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Question #43756

Difficulty: 8/10 | 🟢 Free

The Question

A cylindrical region of radius $R$ contains a uniform magnetic field $\vec{B}$ directed into the plane of the paper. The magnitude of the magnetic field varies with time $t$ as $B = \alpha t$ , where $\alpha$ is a positive constant. A smooth, evacuated, semi-circular insulating tube of radius $R$ is placed along the lower half of the boundary of this region. The tube connects points $P$ and $Q$ , which lie on the right and left ends of the horizontal diameter, respectively. A metal plate located at $P$ has a work function $\phi$ and is illuminated by monochromatic light of frequency $\nu$ . Photoelectrons emitted from the plate are constrained to move through the tube to reach a collector at $Q$ . What is the minimum stopping potential $V_0 = V_P - V_Q$ required to prevent even the most energetic photoelectrons from reaching $Q$ ? (Assume $h$ is Planck's constant and $e$ is the elementary charge).

Options

A.
$\frac{h\nu - \phi}{e} + \frac{\pi R^2 \alpha}{2}$
B.
$\frac{h\nu - \phi}{e}$
C.
$\frac{h\nu - \phi}{e} - \frac{\pi R^2 \alpha}{2}$
D.
$\frac{h\nu - \phi}{e} + \pi R^2 \alpha$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Einstein's Photoelectric Equation|Induced Electric Fields|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the integration of the photoelectric effect with electromagnetic induction, specifically targeting the misconception that stopping potential is exclusively determined by the incident photon energy and work function.

Students often blindly apply $eV_0 = K_{max}$ without accounting for the work done by non-conservative induced electric fields present in time-varying magnetic environments.

The trap lies in recognizing that the induced electric field exerts a force that does path-dependent work on the electron.

By constraining the electron to a specific semi-circular trajectory, the problem forces the application of the generalized work-energy theorem, requiring students to correctly determine the direction of the induced field (Lenz's Law), the resulting force on a negative charge, and the sign of the work done along the specific path.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Dual Nature of Radiation and Matter - Class 12
Concept	Einstein's Photoelectric Equation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free