A-level Physics (Advancing Physics)/Particle Accelerators/Worked Solutions

1. Use the formula for centripetal force to show that the radius of motion depends on the speed of the moving object.

$F=\frac{m{v}^2}{r}$

$r=\frac{m{v}^2}{F}$

So, if F is constant:

$r\propto v^2$

2. A cyclotron with a diameter of 1.5m is used to accelerate electrons (mass 9.11 x 10⁻³¹kg). The maximum force exerted on an electron is 2.4 x 10⁻¹⁸N. What is the maximum velocity of the electrons?

$F=\frac{m{v}^2}{r}$

$v^2=\frac{Fr}{m}=\frac{2.4\times 10^{-18}\times 0.75}{9.11\times 10^{-31}}=1.97\times 10^{12}{\text{ m}}^2{\text{s}}^{-2}$

$v=1.41\times 10^6{\text{ ms}}^{-1}$

3. What are the problems involved in constructing a large cyclotron?

Creating a uniform magnetic field over such a large area (for example, the land area enclosed by the LHC) means that cyclotrons have been largely replaced in experimental particle physics by tubes, instead of flat cylinders.

4. Why don't particles stick to the electrodes when passing through them?

There are two reasons. Firstly, alternating current is sinusoidal, so there is no charge on the electrode when the particles are passing through them. Secondly, even if there were charge on the electrode, the net force on the particle would still be 0, since the force would be equal in every direction.