Question

A circuit has self-inductance 'L' H and carries a current 'I' A. To prevent sparking when the circuit is switched off, a capacitor which can withstand 'V' volt is used. The least capacitance of the capacitor connected across the switch must be equal to

• A. $\frac{IV}{L}$
• B. $L \left( \frac{V}{L} \right)^2$
• C. $L \left( \frac{I}{V} \right)^2$
• D. $\frac{LI}{V}$

Hint:

A dimensional analysis approach works instantly here! Since capacitance $C = \frac{q}{V}$ and charge relates to current via energy identities, checking the unit dimensions reveals that $\frac{I^2}{V^2}$ multiplied by inductance $L$ uniquely matches Farads ($\text{H} \cdot \text{A}^2/\text{V}^2 = \text{F}$).

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
When an inductive circuit carrying a steady current is suddenly broken, the rapid collapse of its magnetic field generates a massive induced EMF that causes an electrical spark across the switch contacts. Connecting a capacitor absorbs this energy safely. We need to find the minimum capacitance needed so that the voltage does not exceed $V$.

Step 2: Key Formula or Approach:
To completely eliminate sparking, the entire magnetic potential energy stored in the inductor's magnetic field must transfer into electrostatic potential energy within the capacitor's electric field. $$\text{Energy Stored in Inductor} = \text{Energy Absorbed by Capacitor}$$ $$U_L = \frac{1}{2} L I^2$$ $$U_C = \frac{1}{2} C V^2$$

Step 3: Detailed Explanation:
Equate the two energy expressions based on the conservation of energy principle:
$$\frac{1}{2} L I^2 = \frac{1}{2} C V^2$$ Cancel the common factor of $\frac{1}{2}$ from both sides of the equation:
$$L I^2 = C V^2$$ To isolate the minimum capacitance $C$, divide both sides by $V^2$:
$$C = \frac{L I^2}{V^2}$$ Grouping the current and voltage terms into a shared exponent bracket:
$$C = L \left(\frac{I}{V}\right)^2$$

Step 4: Final Answer:
The least capacitance required across the switch is $L \left( \frac{I}{V} \right)^2$, matching option (C).