Question

We now have two quite different expressions for the line integral of the magnetic field around the same loop. The point here is to see that they both are intimately related to the charge q(t) on the left capacitor plate. First find the displacement current Idisplacement(t) in terms of q(t).

Answer 1

The displacement current Idisplacement(t) is proportional to the rate of change of the charge q(t) on the capacitor, expressed by 1/( ε0 * A ) * dq(t)/dt where ε0 is the permittivity of free space and A is the area of the capacitor plates.

Step-by-step explanation:

The displacement current Idisplacement(t) is related to the rate at which the charge q(t) on a capacitor changes. According to Maxwell's addition to Ampère's law, the displacement current in the vacuum between the plates of a capacitor when the electric field E changes is given by:

Idisplacement(t) = ε0 * A * dE/dt

where ε0 is the permittivity of free space and A is the area of one of the plates. The electric field E for a parallel-plate capacitor is related to the charge q on the capacitor by:

E = q/( ε0 * A )

Therefore, we can express the displacement current in terms of the charge q(t) by differentiating this equation with respect to time:

Idisplacement(t) = d[q(t)/(ε0 * A)]/dt = 1/( ε0 * A ) * dq(t)/dt

This leads us to the understanding that the displacement current is equal to the rate of change of the charge divided by the permittivity of free space times the capacitor plate area.

Answer 2

q(t) =CV(t) = έAV(0)/d sin ωt

Step-by-step explanation:

Step 1: External electromagnetic force is equal to the voltage of the capacitor . The electric field between the capacitor plates =

E(t) = έ(t)/d = V(0) sin (ωt)/d

Step 2: we calculate the electrical flux between the capacitor= φ(t)=AE(t) = AV(0) sin wt/d

Step 3 : The displacement current can be gotten from the equation below;

I displacement(t) = E(0) dφ(t)/DT= E(0) AV(0)w/d cos wt

Step 4: The current at the outside terminals of the capacitor is the addition of the current used to charge the capacitor and the current in the resistor.

The charge on the capacitor is;

q(t) =CV(t) = έAV(0)/d sin ωt...