Question

Phase difference between conduction and displacement current

Answer 1

Let there be a parallel R-C network with a voltage V as shown in fig .Let the current through resistor R is I and by Ohm’s law it is given by
I =V/R
And current through capacitor C is I and is given by
I =dQ/dt
I =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 actually 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 =I /A=σE (3)
Where A= cross-sectional area
σ=conductivity of resistance element
Also capacitance of a parallel plate capacitor is given by
C=ε A/d (4)
Now rewrite equation(1)
I =C dV/dt
By substituting the values of V and C from equations (2 and 4) in above equation,we get
I = ε A/d( E/t)
I = ε A E/t (5)
Therefore current density J inside capacitor is
J =I /A
Substititing value of I from equation (5) in above equation,we get
J = ε /A E/t
Or J = ε E/t (6a)
Or J = D/t (6b)
Where D= ε E=Electric displacement vector
And J =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)







hope this will help you....