23.41 •• CALC A metal sphere with radius r, is supported on an insu- lating stand at the center of a hollow, metal, spherical shell with radius r,. There is charge +qon the inner sphere and charge –q on the outer spher- ical shell. (a) Calculate the potential V(r) for (i) r < ra; (ii) ra < r < rb; (iii) r> r. (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take V to be zero when r is infinite. (b) Show that the potential of the inner sphere with respect to the outer is 9_(1 1) Vab 4T€0 \ra (c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the spheres has magnitude Vab E(r) %3D (1/ra – 1/rô) r² - (d) Use Eq. (23.23) and the result from part (a) to find the electric field at a point outside the larger sphere at a distancer from the center, where r> rp. (e) Suppose the charge on the outer sphere is not -ų but a nega- tive charge of different magnitude, say -Q. Show that the answers for parts (b) and (c) are the same as before but the answer for part (d) is different.
23.41 •• CALC A metal sphere with radius r, is supported on an insu- lating stand at the center of a hollow, metal, spherical shell with radius r,. There is charge +qon the inner sphere and charge –q on the outer spher- ical shell. (a) Calculate the potential V(r) for (i) r < ra; (ii) ra < r < rb; (iii) r> r. (Hint: The net potential is the sum of the potentials due to the individual spheres.) Take V to be zero when r is infinite. (b) Show that the potential of the inner sphere with respect to the outer is 9_(1 1) Vab 4T€0 \ra (c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the spheres has magnitude Vab E(r) %3D (1/ra – 1/rô) r² - (d) Use Eq. (23.23) and the result from part (a) to find the electric field at a point outside the larger sphere at a distancer from the center, where r> rp. (e) Suppose the charge on the outer sphere is not -ų but a nega- tive charge of different magnitude, say -Q. Show that the answers for parts (b) and (c) are the same as before but the answer for part (d) is different.
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Transcribed Image Text:23.41 • CALC A metal sphere with radius r, is supported on an insu-
lating stand at the center of a hollow, metal, spherical shell with radius r,.
There is charge +qon the inner sphere and charge -q on the outer spher-
ical shell. (a) Calculate the potential V(r) for (i) r < ra; (ii) ra < r < r;;
(iii) r> r. (Hint: The net potential is the sum of the potentials due to
the individual spheres.) Take V to be zero when r is infinite. (b) Show that
the potential of the inner sphere with respect to the outer is
Vab
4TE, \ra
(c) Use Eq. (23.23) and the result from part (a) to show that the electric
field at any point between the spheres has magnitude
Vab
1
E(r) -
(1/ra – 1/r,) r²
(d) Use Eq. (23.23) and the result from part (a) to find the electric field
at a point outside the larger sphere at a distance r from the center, where
r> Tp. (e) Suppose the charge on the outer sphere is not
tive charge of different magnitude, say -Q. Show that the answers for
parts (b) and (c) are the same as before but the answer for part (d) is
different.
–y but a nega-
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