Question

Coaxial cylinders: A long metal cylinder with a radius a is supported on an insulating stand on the axis of a long, hollow, metal tube with radius b.

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asked 2021-01-28

Coaxial cylinders:
A long metal cylinder with a radius a is supported on an insulating stand on the axis of a long, hollow, metal tube with radius b. The positive charge per unit length on the inner cylinder is ? , and there is an equal negative charge per unit length on the outer cylinder. (a) calculate the potential V(r) for:
i) \(r < a\)
ii) \(a < r < b\)
iii) \(r > b\)
Take \(V=0\ at\ r = b\)

Answers (1)

2021-01-29
a) (i) \(\displaystyle{V}={\frac{{\lambda}}{{{2}\pi\epsilon_{{0}}}}}{\left({\ln{{\left({\frac{{{b}}}{{{a}}}}\right)}}}-{\ln{{\left({\frac{{{b}}}{{{b}}}}\right)}}}\right)}={\frac{{\lambda}}{{{2}\pi\epsilon_{{0}}}}}{\ln{{\left({\frac{{{b}}}{{{a}}}}\right)}}}\)
(ii) \(\displaystyle{V}={\frac{{\lambda}}{{{2}\pi\epsilon_{{0}}}}}{\left({\ln{{\left({\frac{{{b}}}{{{r}}}}\right)}}}-{\ln{{\left({\frac{{{b}}}{{{b}}}}\right)}}}\right)}={\frac{{\lambda}}{{{2}\pi\epsilon_{{0}}}}}{\ln{{\left({\frac{{{b}}}{{{r}}}}\right)}}}\)
(iii) \(\displaystyle{V}={0}\)
b) \(\displaystyle{V}_{{{a}{b}}}={V}{\left({a}\right)}-{V}{\left({b}\right)}={\frac{{\lambda}}{{{2}\pi\epsilon_{{0}}}}}{\ln{{\left({\frac{{{b}}}{{{a}}}}\right)}}}\)
c) Between the cylinders:
\(\displaystyle{V}={\frac{{\lambda}}{{{2}\pi\epsilon_{{0}}}}}{\ln{{\left({\frac{{{b}}}{{{r}}}}\right)}}}={\frac{{{V}_{{{a}{b}}}}}{{{\ln{{\left({\frac{{{b}}}{{{a}}}}\right)}}}}}}{\ln{{\left({\frac{{{b}}}{{{r}}}}\right)}}}\)
\(\displaystyle\therefore{E}={\frac{{{d}{V}}}{{{d}{r}}}}=-{\frac{{{V}_{{{a}{b}}}}}{{{\ln{{\left({\frac{{{b}}}{{{a}}}}\right)}}}}}}{\frac{{{d}}}{{{d}{r}}}}{\left({\ln{{\left({\frac{{{b}}}{{{r}}}}\right)}}}\right)}={\frac{{{V}_{{{a}{b}}}}}{{{\ln{{\left({\frac{{{b}}}{{{a}}}}\right)}}}}}}{\frac{{{1}}}{{{r}}}}\)
d) The potential difference between the two cylinders is identical to that in part (b) even if the outer cylinder has no charge.
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