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Question #43552

Difficulty: 8/10 | 🟢 Free

The Question

In a specialized electron microscope, an electron (mass $m$ , charge $-e$ ) is fired from the origin $(0,0,0)$ with an initial velocity $\vec{v} = v_0 \hat{i}$ through a vacuum tube of horizontal length $L$ . The tube is situated above an infinite non-conducting flat sheet lying in the $xz$ -plane with a uniform surface charge density $\sigma > 0$ . The electron's trajectory bends due to the electric field. Defining a dimensionless parameter $\beta = \frac{e \sigma L}{2 m \epsilon_0 v_0^2}$ , what is the exact 'effective tube length' $S$ , defined as the total arc length of the electron's trajectory from $x=0$ to $x=L$ ? Ignore gravity and relativistic effects.

Options

A.
$\frac{L}{2} \left[ \sqrt{1+\beta^2} + \frac{1}{\beta} \ln(\beta + \sqrt{1+\beta^2}) \right]$
B.
$L \sqrt{1 + \frac{\beta^2}{4}}$
C.
$L \left( 1 + \frac{\beta^2}{6} \right)$
D.
$L \left[ \sqrt{1+\beta^2} + \frac{1}{\beta} \ln(\beta + \sqrt{1+\beta^2}) \right]$

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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

Microscope Tube Length|Electric Field Line Geometry|Relative Velocity in 2D

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of electrostatics (Gauss's Law) with exact Lagrangian-style 2D kinematics, specifically targeting the 'Lethal Collision' trap where students inappropriately rely on paraxial optics approximations.

A common mistake is assuming the effective length is simply the straight-line distance from the origin to the exit coordinate (Option B), which ignores the path curvature.

Another frequent error is applying the binomial expansion $\sqrt{1+x^2} \approx 1+x^2/2$ to the arc length integral (Option C), which is only valid for infinitesimally small deflections and fails for exact trajectory tracking in strong fields.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Ray Optics and Optical Instruments - Class 12
Concept	Microscope Tube Length
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free