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A total nuclear charge Ze is distributed non-uniformly inside a nucleus of radius R, with a radial charge density rho(r) that varies linearly with r (rho(r) = a + b*r for r <= R, fixed so that the total enclosed charge is Ze). Which statement is correct?

1. The electric field at r = R is independent of b.
2. The electric potential at r = R is proportional to b.
3. The electric field at r = R depends on the way charge is distributed inside.
4. The electric potential at r = R is independent of the total charge.

Correct answer: The electric field at r = R is independent of b.

Solution

By Gauss's law the flux through the sphere r = R depends only on the total enclosed charge Ze, which is fixed. Hence E(R) = (1/4*pi*epsilon0)*Ze/R² regardless of the values of a and b individually (it is independent of b). The potential at r = R for a spherically symmetric charge equals (1/4*pi*epsilon0)*Ze/R, again set by the total charge.

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