CBSE 2025, Set 3

Question

(a) (i) Write Biot-Savart's law in vector form.
(ii) Two identical circular coils A and B, each of radius R, carrying currents $I$ and $\sqrt{3}I$ respectively, are placed concentrically in XY and YZ planes respectively. Find the magnitude and direction of the net magnetic field at their common centre.

Edvaya · Accepted Answer

The correct answer is Subjective. **(i) Biot-Savart's Law in Vector Form:**
The magnetic field $d\vec{B}$ due to a current element $I d\vec{l}$ at a position vector $\vec{r}$ is given by:
$$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I(d\vec{l} 	imes \vec{r})}{r^3}$$

**(ii) Net Magnetic Field at Centre:**
* **Coil A** is in the XY plane. By the right-hand rule, its magnetic field $\vec{B_1}$ at the centre is along the Z-axis.
 $$B_1 = \frac{\mu_0 I}{2R}$$
* **Coil B** is in the YZ plane. Its magnetic field $\vec{B_2}$ at the centre is along the X-axis.
 $$B_2 = \frac{\mu_0 (\sqrt{3}I)}{2R}$$

![Magnetic Field Vectors](https://www.edvaya.com/images/cbse/physics/2025/set3/q25a-magnetic-field-vectors-answer.webp)

Since $\vec{B_1}$ and $\vec{B_2}$ are perpendicular (along Z and X axes respectively), the magnitude of the net magnetic field $B$ is:
$$B = \sqrt{B_1^2 + B_2^2}$$
$$B = \sqrt{ \left(\frac{\mu_0 I}{2R}ight)^2 + \left(\frac{\mu_0 \sqrt{3}I}{2R}ight)^2 }$$
$$B = \frac{\mu_0 I}{2R} \sqrt{1 + 3}$$
$$B = \frac{\mu_0 I}{2R} (2) = \boxed{\frac{\mu_0 I}{R}}$$

**Direction:**
Let the resultant magnetic field make an angle $	heta$ with $\vec{B_2}$ (the X-axis):
$$	an 	heta = \frac{B_1}{B_2} = \frac{(\mu_0 I / 2R)}{(\sqrt{3} \mu_0 I / 2R)}$$
$$	an 	heta = \frac{1}{\sqrt{3}}$$
$$	heta = 30^\circ$$
The direction of the net magnetic field is $30^\circ$ with the direction of $\vec{B_2}$ (or $60^\circ$ with the direction of $\vec{B_1}$).

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