Consider a circuit consisting of a capacitor of capacitance C and a coil with N turns per unit lengt

Q: Why is the net EMF in the smaller coil zero?

Because its ends are connected by a conducting wire with negligible resistance. Any non-zero EMF would cause an infinite current, which is physically impossible. Thus, the induced EMFs must perfectly cancel out.

Q: Why does the smaller coil reduce the overall inductance?

By Lenz's law, the induced current in the short-circuited coil creates a magnetic field that opposes the change in the main field. This opposing field reduces the total flux linkage for a given current change, effectively lowering the inductance.

Q: How did you calculate the mutual inductance $M$?

Mutual inductance depends on the shared flux. Since the small coil is completely inside, all of its area is subjected to the large coil's uniform magnetic field. We simply multiplied the large coil's field by the small coil's area and total number of turns.

Q: What would happen if the smaller coil was not short-circuited?

If the smaller coil was open-circuited, no current $i'$ could flow in it. The mutually induced EMF would still exist, but it wouldn't create an opposing magnetic field. The equivalent inductance of the circuit would just be $L$.

Q: Why do we use the formula $\omega = 1/\sqrt{L_{eq}C}$?

In an ideal LC circuit, resonance occurs when the inductive reactance equals the capacitive reactance. This balance point is universally given by $\omega = 1/\sqrt{LC}$, where $L$ is the total effective inductance of the system.

Question

Consider a circuit consisting of a capacitor of capacitance $C$ and a coil with $N$ turns per unit length, cross sectional area $S$ and length $d$, where $d^2 \gg S$. There is another coil of length $d/2$, cross sectional area $S/2$ and $2N$ turns per unit length completely inside the larger coil, as shown in the figure. The ends of this smaller coil are connected with each other by an insulated conducting wire. The self-inductance of the larger coil is $L$. Neglecting edge effects and all the Ohmic resistances, the resonant frequency of the circuit is:

Options:

(A) $\frac{4}{\sqrt{15LC}}$

(B) $\frac{6}{\sqrt{5LC}}$

(C) $\frac{2}{\sqrt{3LC}}$

(D) $\sqrt{\frac{2}{3LC}}$

Sigma Percentile · Accepted Answer

### The Beauty of Coupled Oscillators

Imagine a standard LC circuit: a capacitor discharging through an inductor, creating a beautiful, rhythmic oscillation of energy between an electric field and a magnetic field. It is the heartbeat of modern electronics. But what happens when we introduce a twist? What if we place a smaller, short-circuited coil directly inside the main inductor?

This is not just a theoretical curiosity; it is a profound exploration of **Faraday's Law** and **Lenz's Law**. The inner coil is not a passive bystander. It actively fights the changes happening around it, fundamentally altering the behavior of the entire circuit. Let's dive into the mathematics and physics of this fascinating setup.

### Analyzing the Setup

First, we need to understand the individual properties of our two coils. The self-inductance of a long solenoid is given by the classic formula:
$$L = \mu_0 n^2 A \ell$$
where $n$ is the number of turns per unit length, $A$ is the cross-sectional area, and $\ell$ is the length.

For our **larger coil**, the parameters are given directly:
$$L = \mu_0 N^2 S d$$

Now, let's look at the **smaller coil**. It is half as long ($d/2$), has half the cross-sectional area ($S/2$), but its turn density is doubled ($2N$). Let's plug these into our formula to find its self-inductance, $L'$:
$$L' = \mu_0 (2N)^2 \left(\frac{S}{2}ight) \left(\frac{d}{2}ight)$$
$$L' = \mu_0 (4N^2) \frac{S d}{4}$$
$$L' = \mu_0 N^2 S d = L$$

Surprisingly, despite being physically smaller, the doubled turn density perfectly compensates for the reduced volume. The self-inductance of the smaller coil is exactly equal to that of the larger coil!

