54. Two inductors P and Q having inductance ratio 1:2 are connected in parallel in an electric circuit.

The energy stored in the inductors P and Q are in the ratio

(A) 1 : 4

(B) 1 : 2

(C) 2 : 1

(D) 4 : 1

Two Inductors Connected in Parallel: Find the Ratio of Energy Stored with Complete Explanation

Correct Answer

Option (C): 2 : 1

Concept Behind the Question

The energy stored in an inductor is given by the well-known expression

Energy (U) = ½ LI²

where L is the inductance of the coil and I is the current flowing through it.

Since the inductors are connected in parallel, the potential difference across both inductors remains the same. The current through each inductor is inversely proportional to its inductive reactance. As inductive reactance is directly proportional to inductance, the current is inversely proportional to the inductance.

Step-by-Step Solution

Let the inductance of inductor P be L.

Then the inductance of inductor Q is 2L.

Since the inductors are connected in parallel,

Current is inversely proportional to inductance.

Therefore,

IP : IQ = 2 : 1

Now calculate the energy stored.

Energy stored in P:

UP = ½ × L × (2I)²

UP = 2LI²

Energy stored in Q:

UQ = ½ × 2L × I²

UQ = LI²

Hence,

UP : UQ = 2LI² : LI² = 2 : 1

Therefore, the correct answer is:

Option (C) 2 : 1

Why Option (C) is Correct

Although the second inductor has twice the inductance, it carries only half the current because both inductors are connected in parallel. Since the energy stored depends on both inductance and the square of the current, the higher current in the first inductor dominates the calculation. After substituting the values into the energy formula, the final ratio becomes 2 : 1.

Why the Other Options are Incorrect

Option (A): 1 : 4

This option assumes that the energy depends only on inductance or incorrectly squares the inductance ratio. It completely ignores how current divides in a parallel circuit, making the result incorrect.

Option (B): 1 : 2

This ratio would be obtained only if current through both inductors were the same. In a parallel connection, currents are different because they depend on the inductance of each branch. Therefore, this option is incorrect.

Option (D): 4 : 1

This option results from squaring the current ratio but forgetting to include the inductance factor present in the energy formula. Since energy is proportional to both inductance and the square of current, this answer is mathematically incorrect.

Important Formula Used

Energy stored in an inductor:

U = ½ LI²

For inductors connected in parallel:

Current ∝ 1/L

Using these two relations together allows quick calculation of energy ratios without lengthy calculations.

Exam Tips for Similar Questions

Whenever inductors are connected in parallel, first remember that the voltage across every branch is identical. Next, determine how the current divides. Only after finding the current ratio should you apply the energy formula. Students often directly compare inductance values and forget the current term, which leads to incorrect answers. Following the correct sequence of concepts ensures accuracy even in complex numerical problems.

Final Answer

Energy stored in inductors P and Q = 2 : 1

Correct Option: (C)