Let there be a parallel R-C network with a voltage V as shown in fig .Let the current through resistor R is I_c and by Ohm’s law it is given by

I_c=V/R

And current through capacitor C is I_d and is given by

I_d=dQ/dt

I_d=Cdv/dt                                ( dQ=Cdv)(1)

In practice ,the current does not flow through the capacitor . But ,the current that flows out of one electrode of capacitor equals the current that flows in to the other electrode. The net effect is as if there is a current flowing through the path containing the capacitor. But current, I_cactually flows through the resistor.

Hence ,from the above result ,current flowing through the resistor is known as conduction current and it obeys Ohm’s law,while the current flowing through the capacitor is commonly known as Displacement current.

Mathematical Proof.As the electric field inside each element equals the voltage V across the element divided by its length d

That is                           E=V/d or V=ED                                                  (2)

Now the current density in resistor is given by

J_c=I_c/A=σE                                                  (3)

Where  A= cross-sectional area

σ=conductivity of resistance element

Also capacitance of a parallel plate capacitor is given by

C=ε₀A/d                                                                   (4)

Now rewrite equation(1)

I_d=C dV/dt

By substituting the values of V and C from equations (2 and 4) in above equation,we get

I_d= ε ₀A/d( E/t)

I_d= ε₀A E/t                                             (5)

Therefore current density J_d inside capacitor is

J_d=I_d/A

Substititing value of I_d from equation (5) in above equation,we get

J_d= ε ₀/A E/t

Or J_d= ε₀ E/t                                      (6a)

Or                                             J_d= D/t                                                  (6b)

Where                      D= ε ₀E=Electric displacement vector

And                           J_d=Displacement current density

Equation (6a) proves that displacement current density arises whenever there will be change in electric field E that is (E/t≠0)

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