Question

Two circular coils made of similar wires but of radius $20\text{cm}$ and $40\text{cm}$ are connected in parallel. The ratio of magnetic fields at their centre is -

• A. $4:1$
• B. $1:4$
• C. $2:1$
• D. $1:2$

Answer 1

The correct option is A $4:1$

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 i..e,

$A_1=A_2$ and ${\rho }_1={\rho }_2$.

Thus, $\frac{i_1}{i_2}=\frac{l_2}{l_1}=\frac{2\pi r_2}{2\pi r_1}=\frac{r_2}{r_1}$

Hence, magnetic field at centre of a coil having radii $r_1$ and $r_2$ is,

$B_1=\frac{{\mu }_0{i}_1}{2r_1}$

$B_2=\frac{{\mu }_0{i}_2}{2r_2}$

$\therefore \frac{B_1}{B_2}=\frac{i_1{r}_2}{i_2{r}_1}\{\text{as}\frac{i_1}{i_2}=\frac{r_2}{r_1}\}$

$\frac{B_1}{B_2}={\left(\frac{r_2}{r_1}\right)}^2={\left(\frac{40}{20}\right)}^2=4:1$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(A\right)$ is the correct answer.