Question

A long solenoid has 103 turns/cm and carries current i. An electron moves within the solenoid in a circle of radius 2.60 cm perpendicular to the solenoid axis. The speed of the electron is 1.38 × 107 m/s. Find the current i in the solenoid.

Answer 1

0.23348 A

Step-by-step explanation:

B = Magnetic field

v = Velocity of electron =
`1.38* 10^7\ m/s`

q = Charge of electron =
`1.6* 10^(-19)\ C`

`\mu_0` = Vacuum permeability =
`4\pi * 10^(-7)\ H/m`

r = Radius of circle = 0.026 m

N = Number of turns = 103 turns/cm =
`103* 100\ turns/m`

I = Current

The magnetic and centripetal force will be balanced

`Bqv=m(v^2)/(r)\\\Rightarrow B=(mv)/(qr)`

The magnetic field in solenoid is given by

`B=N\mu_0 I\\\Rightarrow I=(B)/(N\mu_0)`

From the first equation

`I=((mv)/(qr))/(N\mu_0)\\\Rightarrow I=(mv)/(N\mu_0qr)\\\Rightarrow I=(9.11* 10^(-31)* 1.38* 10^7)/(103* 100* 4\pi * 10^(-7)* 1.6* 10^(-19)* 0.026)\\\Rightarrow I=0.23348\ A`

The current in the solenoid is 0.23348 A