3. A 1.00 microampere beam of protons is accelerated through a difference of pot

Question

3. A 1.00 microampere beam of protons is accelerated through a difference of potential of 10,000 volts. Calculate the volume charge density in the beam, once the protons have been accelerated, assuming that the current density is uniform within a diameter of 2.00 millimeters, and zero outside this diameter. (The current density in an ion beam is in fact highest in the center and then gradually falls off with increasing radius. Hint: the charge per unit length is equal to the current divided by the velocity of the protons.)

Explanation / Answer

A proton with kinetic energy K of 10000 eV = 1.602 * 10-15 J has a velocity of (2K/m)1/2 = 1.384 * 106 m/s.

Since 1 A is 1 C/s, the charge per meter is (1 * 10-6 C/s)/(1.384 * 106 m/s) = 7.225 * 10-13 C/m.

The volume of a cylinder 1 meter long with a diameter of 2.00 mm is (1 m)()(1.00 * 10-3m)2 = 3.1416 * 10-6 m3.

So the volume charge density is (7.225 * 10-13 C)/(3.1416 * 10-6 m3) = 2.30 * 10-7 C/m3

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