Let's say we have a charged sphere with radius $r$. Usually, the way we define its potential is $V = kQ/r$. In this way, we have defined its potential with infinity as the zero reference point. Let's say I want to change the reference point of 0, so now that reference point is at a distance r(a) from its center. How would we write the equation for V?
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asked Jun 20, 2019 at 2:21
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1$\begingroup$ All you can do to the expression for $V$ is add or subtract a constant. You can figure out what that constant needs to be to make the potential be zero at whatever you radius you want. $\endgroup$– G. SmithCommented Jun 20, 2019 at 2:28
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$\begingroup$ The "electric potential [of] infinity is not [at] the $0$ reference" - This is impossible. No matter what you do, the infinite potential would remain at the origin. $\endgroup$– safesphereCommented Jun 20, 2019 at 5:02
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1 Answer
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Changing reference points amounts to a shifting the potential by a constant. If you want
$$V(r) = \frac{kQ}{r} + C$$ to be equal to zero at $r=r_a$, then
$$V(r_a) = \frac{kQ}{r_a} + C = 0 \implies C = -\frac{kQ}{r_a}$$ and so
$$V(r) = \frac{kQ}{r} - \frac{kQ}{r_a} = kQ \left(\frac{1}{r}-\frac{1}{r_a}\right)$$
answered Jun 20, 2019 at 2:33