Answer exactly one question from this section.

Do not answer more than the required number of questions in this section. Extra answers will not be marked.

1.

X rays can be generated by firing high-energy electrons into a metal target. Only a small fraction of the kinetic energy of the electrons is converted to X rays however. Most of the energy is transformed into heat.

(a)

Suppose that electrons, each with a kinetic energy of $2.5\times 10^{-14}\,\mathrm{J}$, are fired into a copper anode of volume $85\,\mathrm{cm}^3$ (volume measured at $25^\circ\mathrm{C}$) at a rate of $2.5\times 10^{17}\,\mathrm{electrons/s}$. The anode is hollow¹ and contains $30\,\mathrm{cm}^3$ of water (measured at $25^\circ\mathrm{C}$). The anode and water are in good thermal contact. If all the kinetic energy were transformed to heat and assuming that none of the heat is lost to the surroundings, how long would it take for the anode's temperature to increase to $100^\circ\mathrm{C}$ from 25? The density of copper at $25^\circ\mathrm{C}$ is $7.11\,\mathrm{g/cm}^3$ and its specific heat capacity is $0.38\,\mathrm{J\,K^{-1}g^{-1}}$. [8 marks]

(b)

To avoid excessive heating of the anode, the water it contains is constantly replaced by cool water. Suppose that water with an initial temperature of $4^\circ\mathrm{C}$ flows through the anode at a rate of 6L/min. Assuming perfect mixing of the water in the anode and perfectly efficient heat transfer between the copper and water, what steady temperature would the anode reach? [7 marks]

Hint: How much heat must be removed to hold the temperature steady?

Note: The temperature you will calculate here is a lower bound since neither mixing nor heat transfer will be perfectly efficient in the real system.

2.

(a)

In deriving the equation for the energy of a particle in a box, we assumed that the kinetic energy could be treated classically. We can use the relativistic relationship between energy and momentum instead to obtain an equation for a relativistic particle in a box. Carry out this derivation. [8 marks]

Hint: Except for the energy-momentum relation, the derivation is in all respects identical to that for the ordinary particle in a box.  

(b)

Using the equation derived in question 2a, calculate the wavelength of a photon capable of inducing a transition from the ground state to the first excited state of an electron in a 3pm box. [7 marks]