Before You Begin

Question #43403

Difficulty: 8/10 | 🟢 Free

The Question

An ideal LC circuit consists of a capacitor of capacitance $C$ and a long ideal solenoid of inductance $L_1$ , radius $r_1$ , and $n_1$ turns per unit length. The circuit undergoes undamped electromagnetic oscillations with a maximum current $I_0$ in the solenoid. A secondary coil consisting of $N_2$ total turns and radius $r_2$ (where $r_2 < r_1$ ) is placed coaxially inside the central region of the solenoid. What is the magnitude of the induced electromotive force (EMF) in the secondary coil at the instant when the electrical energy stored in the capacitor is exactly three times the magnetic energy stored in the solenoid?

Options

A.
$\frac{\sqrt{3}}{2} \frac{\mu_0 n_1 N_2 \pi r_2^2 I_0}{\sqrt{L_1 C}}$
B.
$\frac{1}{2} \frac{\mu_0 n_1 N_2 \pi r_2^2 I_0}{\sqrt{L_1 C}}$
C.
$\frac{3}{4} \frac{\mu_0 n_1 N_2 \pi r_2^2 I_0}{\sqrt{L_1 C}}$
D.
$\frac{\sqrt{3}}{2} \frac{\mu_0 n_1 N_2 \pi r_1^2 I_0}{\sqrt{L_1 C}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Magnetic Field on Axis vs. Center|LC Oscillations Energy Cycle|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of the phase quadrature between the current and its derivative in an oscillating LC circuit, coupled with Faraday's law.

A common trap is assuming the induced EMF (which depends on $dI/dt$ ) is directly proportional to the instantaneous current $I$ .

When $U_E = 3U_B$ , the current is half its maximum ( $I = I_0/2$ ), leading students to incorrectly choose an EMF that is half the maximum (Option B).

However, because $I$ and $dI/dt$ are out of phase by $\pi/2$ , when $I$ is at $50\%$ of its peak, $dI/dt$ is at $\sqrt{3}/2 \approx 86.6\%$ of its peak.

Another trap (Option D) is using the primary solenoid's radius $r_1$ instead of the secondary coil's radius $r_2$ for the flux area, ignoring that the magnetic flux only links through the smaller cross-sectional area of the secondary coil.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Moving Charges and Magnetism - Class 12
Concept	Magnetic Field on Axis vs. Center
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free