Question

The aurora is caused when electrons and protons, moving in the earth's magnetic field of ≈5.0×10−5T, collide with molecules of the atmosphere and cause them to glow.

A) What is the radius of the circular orbit for an electron with speed 3.0×10⁶ m/s ? Assume that the electron moves in a plane perpendicular to the magnetic field.

Express your answer with the appropriate units.

B) What is the radius of the circular orbit for a proton with speed 3.0×10⁴ m/s ?

Answer 1

Using force balance on electron

Centripetal Force = Force due to magnetic field

Fc = Fm

Fm = q*VxB = q*V*B*sin theta

Since electron moves in a plane perpendicular to the magnetic field, So sin 90 deg = 1

Fm = q*V*B

And Fc = m*V^2/r

So,

mV^2/r = q*V*B

r = mV/qB

Part A.

m = mass of electron = 9.1*10^-31 kg

V = speed of electron = 3.0*10^6 m/sec

q = Charge on electron = -1.6*10^-19 C

B = Magnetic field = 5.0*10^-5 T

So

r = 9.1*10^-31*3.0*10^6/(1.6*10^-19*5.0*10^-5)

r = 0.34 m = radius of circular orbit

Part B.

m = mass of Proton = 1.67*10^-27 kg

V = speed of proton = 3.0*10^4 m/sec

q = Charge on proton = 1.6*10^-19 C

B = Magnetic field = 5.0*10^-5 T

So

r = 1.67*10^-27*3.0*10^4/(1.6*10^-19*5.0*10^-5)

r = 6.26 m = radius of circular orbit