the electric field.
  20. 20. Consider a dielectric material consisting of dipoles with Dipolemoment per unit volume.Field due to a Polarized DielectricdvPPThe potential dV at an external point O due towhere R2= (x-x’)2+(y-y’)2+(z-z’)2and R is thedistance between volume element dv’ and thepoint O.ButApplying the vector identity= -(i)
  21. 21. Put this in (i) and integrate over the entire volume v’ of the dielectricApplying Divergence Theorem to the first termwhere an’ is the outward unit normal to the surface dS’ of the dielectricThe two terms in (ii) denote the potential due to surface and volumecharge distributions with densities(ii)
  22. 22. where ρps and ρpv are the bound surface and volume charge densities.Bound charges are those which are not free to move in the dielectricmaterial.The total positive bound charge on surface S bounding the dielectric isEquation (ii) says that where polarization occurs, an equivalentvolume charge density, ρpv is formed throughout the dielectric whilean equivalent surface charge density, ρps is formed over the surface ofdielectric.while the charge that remains inside S is
  23. 23. Total charge on dielectric remains zero.Total charge =When dielectric contains free chargeIf ρv is the free volume charge density then the total volume chargedensity ρtHenceWhere
  24. 24. The effect of the dielectric on the electric field is to increaseinside it by an amount .PDEThe polarization would vary directly as the applied electric field.Where is known as the electric susceptibility of the materialeχIt is a measure of how susceptible a given dielectric is to electric fields.
  25. 25. We know thatDielectric Constant and Strengthorwhere є is the permittivity of the dielectric, єo is the permittivity of thefree space and єr is the dielectric constant or relative permittivity.andThuswhereroεεε =and
  26. 26. No dielectric is ideal. When the electric field in a dielectric issufficiently high then it begins to pull electrons completely out of themolecules, and the dielectric becomes conducting.When a dielectric becomes conducting then it is called dielectricbreakdown. It depends on the type of material, humidity, temperatureand the amount of time for which the field is applied.The minimum value of the electric field at which the dielectricbreakdown occurs is called the dielectric strength of the dielectricmaterial.orThe dielectric strength is the maximum value of the electric field that adielectric can tolerate or withstand without breakdown.
  27. 27. Continuity Equation and Relaxation TimeAccording to principle of charge conservation, the time rate ofdecrease of charge within a given volume must be equal to the netoutward current flow through the closed surface of the volume.The current Iout coming out of the closed surfacewhere Qin is the total charge enclosed by the closed surface.Using divergence theoremBut(i)
  28. 28. Equation (i) now becomesThis is called the continuity of current equation.Effect of introducing charge at some interior point of aconductor/dielectricorAccording to Ohm’s lawAccording to Gauss’s law(ii)
  29. 29. Equation (ii) now becomesIntegrating both sidesorThis is homogeneous liner ordinary differential equation. By separatingvariables we get
  30. 30. where(iii)Equation (iii) shows that as a result of introducing charge at someinterior point of the material there is a decay of the volume chargedensity ρv.The time constant Tr is known as the relaxation time or the relaxationtime.Relaxation time is the time in which a charge placed in the interior of amaterial to drop to e-1= 36.8 % of its initial value.For Copper Tr = 1.53 x 10-19sec (short for good conductors)For fused Quartz Tr = 51.2 days (large for good dielectrics)
  31. 31. Boundary ConditionsIf the field exists in a region consisting of two different media, theconditions that the field must satisfy at the interface separating themedia are called boundary conditionsThese conditions are helpful in determining the field on one side ofthe boundary when the field on other side is known.We will consider the boundary conditions at an interface separating1. Dielectric (єr1) and Dielectric (єr2)2. Conductor and Dielectric3. Conductor and free spaceFor determining boundary conditions we will use Maxwell’s equationsand
  32. 32. Boundary Conditions (Between two differentdielectrics)Consider the E field existing in a region consisting of two differentdielectrics characterized by є1 = є0 єr1 and є2 = є0 єr2E1 and E2 in the media 1 and 2 canbe written asButnt EEE 111+= nt EEE 222+=andAssuming that the path abcda is verysmall with respect to the variation in E
  33. 33. As Δh 0Thus the tangential components of E are the same on the two sides ofthe boundary. E is continuous across the boundary.ButThusorHere Dt undergoes some change across the surface and is said to bediscontinuous across the surface.
  34. 34. ApplyingPutting Δh 0 givesWhere ρs is the free charge density placed deliberately at the boundaryIf there is no charge on the boundary i.e. ρs = 0 thenThus the normal components of D is continuous across the surface.
  35. 35. Biot-Savart’s LawIt states that the magnetic field intensity dH produce at a point P bythe differential current element Idl is proportional to the product Idland the sine of angle α between the element and line joining P to theelement and is inversely proportional to the square of distance Rbetween P and the element.orThe direction of dH can be determined by the right hand thumb rulewith the right hand thumb pointing in the direction of the current, theright hand fingers encircling the wire in the direction of dH
  36. 36. Ampere’s circuit LawThe line integral of the tangential component of H around a close pathis the same as the net current Iinc enclosed by the path.
