You have been able to get a part-time job in a University laboratory. The group

Question

You have been able to get a part-time job in a University laboratory. The group is planning a set of experiments to study the forces between nuclei in order to understand the energy output of the Sun. To do this experiment, you shoot alpha particles from a Van de Graaf accelerator at a sheet of lead. The alpha particle is the nucleus of a helium atom and is made of 2 protons and 2 neutrons. The lead nucleus is made of 82 protons and 125 neutrons. The mass of the neutron is almost the same as the mass of a proton. To assure that you are actually studying the effects of the nuclear force, an alpha particle should come into contact with a lead nucleus. Assume that both the alpha particle and the lead nucleus have the shape of a sphere. The alpha particle has a radius of 1.0 x 10-15 m and the lead nucleus has a radius 4 times larger. Your boss wants you to make two calculations: (a) What is the minimum speed of such an alpha particle if the lead nucleus is fixed at rest? (b) What is the potential difference between the two ends of the Van de Graaf accelerator if the alpha particle starts from rest at one end (from a bottle of helium gas)?  

Explanation / Answer

a)

Initially, assume the electric potential energy between the alphaparticle and the 82Pb nucleus is zero. For the minimum approach speed, all initial kinetic energy will be converted into potential energy upon contact. Since the nuclei are assumed to be spherical, we can treat all the charge as being located at the center of each. Then contact will be made when the centers of the nuclei are 5ralpha apart. So the potential energy when the two make contact is

Charge of alpha particle Q1=2e

Charge of 82Pb Q2=82e

U=KQ1Q2/r=(1/2)m0vo2

(9*109)(2*1.6*10-19)(82*1.6*10-19)/5*(1*10-5)=(1/2)*(4*1.67*10-27)(vo2)

vo=4.757*107 m/s

b)

KE=q1V

(1/2)movo2=q1V

(1/2)(4*1.67*10-27)(4.756*107)2=(2*1.6*10-19)*V

V=2.36*107 Volts

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