KCET · Physics · Magnetic Effects of Current

An electron enters the space between the plates of a charged capacitor as shown. The charge density on the plate is \(\sigma\). Electric intensity in the space between the plates is E. A uniform magnetic field \(B\) also exists in the space perpendicular to the direction of \(E\)
The electron moves perpendicular to both \(\overrightarrow{\mathrm{E}}\) and \(\overrightarrow{\mathbf{B}}\) without any change in direction. The time taken by the electron to travel a distance \(l\) in the space is

A \(\frac{\sigma l}{\varepsilon_{0} \mathrm{~B}}\)
B \(\frac{\sigma \mathrm{B}}{\varepsilon_{0} l}\)
C \(\frac{\varepsilon_{0} l \mathrm{~B}}{\sigma}\)
D \(\frac{\varepsilon_{0} l}{\sigma \mathrm{B}}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{\varepsilon_{0} l \mathrm{~B}}{\sigma}\)

Step-by-step Solution

Detailed explanation

Here, magnetic force \(=\) electrostatic force
\(\begin{aligned}q v B &=q E \\v &=\frac{E}{B} \\&=\frac{\sigma}{\varepsilon_{0} B}\end{aligned}\)
The time taken by electron to travel a distance 1 in that space
\(\begin{aligned}t=\frac{1}{v} &=\frac{1}{\frac{\sigma}{\varepsilon_{0} B}} \\&=\frac{\varepsilon_{0} l B}{\sigma}\end{aligned}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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