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Question #43069

Difficulty: 8/10 | 🟢 Free

The Question

A parallel plate capacitor consists of two rectangular plates of length $l_0$ and width $w$ , separated by a distance $d$ . A solid dielectric slab of the same dimensions ( $l_0 \times w \times d$ ) and dielectric constant $K$ is inserted a distance $x$ ( $0 < x < l_0$ ) between the plates. The capacitor is connected in series with an inductor of inductance $L$ , a resistor of resistance $R$ , and an AC voltage source of RMS value $V_{\text{rms}}$ and variable angular frequency $\omega$ . For any given position $x$ of the slab, the AC source is tuned to the instantaneous resonant frequency $\omega_0$ of the circuit. Assuming fringing fields are negligible, which of the following expressions correctly gives the time-averaged electrostatic force $\langle F \rangle$ pulling the slab into the capacitor, and the half-power bandwidth $\Delta\omega$ of the circuit at this position $x$ ? (All quantities are in SI units)

Options

A.
$\langle F \rangle = \frac{V_{\text{rms}}^2 L (K-1)}{2 R^2 [l_0 + x(K-1)]}$ , $\Delta\omega = \frac{R}{L}$
B.
$\langle F \rangle = \frac{V_{\text{rms}}^2 \epsilon_0 w (K-1)}{2 d}$ , $\Delta\omega = \frac{R}{L} \sqrt{\frac{l_0}{l_0 + x(K-1)}}$
C.
$\langle F \rangle = \frac{V_{\text{rms}}^2 \epsilon_0 w (K-1)}{2 d}$ , $\Delta\omega = \frac{R}{L}$
D.
$\langle F \rangle = \frac{V_{\text{rms}}^2 L (K-1)}{2 R^2 [l_0 + x(K-1)]}$ , $\Delta\omega = \frac{1}{\sqrt{L \frac{\epsilon_0 w}{d} [l_0 + x(K-1)]}}$

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

Bandwidth in RLC|Dielectric Slab Displacement|Vector Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple AC circuit dynamics with mechanical work-energy principles, while avoiding a fundamental trap regarding resonance characteristics.

The primary trap is assuming that the bandwidth $\Delta\omega$ of the RLC circuit depends on the varying capacitance $C(x)$ , confusing it with the resonant frequency $\omega_0 = 1/\sqrt{LC}$ or the Quality factor $Q = \frac{1}{R}\sqrt{\frac{L}{C}}$ .

In reality, $\Delta\omega = R/L$ remains strictly constant.

Additionally, students often incorrectly use the source voltage $V_{\text{rms}}$ instead of the amplified capacitor voltage $V_{C,\text{rms}} = Q V_{\text{rms}}$ when calculating the electrostatic force $F = \frac{1}{2}V^2 \frac{dC}{dx}$ , leading to the incorrect force expression in the distractors.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Alternating Current - Class 12
Concept	Bandwidth in RLC
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free