### The Master Equation: Mutual Inductance

Because the smaller coil is completely nested inside the larger one, the magnetic field generated by the large coil passes entirely through the small coil. This means they are magnetically coupled. The mutual inductance $M$ is calculated by finding the flux linkage in the small coil due to the current in the large coil.

The magnetic field of the large coil is $B = \mu_0 N i$. The flux through one turn of the small coil is $B 	imes (S/2)$. The total number of turns in the small coil is $(2N) 	imes (d/2) = N d$. Therefore, the total flux linkage $\Phi_{21}$ is:
$$\Phi_{21} = (\mu_0 N i) \left(\frac{S}{2}ight) (N d)$$
$$M = \frac{\Phi_{21}}{i} = \mu_0 N^2 S \frac{d}{2}$$

Comparing this to our expression for $L$, we immediately see that:
$$M = \frac{L}{2}$$

### The Short-Circuit Effect

Here is where the physics gets truly exciting. The smaller coil is short-circuited—its ends are connected by a wire with zero resistance. According to **Faraday's Law**, the total induced EMF in a closed loop with zero resistance must be zero. If it weren't, an infinite current would flow!

The total flux $\Phi'$ in the smaller coil is the sum of the flux from its own current $i'$ and the flux from the larger coil's current $i$:
$$\Phi' = M i + L' i'$$

Since the EMF is the negative rate of change of flux, we have:
$$e' = -\frac{d\Phi'}{dt} = -M \frac{di}{dt} - L' \frac{di'}{dt} = 0$$

Substituting $L' = L$ and $M = L/2$, we get a direct relationship between the changing currents:
$$\frac{L}{2} \frac{di}{dt} + L \frac{di'}{dt} = 0$$
$$\frac{di'}{dt} = -\frac{1}{2} \frac{di}{dt}$$

This negative sign is **Lenz's Law** in action! The induced current in the inner coil flows in the opposite direction, creating a magnetic field that fights the primary field.

### Final Calculation: Equivalent Inductance

Now, let's find the net EMF $e$ across the larger coil. It experiences its own self-induced EMF and a mutually induced EMF from the inner coil:
$$e = -L \frac{di}{dt} - M \frac{di'}{dt}$$

Substitute our relationship for $di'/dt$:
$$e = -L \frac{di}{dt} - \left(\frac{L}{2}ight) \left(-\frac{1}{2} \frac{di}{dt}ight)$$
$$e = -L \frac{di}{dt} + \frac{L}{4} \frac{di}{dt}$$
$$e = -\frac{3L}{4} \frac{di}{dt}$$

The effective inductance $L_{eq}$ of the entire nested system is simply the coefficient of $di/dt$. Therefore:
$$L_{eq} = \frac{3L}{4}$$

The inner coil has effectively "stolen" a quarter of the large coil's inductance! Finally, we substitute this equivalent inductance into the standard formula for the resonant frequency of an LC circuit:
$$\omega = \frac{1}{\sqrt{L_{eq} C}}$$
$$\omega = \frac{1}{\sqrt{\frac{3L}{4} C}}$$
$$\omega = \frac{2}{\sqrt{3LC}}$$

And there we have it. A beautiful interplay of geometry, electromagnetism, and circuit theory leading to an elegant final result.

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Visualized Solution

$C i r c u i t S e t u p$

$S e l f - I n d u c t an ce o f L a r g e C o i l$

$S e l f - I n d u c t an ce o f S ma l l C o i l$

$M u t u a l I n d u c t an ce (M)$

$C u r r e n t in S ma l l C o i l$

$N e t E M F in L a r g e C o i l$

$R eso nan t F r e q u e n cy$

The Sigma Insight: Self and Mutual Inductance

The Beauty of Coupled Oscillators

Analyzing the Setup

The Master Equation

The Short-Circuit Effect

Final Calculation

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Circuit Setup

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Self−Inductance of Large Coil

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Self−Inductance of Small Coil

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Mutual Inductance (M)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Current in Small Coil

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Net EMF in Large Coil

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Resonant Frequency