Relativistic form of Newton's second law

[andresordonez]

SOLVED
(Problem 10, Chapter 2, Modern Physics - Serway)

Homework Statement

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

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

Homework Equations

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

The Attempt at a Solution

The particle moves in a circle then the magnetic field is perpendicular to the velocity and F=qvB.

<br /> f=\frac{v}{2\pi R}<br />

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

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

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

What's wrong?

 

[Maxim Zh]

The relativistic form of Newton's second law is

<br /> \frac{\partial\vec{p}}{\partial t} = \vec{F},<br />

where

<br /> \vec{p} = m\gamma(v)\vec{v}.<br />

The factor

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

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!

 

[andresordonez]

Thanks!