Question

A solenoid of length $1m$, area of cross-section $4.0c{m}^2$ and having $4000$ turns is placed inside another solenoid of $2000$ turns having a cross-sectional area $6c{m}^2$ and length $2m$. The mutual inductance between the solenoids is $x\pi \times {10}^{-5}H$. Findout the value of $x$.

Answer 1

It is given that:

For the solenoid 1

$l_1=1m$

$A_1=4c{m}^2=4\times {10}^{-4}{m}^2$

$n_1=4000$

For the solenoid 2

$l_2=2m$

$A_2=6\times {10}^{-4}{m}^2$

$n_2=2000$

Mutual inductance between the solenoid is $M=x\pi \times {10}^{-5}H$

$x=?$

Let $i$ current is flowing through the outer solenoid 2.

So,$B=n_2{\mu }_{0}i$

Flux

$\varphi =n_{1}B{A}_1$

$\varphi =n_1{n}_2{\mu }_{0}i{a}_1$

$\because E=\frac{d\varphi }{dt}=n_1{n}_2{\mu }_0{a}_1\frac{di}{dt}$

$and,E=M\frac{di}{dt}$

$M=\frac{E}{\frac{di}{dt}}$

$M=n_1{n}_2{\mu }_0{a}_1$

$x\pi \times {10}^{-5}=4000\times 2000\times 4\pi \times {10}^{-7}\times 4\pi \times {10}^{-4}$

$x=128$