Solution Found!
CALC Coaxial Cylinders. A long metal cylinder with radius
Chapter 23, Problem 63P(choose chapter or problem)
CALC Coaxial Cylinders. A long metal cylinder with 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 \(\lambda\), 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) \(r>b\). (Hint: The net potential is the sum of the potentials due to the individual conductors.) Take V = 0 at r = b.
(b) Show that the potential of the inner cylinder with respect to the outer is
\(V_{a b}=\frac{\lambda}{2 \pi \epsilon_0} \ln \frac{b}{a}\)
(c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the cylinders has magnitude
\(E(r)=\frac{V_{a b}}{\ln (b / a)} \frac{1}{r}\)
(d) What is the potential difference between the two cylinders if the outer cylinder has no net charge?
Questions & Answers
QUESTION:
CALC Coaxial Cylinders. A long metal cylinder with 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 \(\lambda\), 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) \(r>b\). (Hint: The net potential is the sum of the potentials due to the individual conductors.) Take V = 0 at r = b.
(b) Show that the potential of the inner cylinder with respect to the outer is
\(V_{a b}=\frac{\lambda}{2 \pi \epsilon_0} \ln \frac{b}{a}\)
(c) Use Eq. (23.23) and the result from part (a) to show that the electric field at any point between the cylinders has magnitude
\(E(r)=\frac{V_{a b}}{\ln (b / a)} \frac{1}{r}\)
(d) What is the potential difference between the two cylinders if the outer cylinder has no net charge?
ANSWER:Solution 63P Step 1 of 6: Given Coaxial cylinders, with the inner cylinder of radius a has the charge per unit length and the outer cylinder has radius b with charge per unit length as as shown in the figure below,