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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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 $$V=0\ at\ r = b$$

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.