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Question #43456

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The Question

A common-emitter NPN transistor amplifier has a low-frequency small-signal current gain $\beta_0$ and a base-emitter dynamic resistance $r_\pi$ . A parasitic parallel-plate capacitor is formed directly across the base-emitter terminals. The capacitor has circular plates of radius $R$ and separation $d$ . The space between the plates is filled with a non-uniform dielectric whose permittivity varies radially as $\epsilon(r) = \epsilon_0(1 + \alpha r)$ , where $r$ is the radial distance from the central axis. An AC input voltage $V_{be}(t) = V_0 \sin(\omega t)$ is applied across the base-emitter terminals. Assuming the intrinsic capacitances of the transistor are negligible, which of the following expressions correctly gives the maximum induced magnetic field $B_0$ at the edge of the capacitor ( $r=R$ ) and the effective high-frequency current gain magnitude $|\beta_{eff}| = |I_c|/|I_{in}|$ of the combined structure? (Here $I_{in}$ is the total current supplied by the source and $I_c$ is the collector current).

Options

A.
$B_0 = \frac{\mu_0 \epsilon_0 \omega V_0 R}{2d} \left( 1 + \frac{2}{3}\alpha R \right)$ , $|\beta_{eff}| = \frac{\beta_0}{\sqrt{1 + \left[ \frac{\pi \omega r_\pi \epsilon_0 R^2}{d} \left( 1 + \frac{2}{3}\alpha R \right) \right]^2}}$
B.
$B_0 = \frac{\mu_0 \epsilon_0 \omega V_0 R}{2d} \left( 1 + \frac{1}{2}\alpha R \right)$ , $|\beta_{eff}| = \frac{\beta_0}{\sqrt{1 + \left[ \frac{\pi \omega r_\pi \epsilon_0 R^2}{d} \left( 1 + \frac{1}{2}\alpha R \right) \right]^2}}$
C.
$B_0 = \frac{\mu_0 \epsilon_0 \omega V_0 R}{2d} \left( 1 + \frac{2}{3}\alpha R \right)$ , $|\beta_{eff}| = \frac{\beta_0}{1 + \frac{\pi \omega r_\pi \epsilon_0 R^2}{d} \left( 1 + \frac{2}{3}\alpha R \right)}$
D.
$B_0 = \frac{\mu_0 \epsilon_0 \omega V_0 R}{2d}$ , $|\beta_{eff}| = \frac{\beta_0}{\sqrt{1 + \left[ \frac{\pi \omega r_\pi \epsilon_0 R^2}{d} \right]^2}}$

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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

Transistor Configuration Gains|Non-Uniform Dielectric Filling|Ampere’s Law Displacement Current

Why This Question Is Tricky

Common Mistakes

This problem tests the coupling of electromagnetic field theory (Maxwell's displacement current) with lumped-element circuit models (small-signal transistor analysis).

Students often fall into two primary traps: (1) incorrectly integrating the spatially varying permittivity by treating it as a simple average (yielding the $1 + \alpha R/2$ term), and (2) failing to recognize that the displacement current acts as a parasitic shunt admittance that is $90^\circ$ out of phase with the resistive base current.

By forcing the calculation of both the induced magnetic field via the Ampere-Maxwell law and the effective AC current gain, the question penalizes those who decouple field theory from circuit analysis or linearly add admittances without proper phasor considerations.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Semiconductor Electronics: Materials, Devices, and Simple Circuits - Class 12
Concept	Transistor Configuration Gains
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free