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Chapter 2: Q43P (page 107)

Find the capacitance per unit length of two coaxial metal cylindrical tubes, of radiaandb.

Short Answer

Expert verified

The capacitance per unit length isCI=2πε0Inba

Step by step solution

01

Determine the expression for the electric field.

Write the expression for electric filed,

E=q2πsIε0s^fora<s<b

Here, qis the charge, Eis the electric filed.

Consider the given figure.

Now considerIis the length of the tube.

Let’s give the chargelocalid="1654330497372" +qto the inner cylinder and-qto the outer cylinder.

02

Determine the expression for potential difference

Now finding the potential difference between inner and outer faces of cylinder and it is given by,

V(a)-V(b)=-∫ba(q2πsIε0)dsV(a)-V(b)=V

Solve further as,

V=-∫baq2πsIε0ds=-q2πε0Iloga-logb=q2πε0IInba

03

Determine Capacitance per unit length

CI=2πε0InbaConsider the expression for the potential.

Now comparing the above equation with localid="1654328482139" VqC, here Cis the capacitance. Then,

localid="1654328603077" CI=2πε0Inba

CI=2πε0Inba

Hence, the capacitance per unit length is given by,

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Most popular questions from this chapter

Find the net force that the southern hemisphere of a uniformly charged solid sphere exerts on the northern hemisphere. Express your answer in terms of the radiusR and the total charge Q.

Two large metal plates (each of area A) are held a small distance d

a part. Suppose we put a chargeQon each plate; what is the electrostatic pressure on the plates?

(a) Check that the results of Exs. 2.5 and 2.6, and Prob. 2.11, are consistent with Eq. 2.33.

(b) Use Gauss's law to find the field inside and outside a long hollow cylindrical

tube, which carries a uniform surface charge σ.Check that your result is consistent with Eq. 2.33.

(c) Check that the result of Ex. 2.8 is consistent with boundary conditions 2.34 and 2.36.

Find the energy stored in a uniformly charged solid sphere of radiusRand charge q.Do it three different ways:

(a)Use Eq. 2.43. You found the potential in Prob. 2.21.

(b)Use Eq. 2.45. Don't forget to integrate over all space.

(c)Use Eq. 2.44. Take a spherical volume of radiusa.What happens as a→∞?

Using Eqs. 2.27 and 2.30, find the potential at a distance zabove the

center of the charge distributions in Fig. 2.34. In each case, compute E⃗=-∇⃗V,and compare your answers with Ex. 2.1, Ex. 2.2, and Prob. 2.6, respectively. Suppose that we changed the right-hand charge in Fig. 2.34a to -q;what then is the potential at P?What field does that suggest? Compare your answer to Pro b. 2.2, and explain carefully any discrepancy.
See all solutions

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