Question:
An electron gun is placed inside a long solenoid of radius R on its axis. The solenoid has n turns/length and carries a current I. The electron gun shoots an electron along the radius of the solenoid with speed v. If the electron does not hit the surface of the solenoid, maximum possible value of v is (all symbols have their standard meaning) :
An electron gun is placed inside a long solenoid of radius R on its axis. The solenoid has n turns/length and carries a current I. The electron gun shoots an electron along the radius of the solenoid with speed v. If the electron does not hit the surface of the solenoid, maximum possible value of v is (all symbols have their standard meaning) :
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Inside the solenoid \(B = \mu_0 nI\) along the axis, perpendicular to the velocity; the electron moves on a circle passing through the axis whose diameter must not exceed R.
Updated On: Nov 5, 2026
- \(\frac{e\mu_0 nIR}{4m}\)
- \(\frac{e\mu_0 nIR}{m}\)
- \(\frac{e\mu_0 nIR}{2m}\)
- \(\frac{2e\mu_0 nIR}{m}\)
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The Correct Option is C
Solution and Explanation
Field inside the long solenoid: \(B = \mu_0 nI\), directed along the axis. The electron is shot along a radius, so \(\vec{v} \perp \vec{B}\) and it moves on a circle of radius \(r = \frac{mv}{eB}\) lying in a plane perpendicular to the axis.
The circle passes through the starting point on the axis. The farthest point of the path from the axis is the diametrically opposite point at distance \(2r\).
To avoid hitting the surface: \(2r \le R \Rightarrow r \le \frac{R}{2}\).
\(\frac{mv}{e\mu_0 nI} \le \frac{R}{2} \Rightarrow v \le \frac{e\mu_0 nIR}{2m}\).
Answer: (3) \(\frac{e\mu_0 nIR}{2m}\)
The circle passes through the starting point on the axis. The farthest point of the path from the axis is the diametrically opposite point at distance \(2r\).
To avoid hitting the surface: \(2r \le R \Rightarrow r \le \frac{R}{2}\).
\(\frac{mv}{e\mu_0 nI} \le \frac{R}{2} \Rightarrow v \le \frac{e\mu_0 nIR}{2m}\).
Answer: (3) \(\frac{e\mu_0 nIR}{2m}\)
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