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Question #43769

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The Question

A photoelectric setup consists of a solid conducting sphere of radius $a$ placed concentrically inside a thin conducting spherical shell of radius $b$ ( $b > a$ ). The inner sphere carries a charge $+Q$ and the outer shell carries a charge $-q$ ( $q > 0$ ). Monochromatic light of wavelength $\lambda$ is incident on the inner sphere, causing the emission of photoelectrons. The work function of the inner sphere material is $\phi$ . What is the minimum value of $Q$ required to completely stop the fastest photoelectrons from reaching the outer shell? (Assume $\frac{hc}{\lambda} > \phi$ , and $e$ is the magnitude of the elementary charge).

Options

A.
$\frac{4\pi\epsilon_0 a b}{e(b-a)} \left( \frac{hc}{\lambda} - \phi \right)$
B.
$\frac{4\pi\epsilon_0 a^2}{e(b-a)} \left( \frac{hc}{\lambda} - \phi \right)$
C.
$\frac{4\pi\epsilon_0 a}{e} \left( \frac{hc}{\lambda} - \phi \right) + q\frac{a}{b}$
D.
$\frac{4\pi\epsilon_0 b^2}{e(b-a)} \left( \frac{hc}{\lambda} - \phi \right)$

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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

Stopping Potential Sign|Sign Convention Dynamics|Potential of Concentric Shells

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply the Work-Energy Theorem in a non-uniform electric field and their understanding of Gauss's Law in concentric geometries.

A common trap is assuming the electric field between the shells is uniform ( $E = kQ/a^2$ ), which incorrectly simplifies the potential difference to $\Delta V = E(b-a)$ .

Another major misconception is incorrectly applying the superposition principle by assuming the charge $-q$ on the outer shell alters the potential difference or the electric field in the region $a < r < b$ .

Finally, students often confuse absolute electrostatic potential with potential difference, mistakenly equating the stopping potential to the absolute potential of the inner shell $V_a$ rather than the path-dependent integral $V_a - V_b$ .

Quick Facts About This Question

Question Information

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