Question

A cylindrical capacitor has radii a= 1 cm and b = 2.5 cm. If the space between the plates is filled with an lnhomogenous dielectric with ${ϵ}_r=\frac{(10+\rho )}{\rho }$ where $\rho$ is in centimeters, then the capacitance per meter of the capacitor is pF/m.

• A. 434.6

Answer 1

The correct option is A 434.6

$V=-{\int }_b^a\frac{Q}{2\pi {ϵ}_0{ϵ}_r\rho L}d\rho$

$=\frac{Q}{2\pi {ϵ}_0{ϵ}_r\rho L}{\int }_b^a\frac{d\rho }{\rho \left(\frac{10+\rho }{\rho }\right)}=\frac{Q}{2\pi {ϵ}_{0}L}d\rho {\int }_b^a\frac{d\rho }{10+\rho }$

$V=\frac{Q}{2\pi {ϵ}_{0}L}\ln (10+\rho ){|}_b^a=\frac{Q}{2\pi {ϵ}_{0}L}\ln \left(\frac{10+b}{10+a}\right)$

Thus the capacitance per meters is , (L= 1 m)

$C=\frac{Q}{V}=\frac{2\pi {ϵ}_0}{\ln \left(\frac{10+b}{10+a}\right)}=2\pi .\frac{{10}^{-9}}{36\pi }.\frac{1}{\ln \frac{12.5}{11}}$

$=434.6pF/m$