Question

A solenoid of inductance $L$ and length $l$ and mass $m$, whose windings are made of material of density $D$ and resistivity is $\rho$,(the winding resistance is $R$). Find out the inductance in terms of others.

• A. $\frac{{\mu }_0}{4\pi l}\frac{Rm}{\rho D}$
• B. $\frac{{\mu }_0}{4\pi R}\frac{lm}{\rho D}$
• C. $\frac{{\mu }_0}{4\pi l}\frac{R^{2}m}{\rho D}$
• D. $\frac{{\mu }_0}{2\pi R}\frac{lm}{\rho D}$

Answer 1

Answer: (A)

$\frac{{\mu }_0}{4\pi l}\frac{Rm}{\rho D}$

For a solenoid, $L={\mu }_0{N}^2\frac{A}{l}$. If $x$ is the length of the wire and $a$ is the area of cross-section, then

Resistance $R=\frac{\rho x}{a}$ and mass $m=axD$

$Rm=\frac{\rho x}{a}axD,x=\sqrt{\frac{Rm}{\rho D}}$

Also, $x=2\pi rN,N=\frac{x}{2\pi r}(\therefore L=\frac{{\mu }_0{N}^{2}A}{l})$

$\therefore L={\mu }_0{\left(\frac{x}{2\pi r}\right)}^2\frac{\pi r^2}{l}=\frac{{\mu }_0}{4\pi l}\frac{Rm}{\rho D}$