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) :

Solution

Physics sir · 3 steps

The electron's circle starts on the axis, so its farthest point is a full diameter away: keep 2r within R.

1. Field and force. Inside a long solenoid the field is uniform and along the axis, B = μ₀ n I. The electron is fired along a radius, so its velocity is perpendicular to B. It therefore moves in a circle in the cross-sectional plane, with radius r = mv/eB.
2. Where it reaches. The circle passes through the starting point on the axis. Its point farthest from the axis is diametrically opposite, at a distance 2r. The electron misses the surface only if 2r ≤ R, so r ≤ R/2.

   mv/(eμ₀ n I) ≤ R/2 ⇒ v ≤ (eμ₀ n I R)/2m

3. So. The maximum speed is v = (eμ₀ n I R)/2m, which is option (c).

The idea to keep. When a charge starts on the axis and circles in a uniform field, its farthest reach from the start is the diameter 2r, not r. Compare 2r with the boundary distance.

Answer from the NTA final answer key. Solution by PrepWiser.