Question

Suppose that a parallel-plate capacitor has circular plates with radius R = 43 mm and a plate separation of 5.1 mm. Suppose also that a sinusoidal potential difference with a maximum value of 170 V and a frequency of 47 Hz is applied across the plates; that is, V = (170 V) sin[2?(47 Hz)t]. Find Bmax(R), the maximum value of the induced magnetic field that occurs at r = R.

Answer 1

Here radius of the circular plates is R = 43 mm = 43*10^-3m

seperation between the plates is d = 5.1 mm = 5.1*10^-3m

frequency of voltage f = 47 Hz

Then angular frequency ? = 2?f = 2?*47 rad/sec

We know that magnetic field between the plates when r < = R is

      B_in=(?₀?₀r/2)dE/dt

E = V/d

Then B_in=(?₀?₀r/2d)dV/dt

Here V = ( Vmax) sin?t

dV/dt = ( Vmax)?cos?t

dV/dt = (170 V) cos[2?(47 Hz)t]*2?(47Hz)

?₀= 4?*10^-7H/m

?₀= 8.85*10^-12C²/Nm²

Here Vmax = 170 V

This grows until r = R = 43 mm = 0.043 m

Then Bmax =(?₀?₀R?/2d)Vmax

                  = [(4?*10^-7H/m) * (8.85*10^-12C²/Nm²)* 0.043 m *(2?*47 rad/sec )/2* 5.1 *10^-3m]*170Volts

                   = 2.35*10^-12T