1. (i) Divergence Divergence, Curl and Gradient Operations
The divergence of a vector V written as div V represents the
scalar quantity.
div V =   V =
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2. Example for Divergence
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3. 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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4. 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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5. 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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6. 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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7. 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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8. Important Vector notations in electromagnetism
div grad S = 1. 2. 3. 4.
curl grad  = 0
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9. 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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10. 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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11. 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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12. 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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13. 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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14. The integral form of Gauss law is,
(1)
By using divergence theorem
(2)
From equations (1) and (2),
(3)
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15. 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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16. Eqn(5) ×
(or) div
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17. 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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18. 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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19. [ 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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20. 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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21. 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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22. If the work is done along a closed path,
emf =
The magnetic flux linked with closed area S due to the Induction B =
(3)
(4)
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23. (Integral form of Maxwell’s third equation)
(5)
= 
(6)
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24. (Using Stokes’ theorem ) = (7)
Hence,
(8)
Maxwell’s’ third equation in differential form
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25. 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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26. 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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27. Using Stokes theorem, (8) (8) (Using (7) and (8)) (Using (7) and (8))
(9)
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28. The above equation can also be written as
(10)
(11)
Differential form of Maxwell fourth equation
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29. Hall Effect
When a piece of conductor (metal or semi conductor) carrying a current is placed in a transverse magnetic field, an electric field is produced inside the conductor in a direction normal to both the current and the magnetic field.
This phenomenon is known as the Hall Effect and the generated voltage is called the Hall voltage.
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30. Consider a conventional current flow through the strip along OX and a magnetic field of induction B is applied along axis OY.
If the strip is made up of metal ,the charge carriers in the strip will be electrons.
As conventional current flows along OX, the electrons must be moving along XO.
If the velocity of the electrons is `v’ and charge of the electrons is `e’, the force on the electrons due to the magnetic field
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31. F =  Bev, which acts along OZ.
This causes the electrons to be deflected and the electrons accumulate at the face ABEF.
Face ABEF will become negative and the face OCDG becomes positive.
A potential difference is established across faces ABEF and OCDG, causing a field EH.
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32. At equilibrium, eEH = Be (or) EH = B
This field gives rise to a force of `eEH’ on the electrons in the opposite direction. (i.e, in the negative Z direction)
At equilibrium, eEH = Be (or) EH = B
If J is the current density, then, J =  ne
where `n’ is the concentration of current carriers.
v =
Substitute the value of `’ in eqn
EH =
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33. The Hall Effect is described by means of the Hall coefficient `RH’ in terms of current density `J’ by the relation,
EH = RHBJ (or) RH = EH/ BJ
All the three quantities EH, J and B are measurable, the Hall coefficient RH and hence the carrier density `n’ can be found out.
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