Question

Find the inductance $\ell$ of a solenoid of length whose winding are made of material of density D and resistivity $\rho$.The winding resistance is R:

• A. $\frac{{\mu }_{0}Rm}{4\pi \ell \rho D}$
• B. $\frac{{\mu }_0\ell m}{4\pi R\rho D}$
• C. $\frac{{\mu }_0}{4\pi \ell }\frac{R^{2}m}{\rho D}$
• D. $\frac{{\mu }_0}{2\pi R}\frac{\ell m}{\rho D}$

Answer 1

The correct option is A

$\frac{{\mu }_{0}Rm}{4\pi \ell \rho D}$

In a solenoid $L=\frac{{\mu }_0{N}^{2}A}{\ell }$ if x is the length of the wire and ‘a’ is cross section area

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

$Rm=\frac{\rho xaxD}{a}$

$x=\sqrt{\frac{Rm}{\rho D}}$

Also $x=2\pi rN,N=\frac{x}{2\pi r}putthis\left[\frac{L={\mu }_0{N}_{\ell }^{2}A}{\ell }\right]$

We get

$L={\mu }_0[\frac{x}{2\pi r}{]}^2\frac{\pi r^2}{\tau }=\frac{{\mu }_0}{4\pi \ell }\frac{Rm}{\rho D}$