Question

For your senior project, you would like to build a cyclotron that will accelerate protons to $10\%$of the speed of light. The largest vacuum chamber you can find is $50cm$in diameter. What magnetic field strength will you need?

Short answer

You will need magnetic field strength is$1.25T$

Step-by-step solution

Step :1 Introduction

The given is that the speed of the particle is $10\%$of the speed of sunshine and therfore the diameter of the chamber is the target is to $50cm$. The objective is to find the magnetic field strength.

Step :2 Cyclotron 

The cyclotron motion is defined as a particle traveling in an exceedingly circular motion perpendicular to the force field at a constant speed. The radius of the circular motion produced by the magnetic field is related to it by an equation in the form

$r_{cyc}=\frac{mv}{qB}$

Where $q$denotes the particle's charge, $v$denotes its speed, $B$denotes the magnetic field, and $m$denotes the particle's mass. Because the particle's speed is $10\%$that of light, we can calculate it

$v=0.13(3\times {10}^{8}m/s)=3\times {10}^{7}m/s$

The chamber's diameter is $50cm$. We can calculate its radius by

$r_{cyc}=\frac{50cm}{2}=25cm=0.25m$

Our goal is to determine the appropriate magnetic field, thus we change the equation for $B$to be

$B=\frac{mv}{q{r}_{cyc}}$

Now we can get $B$by plugging the numbers for $m,v,q$and $r_{cyc}$into the equation

$B=\frac{mv}{q{r}_{cyc}}$

$=\frac{(1.67\times {10}^{-27}kg)(3\times {10}^{7}m/s)}{(1.6\times {10}^{-19}C)(0.25m)}$

$=1.25T$