Physics Magnetic field

Question

An electron and a proton have the same kinetic energy upon entering a region of constant magnetic field and their velocity vectors are perpendicular to the magnetic field. Suppose the magnetic field is strong enough to allow the particles to circle in the field.

What is the ratio of the radii of their circular paths rp/re?

Answer 1

Hello Annie!

First, we'll derive an expression for radius, then we'll use it to come up with an answer to the question.

The magnetic field is perpendicular to the velocity, and this makes the force from the field on the particle (called the Lorenz force) act as a centripetal force.

F(magnetic) = Fc

qvB = mv² / r

r = mv / qB

This is where it gets tricky.

rp/re = (mp*vp)/qB) / (me*re/qB) = (mp*vp)/(me*ve)

Since KE = 0.5 mv², v = (2KE/m)^1/2, and v² = 2KE / m

Now replace them in the equation:

rp/re = (mp*2KE/mp) / (me*2KE/me) = 1

Woah! The answer is 1!