Question

An ideal parallel plate capacitor of capacitance C has circular plates located at z = 0 and z = d respectively. The medium between the plates is a linear, homogeneous, isotropic dielectric of dielectric constant K. The capacitor is connected to a resistance R in series, and a voltage V is applied to the circuit. The charge q on the capacitor plates increases with time according to q = C V (1-e−t/RC). Find the magnitude of the magnetic field H inside the dielectric.

Answer 1

Given that,

Capacitance of the parallel plate capacitor = c

Distance between the parallel plate capacitor = d

Dielectric constant of the material = k

Resistance of the resistor = R

Voltage = V

We know that,

The capacitor is

$C=\dfrac{k\epsilon_{0}A}{d}$

We need to calculate the value of $\dfrac{dQ}{dt}$

Using equation of charge

$Q=CV(1-e^{-\frac{t}{RC}})$

On differentiating

$\dfrac{dQ}{dt}=\dfrac{CVt}{RC}e^{\dfrac{-t}{RC}}$

$\dfrac{dQ}{dt}=\dfrac{Vt}{R}e^{\frac{-t}{RC}}$

We need to calculate the magnetic field inside the dielectric

Using formula of magnetic field

$H=\dfrac{\mu_{0}}{2\pi r}\dfrac{dQ}{dt}$

Put the value of $\dfrac{dQ}{dt}$

$H=\dfrac{\mu_{0}}{2\pi r}\times\dfrac{Vt}{R}e^{\frac{-t}{RC}}$

$H=\dfrac{\mu_{0}}{2\pi r}\times\dfrac{Vt}{R}e^{\frac{-t}{R(\dfrac{k\epsilon_{0}A)}{d}}}$

$H=\dfrac{\mu_{0}}{2\pi r}\times\dfrac{Vt}{R}e^{\frac{-td}{Rk\epsilon_{0}A}}$

Hence, The magnetic field inside the dielectric $\dfrac{\mu_{0}}{2\pi r}\times\dfrac{Vt}{R}e^{\frac{-td}{Rk\epsilon_{0}A}}$