Animated Solution for Physics - Electrostatics: A simple pendulum of mass $m$, length $l$ and charge $+q$ suspended in the electric field produced by two conducting parallel plates as shown. The value of deflection of pendulum in equilibrium position will be

Imagine you are looking at the space between two parallel conducting plates. It's not just empty space! A dielectric medium of thickness $t$ is attached to the left plate, and the rest of the space is an air gap. A simple pendulum is suspended right in the middle of this air gap.

To solve this, we need to model the physical space as an electrical circuit. The space between the plates acts exactly like two separate capacitors connected in series. Let's call the dielectric portion $C_2$ and the air gap portion $C_1$.

Since these two imaginary capacitors are in series, their equivalent capacitance is given by the standard formula:

The total potential difference across the entire setup is the potential of the positive plate minus the potential of the negative plate. This gives us $\Delta V=V_2-(-V_1)=V_1+V_2$.

In a series combination, the charge on each capacitor is identical. Therefore, the total charge $Q$ is simply the equivalent capacitance multiplied by the total potential difference:

Now, the pendulum is hanging specifically in the air gap, which corresponds to capacitor $C_1$. The potential difference across just this air gap, let's call it $V_1^{\prime }$, is $Q$ divided by $C_1$. Substituting our expression for $Q$, we get:

To find the force on the pendulum, we need the electric field $E$ in the air gap. The electric field is simply the potential difference across the air gap divided by its thickness. The total distance is $d$, and the dielectric thickness is $t$, so the air gap thickness is $(d-t)$.

Let's shift our focus to the pendulum bob and draw its free body diagram. Gravity pulls it downwards with a force $mg$. The electric field pushes the positive charge horizontally with a force $qE$. The string pulls it back with a tension $T$.

In the equilibrium position, the bob is perfectly still, meaning all forces must balance out perfectly. If the string makes an angle $\theta$ with the vertical, we can resolve the tension into horizontal and vertical components:

By dividing the first equation by the second, the tension $T$ beautifully cancels out, leaving us with:

Finally, we substitute the massive expression for the electric field $E$ that we derived earlier. This gives us the exact angle of deflection in equilibrium: