Answer to Question #123857 in Electricity and Magnetism for Perry

Question

ω, α and a are positive constants. Find the induced electric field at all points in space.
Q4. 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 (5)
magnetic field H inside the dielectric. (5)

Answer 1

As per the given question,

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 across the circuit=V

we know that $c=\frac{k\epsilon_o A}{d}$

charge increase with respect to the time $Q = C V (1-e^{−\frac{t}{RC}})$

Hence, $\frac{dQ}{dt}=\frac{CVt}{RC} e^{\frac{-t}{Rc}}= \frac{Vt}{R}e^{\frac{-t}{Rc}}$

We know that magnetic field at a distance r from the plate is,

$B=\frac{\mu_o}{2\pi r}\frac{dQ(t)}{dt}$

Now, substituting the values,

$\Rightarrow B=\frac{\mu_o}{2\pi r}\frac{Vt}{R}e^{\frac{-t}{Rc}} =\frac{\mu_o}{2\pi r}\frac{Vt}{R}e^{\frac{-t}{R\frac{k\epsilon_o A}{d}}}$

$\Rightarrow B=\frac{\mu_o}{2\pi r}\frac{Vt}{R}e^{\frac{-td}{Rk\epsilon_o A}}$