Question

I need a little clearing in the potential difference in this question...

A small sphere of radius r1 and charge q1 is enclosed by a spherical shell of radius r2 and charge q2. Show that if q1 is positive, charge will necessarily flow from the sphere to the shell (when the two are connected by a wire) no matter what the charge q2 on the shell is.

Answer 1

If we apply the law of Gauss, we can say that the electric field between the sphere and the shell is:
\[{E_1} = \frac{{{q_1}}}{{{r^2}}},\quad a \leqslant r \leqslant b\]
next I apply the equation:
\[{\mathbf{E}} = - {\text{grad}}\varphi \]
wherein \(\varphi\) is the potential function, and I get:
\[{\varphi _1} = \frac{{{q_1}}}{r} + {c_1},\quad {c_1} \in \mathbb{R}\]
wherein \(c_1\) is a real constant
Now, following the same procedure in the space outside the shell of radius \(b\), and I get:
\[{E_2} = \frac{{{q_1} + {q_2}}}{{{r^2}}},\quad b \leqslant r < \infty \]
\[{\varphi _2} = \frac{{{q_1} + {q_2}}}{r} + {c_2},\quad {c_2} \in \mathbb{R}\]
here is the situation: