Question

In the figure shown the plates of a parallel plate capacitor have unequal charges. Its capacitance is 'C. P is a point outside the capacitor and close to the plate of charge -Q. The distance between the plates is 'd' then which statement is correct

• A. A point charge at point 'P will experience electric force due to capacitor
• B. The potential difference between the plates will be $\frac{Q}{2C}$
• C. The energy stored in the electric field in the region between the plates is $\frac{9Q^2}{4C}$
• D. The force on one plate due to the other plate is $\frac{Q^2}{2\pi \epsilon_0 d^2}$

Answer 1

Answer: (A)

A point charge at point 'P will experience electric force due to capacitor

Answer 2

Let the charges on the plates be $Q_1 = 2Q$ and $Q_2 = -Q$. The area of the plates is $A$. The capacitance is $C = \frac{\epsilon_0 A}{d}$.

The electric field outside the capacitor (at point P) is given by $E_{out} = \frac{Q_1 + Q_2}{2A\epsilon_0}$. Substituting the charges: $E_{out} = \frac{2Q + (-Q)}{2A\epsilon_0} = \frac{Q}{2A\epsilon_0}$. Since $E_{out} \neq 0$, a point charge placed at P will experience an electric force. Thus, statement (1) is correct.

The electric field between the plates is $E_{in} = \frac{Q_1 - Q_2}{2A\epsilon_0} = \frac{2Q - (-Q)}{2A\epsilon_0} = \frac{3Q}{2A\epsilon_0}$. The potential difference between the plates is $V = E_{in} \cdot d = \frac{3Q}{2A\epsilon_0} \cdot d$. Using $C = \frac{\epsilon_0 A}{d}$, we get $A\epsilon_0 = Cd$. So, $V = \frac{3Qd}{2Cd} = \frac{3Q}{2C}$. Statement (2) is incorrect.

The energy stored in the electric field between the plates is $U = \frac{1}{2} C V^2 = \frac{1}{2} C \left(\frac{3Q}{2C}\right)^2 = \frac{1}{2} C \frac{9Q^2}{4C^2} = \frac{9Q^2}{8C}$. Statement (3) is incorrect.

The force on one plate (say, $Q_1$) due to the other plate ($Q_2$) is $F = |Q_1 E_{Q_2}|$. The electric field due to a single plate $Q_2$ is $E_{Q_2} = \frac{|Q_2|}{2A\epsilon_0} = \frac{Q}{2A\epsilon_0}$. So, $F = |2Q \cdot \frac{Q}{2A\epsilon_0}| = \frac{Q^2}{A\epsilon_0} = \frac{Q^2}{Cd}$. Statement (4) is incorrect.