Question 24.4: A cylindrical capacitor Two long, coaxial cylindrical conduc......

IDENTIFY and SET UP As in Example 24.3, we use the definition of capacitance, C=Q / V_{a b}. We use the result of Example 23.10 (Section 23.3) to find the potential difference V_{a b} between the cylinders, and find the charge Q on a length L of the cylinders from the linear charge density. We then find the corresponding capacitance C from Eq. (24.1).

C={\frac{Q}{V_{ab}}}      (24.1)

Our target variable is this capacitance divided by L.

EXECUTE As in Example 24.3, the potential V between the cylinders is not affected by the presence of the charged outer cylinder. Hence our result in Example 23.10 for the potential outside a charged conducting cylinder also holds in this example for potential in the space between the cylinders:

V=\frac{\lambda}{2 \pi \epsilon_{0}} \ln \frac{r_{0}}{r}

Here r_{0} is the arbitrary, finite radius at which V=0. We take r_{0}=r_{b}, the radius of the inner surface of the outer cylinder. Then the potential at the outer surface of the inner cylinder (at which r=r_{a} ) is just the potential V_{a b} of the inner (positive) cylinder a with respect to the outer (negative) cylinder b :

V_{a b}=\frac{\lambda}{2 \pi \epsilon_{0}} \ln \frac{r_{b}}{r_{a}}

If \lambda is positive as in Fig. 24.6, then V_{a b} is positive as well: The inner cylinder is at higher potential than the outer.

The total charge Q in a length L is Q=\lambda L, so from Eq. (24.1) the capacitance C of a length L is

C=\frac{Q}{V_{a b}}=\frac{\lambda L}{\frac{\lambda}{2 \pi \epsilon_{0}} \ln \frac{r_{b}}{r_{a}}}=\frac{2 \pi \epsilon_{0} L}{\ln \left(r_{b} / r_{a}\right)}

The capacitance per unit length is

\frac{C}{L}=\frac{2 \pi \epsilon_{0}}{\ln \left(r_{b} / r_{a}\right)}

Substituting \epsilon_{0}=8.85 \times 10^{-12} \mathrm{~F} / \mathrm{m}=8.85 \mathrm{pF} / \mathrm{m}, we get

\frac{C}{L}=\frac{55.6 \mathrm{pF} / \mathrm{m}}{\ln \left(r_{b} / r_{a}\right)}

EVALUATE The capacitance of coaxial cylinders is determined entirely by their dimensions, just as for parallel-plate and spherical capacitors. Ordinary coaxial cables are made like this but with an insulating material instead of vacuum between the conductors. A typical cable used for connecting a television to a cable TV feed has a capacitance per unit length of 69 \mathrm{pF} / \mathrm{m}.

KEYCONCEPT The capacitance per unit length of two long, coaxial, conducting cylinders depends on the ratio of the inner radius of the outer conductor to the outer radius of the inner conductor.