# Conduction current , displacement current

Conduction current & displacement current :

Conduction current is the electric current which exists in a conductor when the electrons flow in the conductor at a uniform rate. So, it is the current in a conductor when the electric field remains constant with respect to time.

Suppose E  is changing w.r.t. time, then Electric Flux ( φE ) is also changing w.r.t. time. In this case the current in the region is a displacement current (ID).

$\displaystyle I_d = \epsilon_0 \frac{d\phi_E}{dt}$

Understanding displacement current :

The current existing between the plates of a parallel plate capacitor is the displacement current.    Note that the conduction current is due to the flow of electrons, whereas the displacement current is due to displacement of electrons in a time-varying electric field.

The importance of displacement current lies in maintaining a continuity to the path of current flow through the capacitor.

Solved Example :In a region in space, closed loops of B are found. What is your conclusion regarding flowing of actual charges across the area bounded by the loop.

Solution : Just because there are closed loops of B present in a region in space, it does not necessarily imply that actual charges flow across the area bounded by the loop. A displacement current (such as that between the plates of a parallel-plate capacitor) can also produce loops of  B .

Exercise :  A changing electric field in a region produces closed loops of B . Comment on E and  dE/dt   regarding whether they are non-zero at all points on the loop and in the area enclosed by the loop.

[Hint: The basic idea is to have a time-varying total electric flux through the area enclosed by the loop. On the loop, in particular, there should be no electric field]

Exercise :  A variable frequency AC source is connected to a capacitor. Comment on the displacement  current   whether will it increase or decrease.

[Hint: The reactance of a capacitor is given by

$\displaystyle X_C = \frac{1}{2\pi f C}$

$\displaystyle X_C \propto \frac{1}{ f }$

Solved Example :  How would you establish an instantaneous displacement current of 2 mA in the space between parallel plates having capacitance 2 μF ?

Sol: C = 2 × 10-6 F , Id = 2 × 10-3 A

$\displaystyle I_d = \epsilon_0 \frac{d\phi_E}{dt}$

$\displaystyle I_d = \epsilon_0 \frac{d (E A ) }{dt}$

$\displaystyle I_d = \epsilon_0 A \frac{d (V/d ) }{dt}$ ( since V = E d )

$\displaystyle I_d = \frac{\epsilon_0 A}{d} \frac{d V }{dt}$

$\displaystyle I_d = C \frac{dV}{dt}$

$\displaystyle \frac{dV}{dt} = \frac{I_d}{C}$

$\displaystyle \frac{dV}{dt} = \frac{2\times 10^{-3}}{2\times 10^{-6}}$

= 1000 V/s

So, by applying a varying potential difference of 500 V/s, we would produce a displacement current of desired value.