Question

A copper wire having a resistance of $0.01$ ohm in each metre is used to wind a $400$ turn solenoid of radius $1.0cm$ and length $20cm$. Find the emf of a battery which when connected across the solenoid will cause a magnetic field of $1.0\times {10}^{-2}$ T near the centre of the solenoid.

Answer 1

As shown in the figure,radius of each coil=$1cm$

Number of turn=$400$

Total length of winding =N(perimeter of one coil)

$=400\left(\pi \right(1\times {10}^{-2}{)}^2)$

$=400\pi \times {10}^{-4}=4\pi \times {10}^{-2}m$

Resistance of winding =$1\times$length of winding coils

=$1\times (4\pi \times {10}^{-2})ohm$

External battery cause a magnetic field .

Let us consider,current flowing in coil is $i$.

Magnetic field due to current at center O.

$B=N\frac{{\mu }_{0}l}{2r}$

$1.0\times {10}^{-}2=N\frac{{\mu }_{0}l}{2r}$

$I=\frac{2\times 1\times {10}^{-2}\times {10}^{-2}}{4\pi \times {10}^{-7}\times N}=\frac{2\times {10}^3}{4\pi N}$

$I=\frac{159.23}{N}A$

$I=0.4A$

Now circuit is shown in fig.3,

$V=iR$

$=0.4\times 4\pi \times {10}^{-2}$

$=5\times {10}^{-2}V$

$V=50mV$