Magnetic field electron question

[lemaire]

Homework Statement

In 2.2 µs, an electron moves 11 cm in the direction of a 0.07 T magnetic field. The electron's velocity components perpendicular and parallel to the field are equal.
What is the length of its actual spiral trajectory and How many orbits about the field direction does it complete?

Homework Equations

T = 2pir/v = 2pim/qB

The Attempt at a Solution

I was thinking of using the period, but the velocity is unknown.

 

[marcusl]

You know the speed because it tells you that it moves 11 cm in 2.2us along the field direction, and it also tells you the split of velocity along and normal to the field.

 

[lemaire]

so which formula should i use in order to solve the problem.

 

[marcusl]

Assume that B is along z. The electron speed is 11 cm/2.2 us = 5e6 cm/s, and the transverse component of velocity v_t is $$\sqrt{2}$$ times smaller. v_z is the same.

The Lorentz force is

$$\vec{F}=q\vec{v}\times\vec{B}$$.

Only the transverse velocity components contribute to the force. Divide v_t into x and y components and solve the equation for each one. You'll see that, in the transverse plane, the particle executes a circular motion at the cyclotron frequency

$$\omega=\frac{qB}{m}.$$

Adding the z velocity gives a spiral. That gives you enough information to answer the questions.