(i) Divergence Divergence, Curl and Gradient Operations The divergence of a vector V written as div V represents the scalar quantity. div V =   V = PGGC-11 DR BHANDARI

Example for Divergence PGGC-11 DR BHANDARI

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 PGGC-11 DR BHANDARI

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 PGGC-11 DR BHANDARI

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 PGGC-11 DR BHANDARI

Gradient Operator The Gradient of a scalar function  is a vector whose Cartesian components are, Then grad φ is given by, PGGC-11 DR BHANDARI

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. PGGC-11 DR BHANDARI

Important Vector notations in electromagnetism div grad S = 1. 2. 3. 4. curl grad  = 0 PGGC-11 DR BHANDARI

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) PGGC-11 DR BHANDARI

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 PGGC-11 DR BHANDARI

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. PGGC-11 DR BHANDARI

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 PGGC-11 DR BHANDARI

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 = PGGC-11 DR BHANDARI

The integral form of Gauss law is, (1) By using divergence theorem (2) From equations (1) and (2), (3) PGGC-11 DR BHANDARI

div div Electric displacement vector is (4) (4) (5) (5) (6) e r = · Ñ e r = · Ñ E (4) (4) (5) div div (5) (6) PGGC-11 DR BHANDARI

Eqn(5) × (or) div PGGC-11 DR BHANDARI

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 PGGC-11 DR BHANDARI

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. PGGC-11 DR BHANDARI

[ 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) PGGC-11 DR BHANDARI

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. PGGC-11 DR BHANDARI

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) PGGC-11 DR BHANDARI

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) PGGC-11 DR BHANDARI

(Integral form of Maxwell’s third equation) (5) =  (6) PGGC-11 DR BHANDARI

(Using Stokes’ theorem ) = (7) Hence, (8) Maxwell’s’ third equation in differential form PGGC-11 DR BHANDARI

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 = PGGC-11 DR BHANDARI

Using (3) and (4) (5) We Know that We Know that (6) (6) (7) (7) (Maxwell’s fourth equation in integral form) PGGC-11 DR BHANDARI

Using Stokes theorem, (8) (8) (Using (7) and (8)) (Using (7) and (8)) (9) PGGC-11 DR BHANDARI

The above equation can also be written as (10) (11) Differential form of Maxwell fourth equation PGGC-11 DR BHANDARI

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. PGGC-11 DR BHANDARI

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 PGGC-11 DR BHANDARI

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. PGGC-11 DR BHANDARI

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 = PGGC-11 DR BHANDARI

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. PGGC-11 DR BHANDARI