Question

A beam of protons is moving toward a target in a particle accelerator. this beam constitutes a current whose value is 0.50 μa. (a) how many protons strike the target in 15 s? (b) each proton has a kinetic energy of 4.9 x 10-12 j. suppose the target is a 15 gram block of aluminum, and all the kinetic energy of the protons goes into heating it up. what is the change in temperature of the block that results from 15 s bombardment of protons?

Answer 1

(a) The current is defined as the amount of charge that strikes the target per unit time:

`I= (Q)/(t)`
where Q is the total charge and t the time.
We know the current,
`I=0.50 \mu A=0.5 \cdot 10^(-6)A`, and the time,
`t=15 s`, so we can calculate the amount of charge that strikes the target during this time interval:

`Q=It=(0.5 \cdot 10^(-6) A)(15 s)=7.5 \cdot 10^(-6)C`

We know that each proton has a charge of
`q=1.6 \cdot 10^(-19)C`, so if we divide the total charge Q by the charge of one proton q, we find the number of protons that strike the target:

`N= (Q)/(q)= (7.5 \cdot 10^(-6) C)/(1.6 \cdot 10^(-19)C)=4.7 \cdot 10^(13)`

(b) The total energy given by the beam of protons to the block of aluminium is equal to the kinetic energy of one proton times the number of protons:

`E=NK=(4.7 \cdot 10^(13))(4.9 \cdot 10^(-12)J)=230.3 J`

This energy is given as heat to the block of aluminium, and the increase in temperature of the block is related to this energy by

`E=mC_s \Delta T`
where m=15 g=0.015 kg is the mass of the block,
`C_s=900 J/kg^(\circ)C` is the specific heat capacity of aluminium, and
`\Delta T` is the increase in temperature that we want to find.

If we rearrange the equation, we find
`\Delta T`:

`\Delta T= (E)/(mC_s)= (230.3 J)/((900 J/kg^(\circ))(0.015 kg)) =17.0^(\circ)C`