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Question #43614

Difficulty: 8/10 | 🟢 Free

The Question

A monochromatic light pulse of wavelength $\lambda$ instantaneously strikes a metal plate of work function $\Phi$ at $t=0$ , emitting photoelectrons. Simultaneously, a time-varying uniform electric field $\vec{E}(t) = E_0 \left(1 - \frac{t^2}{\tau^2}\right) \hat{i}$ is applied in the region for a duration $0 \le t \le \tau$ , after which the field becomes zero. The most energetic photoelectrons are emitted at $t=0$ and travel along the $+\hat{i}$ direction. What is the minimum value of $E_0$ required to ensure that no photoelectron escapes to infinity? (Let $m$ and $e$ be the mass and magnitude of charge of an electron, respectively, and $c$ be the speed of light. Assume non-relativistic mechanics.)

Options

A.
$\frac{3}{2 e \tau} \sqrt{2m \left( \frac{hc}{\lambda} - \Phi \right)}$
B.
$\frac{2}{e \tau} \sqrt{2m \left( \frac{hc}{\lambda} - \Phi \right)}$
C.
$\frac{1}{e \tau} \sqrt{2m \left( \frac{hc}{\lambda} - \Phi \right)}$
D.
$\frac{3}{e \tau} \sqrt{m \left( \frac{hc}{\lambda} - \Phi \right)}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Instantaneous Nature|Variable Force Integration|Coulombic Central Force

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of the photoelectric effect with the impulse-momentum theorem in the presence of a time-varying force.

A common trap is attempting to use the work-energy theorem by incorrectly assuming the stopping potential is simply $E_0 d$ or integrating the field over time as if it were space (confusing $\int E dt$ with $\int E dx$ ).

Another trap is assuming a linear average field $E_0/2$ instead of correctly integrating the quadratic time dependence.

By forcing the student to recognize that a time-bounded field requires an impulse analysis rather than a spatial potential analysis, it rigorously probes their fundamental understanding of Newton's Second Law applied to quantum emission scenarios.

Quick Facts About This Question

Question Information

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