A Van de Graaff generator is one of the original particle accelerators and can be used to accelerate charged particles like protons or electrons. You may have seen it used to make human hair stand on end or produce large sparks. One application of the Van de Graaff generator is to create x-rays by bombarding a hard metal target with the beam. Consider a beam of protons at 2.00 keV and a current of 5.05 mA produced by the generator. (a) What is the speed of the protons (in m/s) ? b) How many protons are produced each second?

v = 1.95*10^5 m/s

3.13 x 10^16 proton

Explanation:

Identify the unknown:

The speed of the protons

List the Knowns:

Energy of protons KE = 2 keV = 2*10^3 eV

Current produced by the generator: I = 5.05 mA = 5 x 10^-3 A

1 eV = 1.6 x 10-19 Joule

Mass of proton: m = 1.67 x 10^-27 kg

Charge of proton: q_p = 1.6 x 10^-19 C

Set Up the Problem:

Kinetic energy is given by:

KE = 1/2mv^2

v = √2KE/m

Solve the Problem:

v = √2 x 2*10^3 x 1.6 x 10^-19 / 1.67 x 10^-27

v = 1.95*10^5 m/s

b. Identify the unknown:

Number of protons produced each second

Set Up the Problem:

Current is given by:

I = ΔQ/Δt

So, the total charge in one second:

ΔQ = I*Δt = 5 x 10^-3 x 1 = 5 x 10^-3 C

Number of protons in this charge:

n = ΔQ/q_p

Solve the Problem:

n = 5 x 10^-3/1.6 x 10^-19

=3.13 x 10^16 proton