Question 21: Magnetism and Matter

Question

Two identical circular loops $P$ and $Q$ each of radius $r$ are lying in parallel planes such that they have common axis. The current through $P$ and $Q$ are $I$ and $4 I$ respectively in clockwise direction as seen from $O$. The net magnetic field at $O$ is :

Accepted Answer

The magnetic field (B) on the axis of a circular loop of radius $r$ at a distance $x$ from its center is given by :
$$ \mathrm{B}=\frac{\mu_0 \mathrm{Ir}^2}{2\left(\mathrm{r}^2+\mathrm{x}^2\right)^{3 / 2}} $$
The point O is at a distance $\mathrm{x}=\mathrm{r}$ from the center of both loops P and Q .
For loop P :
Current is $\mathrm{i}=\mathrm{I}$.
The magnetic field $\mathrm{B}_{\mathrm{P}}$ at O due to loop P is :
$$ \mathrm{B}_{\mathrm{P}}=\frac{\mu_0 \mathrm{Ir}^2}{2\left(\mathrm{r}^2+\mathrm{r}^2\right)^{3 / 2}}=\frac{\mu_0 \mathrm{I}}{4 \sqrt{2} \mathrm{r}} $$
Using the right-hand thumb rule, the current is clockwise as seen from O , so the field $\mathrm{B}_{\mathrm{P}}$ due to loop P points towards P.
For loop Q :
Current is $\mathrm{i}=4 \mathrm{I}$.
The magnetic field $\mathrm{B}_{\mathrm{Q}}$ at O due to loop Q is:
$$ B_Q=\frac{\mu_0(4 I) r^2}{2\left(r^2+r^2\right)^{3 / 2}}=\frac{4 \mu_0 I}{4 \sqrt{2} r} $$
Using the right-hand thumb rule, the current is clockwise as seen from O , so the field $\mathrm{B}_{\mathrm{Q}}$ due to loop Q points towards Q.
The net magnetic field $\left(B_{n e t}\right)$ at $O$ is the vector sum of these opposing fields. Since $B_Q>B_P$, the net field will be in the direction of $\mathrm{B}_{\mathrm{Q}}$ (towards Q ):
$$ B_{n e t}=B_Q-B_P=\frac{4 \mu_0 I}{4 \sqrt{2} r}-\frac{\mu_0 I}{4 \sqrt{2} r}=\frac{3 \mu_0 I}{4 \sqrt{2} r} $$
Therefore, the resulting magnetic field at the midpoint is $\frac{3 \mu_0 I}{4 \sqrt{2} r}$ towards $Q$.
Hence, the correct option is (B).

Answer

$\frac{\mu_{\mathrm{o}} I}{4 \sqrt{2} r}$ towards $Q$

Answer

$\frac{3 \mu_{\mathrm{o}} I}{4 \sqrt{2} r}$ towards $Q$

Answer

$\frac{3 \mu_0 I}{4 \sqrt{2} r}$ towards $P$

Answer

$\frac{\mu_{\mathrm{o}} I}{4 \sqrt{2} r}$ towards $P$

Magnetism and Matter

Two identical circular loops P and Q each of radius r are lying in parallel planes such that they have common axis. The current through P and Q are I and 4 I respectively in clockwise direction as seen from O. The net magnetic field at O is :

Choose the correct answer: