# Homework Help: Relativistic form of Newton's second law

1. Jan 24, 2010

### andresordonez

SOLVED
(Problem 10, Chapter 2, Modern Physics - Serway)
1. The problem statement, all variables and given/known data
Recall that the magnetic force on a charge q moving with velocity $$\vec{v}$$ in a magnetic field $$\vec{B}$$ is equal to $$q\vec{v}\times\vec{B}$$. If a charged particle moves in a circular orbit with a fixed speed $$v$$ in the presence of a constant magnetic field, use the relativistic form of Newton's second law to show that the frequency of its orbital motion is

$$f=\frac{qB}{2\pi m}(1-\frac{v^2}{c^2})^{1/2}$$

2. Relevant equations

$$F=\frac{ma}{(1-v^2/c^2)^{3/2}}$$

3. The attempt at a solution
The particle moves in a circle then the magnetic field is perpendicular to the velocity and $$F=qvB$$.

$$f=\frac{v}{2\pi R}$$

$$qvB=\frac{ma}{(1-v^2/c^2)^{3/2}} =\frac{m}{(1-v^2/c^2)^{3/2}}\frac{v^2}{R}$$

$$R=\frac{mv}{(1-v^2/c^2)^{3/2}qB}$$

$$f=\frac{(1-v^2/c^2)^{3/2}qB}{2\pi m}$$

What's wrong?

Last edited: Jan 24, 2010
2. Jan 24, 2010

### Maxim Zh

The relativistic form of Newton's second law is

$$\frac{\partial\vec{p}}{\partial t} = \vec{F},$$

where

$$\vec{p} = m\gamma(v)\vec{v}.$$

The factor

$$\gamma(v) = \frac{1}{\sqrt{1-(v/c)^2}}$$

is constant in this task. Do not differentiate it!
Write the differential equation system for $$v_x$$ and $$v_y$$ and derive the frequency.

Good luck!

3. Jan 24, 2010

Thanks!