First slide

Magnetic field due to current element

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 $R$ is (all symbols have their standard meaning):

Moderate

A

$\frac{e μ_{0} n I R}{4 m}$
B

$\frac{2 e μ_{0} n I R}{m}$
C

$\frac{e μ_{0} n I R}{m}$
D

$\frac{e μ_{0} n I R}{2 m}$

Solution

Electron does not hit the cylinder if it describes the circular path of radius $r = \frac{R}{2}$
We known that
$\begin{array}{l} r = \frac{m v}{B q} \\ \Rightarrow \frac{R}{2} = \frac{m v}{B q} \\ A l s o, B = μ_{0} n I \\ \Rightarrow v = \frac{B q R}{2 m} = \frac{μ_{0} n I e R}{2 m} \end{array}$

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