1. PH0101 UNIT 2 LECTURE 2 Biot Savart law Ampere’s circuital law
Faradays laws of Electromagnetic induction
Electromagnetic waves,
Divergence, Curl and Gradient
Maxwell’s Equations
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2. Biot – Savart Law
Biot - Savart law is used to calculate the magnetic field due to a current carrying conductor.
According to this law, the magnitude of the magnetic field at any point P due to a small current element I.dl ( I = current through the element, dl = length of the element) is,
In vector notation,
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3. Ampere’s circuital law
It states that the line integral of the magnetic field (vector B) around any closed path or circuit is equal to μ0 (permeability of free space) times the total current (I) flowing through the closed circuit.
Mathematically,
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4. Faraday’s Law of electromagnetic induction
Michael Faraday found that whenever there is a change in magnetic flux linked with a circuit, an emf is induced resulting a flow of current in the circuit.
The magnitude of the induced emf is directly proportional to the rate of change of magnetic flux.
Lenz’s rule gives the direction of the induced emf which states that the induced current produced in a circuit always in such a direction that it opposes the change or the cause that produces it.
d is the change magnetic flux linked with a circuit
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5. Electromagnetic waves
According to Maxwell’s modification of Ampere’s law, a changing electric field gives rise to a magnetic field.
It leads to the generation of electromagnetic disturbance comprising of time varying electric and magnetic fields.
These disturbances can be propagated through space even in the absence of any material medium.
These disturbances have the properties of a wave and are called electromagnetic waves.
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6. Representation of Electromagnetic waves
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7. Description of EM Wave
The variations of electric intensity and magnetic intensity are transverse in nature.
The variations of E and H are perpendicular to each other and also to the directions of wave propagation.
The wave patterns of E and H for a traveling electromagnetic wave obey Maxwell’s equations.
EM waves cover a wide range of frequencies and they travel with the same velocity as that of light i.e. 3  10 8 m s – 1.
The EM waves include radio frequency waves, microwaves, infrared waves, visible light, ultraviolet rays, X – rays and gamma rays.
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8. Divergence, Curl and Gradient Operations
(i) Divergence
The divergence of a vector V written as div V represents the
scalar quantity.
div V =   V =
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9. Example for Divergence
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10. Physical significance of divergence
Physically the divergence of a vector quantity represents the rate of change of the field strength in the direction of the field.
If the divergence of the vector field is positive at a point then something is diverging from a small volume surrounding with the point as a source.
If it negative, then something is converging into the small volume surrounding that point is acting as sink.
if the divergence at a point is zero then the rate at which something entering a small volume surrounding that point is equal to the rate at which it is leaving that volume.
The vector field whose divergence is zero is called solenoidal
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11. Curl of a Vector field Curl V =
Physically, the curl of a vector field represents the rate of change of the field strength in a direction at right angles to the field and is a measure of rotation of something in a small volume surrounding a particular point.
For streamline motions and conservative fields, the curl is zero while it is maximum near the whirlpools
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12. Representation of Curl
(I) No rotation of the paddle wheel represents zero curl
(II) Rotation of the paddle wheel showing the existence of curl
(III) direction of curl
For vector fields whose curl is zero there is no rotation of the paddle wheel when it is placed in the field, Such fields are called irrotational
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13. Gradient Operator
The Gradient of a scalar function  is a vector whose Cartesian components are,
Then grad φ is given by,
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14. The electric field intensity at any point is given by,
The magnitude of this vector gives the maximum rate of change of the scalar field and directed towards the maximum change occurs.
The electric field intensity at any point is given by,
E =  grad V = negative gradient of potential
The negative sign implies that the direction of E opposite to the direction in which V increases.
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15. Important Vector notations in electromagnetism
div grad S = 1. 2. 3. 4.
curl grad  = 0
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16. Theorems in vector fields
Gauss Divergence Theorem
It relates the volume integral of the divergence of a vector V to the surface integral of the vector itself.
According to this theorem, if a closed S bounds a volume , then
div V) d =
V  ds
(or)
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17. Stoke’s Theorem
It relates the surface integral of the curl of a vector to the line integral of the vector itself.
According to this theorem, for a closed path C bounds a surface S,
(curl V)  ds =
V dl
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18. Maxwell’s Equations
Maxwell’s equations combine the fundamental laws of electricity and magnetism .
The are profound importance in the analysis of most electromagnetic wave problems.
These equations are the mathematical abstractions of certain experimentally observed facts and find their application to all sorts of problem in electromagnetism.
Maxwell’s equations are derived from Ampere’s law, Faraday’s law and Gauss law.
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19. Maxwell’s Equations Summary
Differential form
Integral form
Equation from electrostatics
2. Equation from magnetostatics
3. Equation from Faradays law
4. Equation from Ampere circuital law
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20. Maxwell’s equation: Derivation
Maxwell’s First Equation
If the charge is distributed over a volume V. Let  be the volume density of the charge, then the charge q is given by,
q =
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21. The integral form of Gauss law is,
(1)
By using divergence theorem
(2)
From equations (1) and (2),
(3)
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22. div div Electric displacement vector is (4) (4) (5) (5) (6) e r = · Ñe r = Ñ E
(4)
(4)
(5)
div
div
(5)
(6)
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23. Eqn(5) ×
(or) div
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24. From Gauss law in integral form
This is the differential form of Maxwell’s I Equation.
(7)
From Gauss law in integral form
This is the integral form of Maxwell’s I Equation
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25. Maxwell’s Second Equation
From Biot -Savart law of electromagnetism, the magnetic induction at any point due to a current element,
dB =
(1)
In vector notation, =
Therefore, the total induction
(2) =
This is Biot – Savart law.
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26. [ i =J . A and I . dl = J(A . dl) = J .dv] (3)
replacing the current i by the current density J, the current per unit area is
(3)
[ i =J . A and I . dl = J(A . dl)
= J .dv]
Taking divergence on both sides,
(4)
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27. Differential form of Maxwell’s’ second equation
For constant current density
(5)
Differential form of Maxwell’s’ second equation
By Gauss divergence theorem,
(6)
Integral form of Maxwell’s’ second equation.
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28. Maxwell’s Third Equation
By Faradays’ law of electromagnetic induction,
(1)
By considering work done on a charge, moving through a distance dl.
W =
(2)
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29. The magnetic flux linked with closed area S due to the Induction B =
If the work is done along a closed path,
emf =
(3)
(4)
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30. (Integral form of Maxwell’s third equation)
(5)
= 
(6)
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31. (Using Stokes’ theorem ) = (7)
Hence,
(8)
Maxwell’s’ third equation in differential form
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32. Maxwell’s Fourth Equation
By Amperes’ circuital law,
(1)
We know,
(or) B = μ0 H
(2)
Using (1) and (2)
(3)
(4)
We know i =
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33. Using (3) and (4) (5) We Know that We Know that (6) (6) (7) (7)
(Maxwell’s fourth equation in integral form)
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34. Using Stokes theorem, (8) (8) (Using (7) and (8)) (Using (7) and (8))
(9)
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35. The above equation can also be written as
(10)
(11)
Differential form of Maxwell fourth equation
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36. Have a nice day.
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