Question

Two circular coils made of similar wires but of radii $20$ and $40$ $cm$ are connected in parallel. Find the ratio of the magnetic fields at their centers.

Answer 1

As the coils are connected in parallel so voltage across them should be same. i.e,
$i_1{R}_1=i_2{R}_2\Rightarrow \frac{i_1}{i_2}=\frac{R_2}{R_1}=\frac{\left({\rho }_2{l}_2\right)/A_2}{\left({\rho }_1{l}_1\right)/A_1}$
As they are made from same wire so cross section (A) and resistivity are same for both coil.i.e, $A_1=A_2$ and ${\rho }_1={\rho }_2$.
thus, $\frac{i_1}{i_2}=\frac{l_2}{l_1}=\frac{2\pi r_2}{2\pi r_1}=\frac{r_2}{r_1}$
The magnetic field at center of a coil of radius $r_1$ is $B_1=\frac{{\mu }_0{i}_1}{2r_1}$
The magnetic field at center of a coil of radius $r_2$ is $B_2=\frac{{\mu }_0{i}_2}{2r_2}$
$\frac{B_1}{B_2}=\frac{i_1{r}_2}{i_2{r}_1}=(\frac{r_2}{r_1}{)}^2=(40/20{)}^2=4$ as $i_1/i_2=r_2/r_1$