Maxwell Equation for Static Electric and Magnetic Field

Differential Form

Integral Form Remarks

Derived from Gauss Law. Applying divergence theorem,

Hence,

Non-existence of magnetic monopole.Applying divergence theorem,

Hence,

Conservative nature of electrostatic field.Applying Stokes theorem,

Hence,

Ampere’s Law.Applying Stokes theorem,

Hence,

JH

0 E

0 B

D v vSenc dvdSDQ

0S dSB

0L dlE

SLdSJIdlH

vS

dvDdSD

v vv

dvdvD vD

0 SLdSEdlE

0 E

vS

dvBdSB 0

0 B

SL

dSHdlH

SSdSJdSH JH

Maxwell Equation for Time Varying Electric and Magnetic Field

Differential Form

Integral Form Remarks

Derived from Gauss Law. Applying divergence theorem,

Hence,

Non-existence of magnetic monopole.Applying divergence theorem,

Hence,

Derived from Faradays Law (case of stationary loop in a time-varying B field).Applying Stokes theorem,

Hence,

Derived from Ampere’s Law in time varying field. Principle of continuity of current eq. leads to the addition of Jd in the original Maxwell eq.To satisfy the continuity of current eq.,

t

DJH

t

BE

0 B

D v vSenc dvdSDQ

0S dSB

SL

dSt

BdlE

SL

dSt

DJIdlH

vS

dvDdSD

v vv

dvdvD vD

SL

dSEdlE

vS

dvBdSB 0

0 B

SS

dSt

BdSE

t

BE

dJJH

t

DJ d

Displacement current densityt

DJ d

Cont

Based on the displacement current density , we can define the displacement current as

Displacement current is a result of a time-varying electric field. An example of such current is the current through a capacitor when AC voltage is applied.

The displacement current is equal to the conduction current Ic, that flow through the conductor.

3

t

DJ d

S Sd dS

t

DdSJI