CBSE Class 12-science Answered

Let us consider a capacitor is charged as shown in figure . Charging current varies with time.

Magnetic field at a point P which is at a distance r from conductor through which charging current is passing

is obtained by considering an amperian loop of radius r as shown in figure that is centered at the point where

the conducting wire is passing through.

In Ampere law, the amperian loop is surface boundary and the current is passing through surface.

In top figure we have considered the flat surface with amperian loop is surface boundary and charging current

is passing through this flat surface. Hence magnetic field induction B at P due to current is calculated as

B × (2πr ) = μ_o

B = ( μ_o ) /  (2 π r )  ............................(1)

But if the surface is not flat and it is balloon like shape as shown in bottom figure still the amperian loop is surface boundary ,

then we have difficulty of finding the magnetic field induction B at P becuase now the current is not passing through surface .

But magnetic field at P is unique and it is not related to the surface connected to amperian loop.

In order to remove this difficulty the concept of displacement current is introduced between capacitor plates.

Displacement current is considered due to charge Q on capacitor plates that varies with time .

If C is the capacitance and V is potential difference across plates of capacitor , then we have Q = C V .

Hence above expression for displacement current is written as

Capacitance C = ε_o (A/d)  and  electric field E between capacitor plates is E = V/d . By using this substitutions ,

above expression for displacement current is written as

Hence to determine magnetic field induction , eqn.(1) should be modified as shown below

It can be proved that displacement current equals conduction current , i.e. .

Hence if we consider flat surface for amperian loop as shown in top figure ,

displacement current  is zero and only we have conduction current .

Then magnetic field induction at p is sames as given in eqn.(1)

if we consider balloon like surface for amperian loop as shown in bottom figure ,

then magnetic field induction is

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