  37. 37. Application of Ampere’s law : Infinite SheetCurrentConsider an infinite current sheet in z = 0 plane.To solve integral we need to know how H is likeIf the sheet has a uniform current density then^yy aKK =Applying Ampere’s Law on closedrectangular path (Amperian path) wegetWe assume the sheet comprising of filaments dH above and below thesheet due to pair of filamentary current.(i)
  38. 38. The resultant dH has only an x-component.where Ho is to be determined.Also H on one side of sheet is the negative of the other.Due to infinite extent of the sheet, it can be regarded asconsisting of such filamentary pairs so that the characteristic ofH for a pair are the same for the infinite current sheets(ii)
  39. 39. Comparing (i) and (iii), we getEvaluating the line integral of H along the closed path(iii)Using (iv) in (ii), we get(iv)
  40. 40. where an is a unit normal vector directed from the current sheet to thepoint of interest.Generally, for an infinite sheet of current density K
  41. 41. Magnetic Flux DensityThe magnetic flux density B is similar to the electric flux density Dwhere µo is a constant and is known as the permeability of free space.Therefore, the magnetic flux density B is related to the magnetic fieldintensity HThe magnetic flux through a surface S is given byIts unit is Henry/meter (H/m) and has the valuewhere the magnetic flux ψ is in webers (Wb) and the magnetic fluxdensity is in weber/ square meter or Teslas.
  42. 42. Magnetic flux lines due to a straightwire with current coming out of thepageEach magnetic flux line is closedwith no beginning and no end andare also not crossing each other.In an electrostatic field, the flux passing through a closed surface isthe same as the charge enclosed.Thus it is possible to have an isolatedelectric charge.Also the electric flux lines are notnecessarily closed.
  43. 43. Magnetic flux lines are always closeupon themselves,.So it is not possible to have an isolatedmagnetic pole (or magnetic charges)An isolated magnetic charge does not exist.Thus the total flux through a closed surface in a magnetic field mustbe zero.This equation is known as the law of conservation of magnetic flux orGauss’s Law for Magnetostatic fields.Magnetostatic field is not conservative but magnetic flux is conserved.
  44. 44. This is Maxwell’s fourth equation.Applying Divergence theorem, we getorThis equation suggests that magnetostatic fields have no source orsinks.Also magnetic flux lines are always continuous.
  45. 45. Faraday’s lawAccording to Faraday a time varying magnetic field produces aninduced voltage (called electromotive force or emf) in a closed circuit,which causes a flow of current.The induced emf (Vemf) in any closed circuit is equal to the time rate ofchange of the magnetic flux linkage by the circuit. This is Faraday’sLaw and can be expressed aswhere N is the number of turns in the circuit and ψ is the flux througheach turn.The negative sign shows that the induced voltage acts in such a way tooppose the flux producing in it. This is known as Lenz’s Law.
  46. 46. Transformer and Motional EMFFor a circuit with a single turn (N = 1)In terms of E and B this can be written aswhere ψ has been replaced by and S is the surface area ofthe circuit bounded by a closed path L..The equation says that in time-varying situation, both electric andmagnetic fields are present and are interrelated.(i)
  47. 47. The variation of flux with time may be caused in three ways.1. By having a stationary loop in a time-varying B field.2. By having a time-varying loop area in a static B field.3. By having a time-varying loop area in a time-varying B field.Consider a stationary conductingloop in a time-varying magnetic Bfield. The equation (i) becomesStationary loop in a time-varying B field(Transformer emf)
  48. 48. This emf induced by the time-varying current in a stationary loop isoften referred to as transformer emf in power analysis since it is due tothe transformer action.By applying Stokes’s theorem to the middle term, we getThis is one of the Maxwell’s equations for time-varying fields.ThusIt shows that the time-varying field is not conservative.
  49. 49. 2. Moving loop in static B field (Motional emf)The motional electric field Em is defined asWhen a conducting loop is moving in a static B field, an emf isintroduced in the loop.The force on a charge moving with uniform velocity u in a magneticfield B isConsider a conducting loop moving with uniform velocity u, the emfinduced in the loop isThis kind of emf is called the motional emf or flux-cutting emf.Because it is due to the motional action. eg,. Motors, generators(i)
  50. 50. By applying Stokes’s theorem to equation (i), we get
  51. 51. Consider a moving conducting loop in a time-varying magnetic fieldalso3. Moving loop in time-varying fieldThen both transformer emf and motional emf are present.Thus the total emf will be the sum of transformer emf and motionalemf
  52. 52. For static EM fieldsDisplacement CurrentBut the divergence of the curl of a vector field is zero. SoBut the continuity of current requires(ii)(i)(iii)Equation (ii) and (iii) are incompatible for time-varying conditionsSo we need to modify equation (i) to agree with (iii)Add a term to equation (i) so that it becomeswhere Jd is to defined and determined.(iv)
  53. 53. Again the divergence of the curl of a vector field is zero. SoIn order for equation (v) to agree with (iii)(v)Putting (vi) in (iv), we getThis is Maxwell’s equation (based on Ampere Circuital Law) for atime-varying field. The term is known as displacementcurrent density and J is the conduction current density .or (vi)
  54. 54. Maxwell’s Equations in Final Form