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FranzDiCoccio
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Homework Statement
Two conducting concentric spheres of negligible thickness. The radii of the spheres are R_1 and R_2, respectively, with R_2>R_1. A charge q_2 is placed on the external sphere.
A charge q_1 is placed on the internal sphere.
Assume that the electric potential is zero infinitely far from the center of the spheres.
Find q_1 such that the potential on the inner sphere is zero.
Homework Equations
- Gauss law: \Phi(\vec{E}) = \frac{Q}{\varepsilon_0}
- Potential of a point charge (such that V=0 for d=\infty): \displaystyle V= k \frac{Q}{d}.
The Attempt at a Solution
Using Gauss' law I can say that, if only the outer sphere was present,
V_2 = k \frac{q_2}{R_2}
on its surface and inside it. Hence, at radius R_1,
V_1 = k \frac{q_2}{R_2}
If only the inner sphere was present
V_1 = k \frac{q_1}{R_1}
on its surface
When both spheres are present, on the surface of the inner sphere
V_1 = k \frac{q_2}{R_2}+k \frac{q_1}{R_1}
If we want V_1=0, it should be
\frac{q_2}{R_2}=- \frac{q_1}{R_1}
and hence
q_1=- \frac{R_1}{R_2} q_2
It seems to me that this makes sense. The potential would be
V_2 = \frac{k} q_2 \left\{<br /> \begin{array}{cc}<br /> \frac{1}{r}\left(1-\frac{R_1}{R_2}\right) & r\geq R_2 \\<br /> \frac{1}{R_2}-\frac{1}{r}\frac{R_1}{R_2} & R_1 \leq r \leq R_2<br /> \end{array}<br /> \right.<br />
Assuming that q_2>0, the potential grows when the outer sphere is approached from outside, reaches its maximum on the surface of the spheres and drops to 0 when it reaches the inner radius...
Inside that, it stays zero.I'm asking here just to be on the safe side.
I came across a (kind of "official") solution for this exercise that in my opinion does not make any sense. It concludes q_1=0.
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