Question

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

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

Answer 1

The correct option is A $\frac{{\mu }_0}{4\pi l}.\frac{R_m}{\rho D}$

For a solenoid $L={\mu }_0{N}^2\frac{A}{l}2yx$ is the length of the wire and a is the of cross-section, then
$R=\frac{px}{a}$
and $m=axD$
$R_m=\rho \frac{x}{a}=axD$
$x=\sqrt{\frac{R_m}{\rho D}}$
$x=2\pi rN,N=\frac{x}{2\pi r}$
$L={\mu }_0{\left(\frac{x^2}{2\pi r}\right)}^2\frac{\pi r^2}{l}$
$=\frac{{\mu }_0}{4\pi l}.\frac{R_m}{\rho D}$