Figure 9 shows a cyclotron - AQA - A-Level Physics - Question 5 - 2021 - Paper 2

To find the number of times the proton moves across the gap, we can use the relationship between the kinetic energy and the potential difference:

The kinetic energy (Ek) gained by the proton is given by:

$E_k = qV$

Where q is the charge of the proton (1.6 x 10^-19 C) and V is the potential difference (10.0 kV = 10000 V). Thus, we find:

$14 MeV = qV = 1.6 \times 10^{-19} \times 10000$

Calculating gives:

$\frac{14000000}{10000} = 1400$

Therefore, the proton moves across the gap approximately 1400 times.

To show that the energy is given by the formula:

Starting with the expression for the centripetal force due to the magnetic field, we have:

$F = qvB$

Setting this equal to the centripetal force:

$qvB = \frac{mv^2}{R}$

From this, we can isolate v:

$v = \frac{qBR}{m}$

Now substituting this expression for v into the kinetic energy equation:

$E_k = \frac{1}{2}mv^2 = \frac{1}{2}m\left(\frac{qBR}{m}\right)^2$

We can simplify this to:

$E_k = \frac{q^2B^2R^2}{2m}$

Identifying q as the charge of the proton (e) and m as the mass of the proton (mp), we arrive at the required formula:

$E_k = \frac{eB^2R^2}{2m_p}$

We need to find the cyclotron that provides at least 11 MeV of energy.

• For cyclotron X:

  • Minimum kinetic energy: 11 MeV / Cost: £2.3 million

• For cyclotron Y:

  • Minimum kinetic energy: greater than 11 MeV

• For cyclotron Z:

  • Minimum kinetic energy: less than 11 MeV

Thus, cyclotron X and Y are potential candidates, but since Y has a larger capacity, it is more suitable. Now we evaluate the costs:

Since the cost increases with energy requirement:

$\text{Cost} \propto E_k^{1.5}$

Calculating for both:

For Y: Potential costs based on its energy yield will be higher than for cyclotron X. Therefore, cyclotron X will satisfy the energy requirement at lowest cost.

To determine the approximate cost of the chosen cyclotron, we apply the scaling relation:

Given the cost of a 10 MeV cyclotron is £2.3 million, we calculate:

$\text{Cost} = 2.3 \times (\frac{11}{10})^{1.5}$

Calculating this:

• First calculate $\left(\frac{11}{10}\right)^{1.5} \approx 1.1447$
• Then multiply it by £2.3 million:
  • Cost approximately: £2.3 million * 1.1447 = £2.63 million

Hence, the approximate cost of this cyclotron providing 11 MeV would be around £2.63 million.