Question

Imagine two bar magnets having same magnetic dipole moment $M$, are inclined with each other as shown in figure. Their resultant magnetic moment will be

• A. $\dfrac{M}{2}$
• B. $M$
• C. $\sqrt{2}M$
• D. $\sqrt{3}M$

Hint:

Always check the actual orientation of magnetic dipoles, not just the angle between rods.

Answer 1

The Correct Option is B

Step 1: Identify the angle between magnetic moments.
From the figure, the angle between the two magnetic dipole moments is $60^\circ$.
Step 2: Write formula for resultant magnetic moment.
For two equal magnetic moments inclined at an angle $\theta$: \[ M_R = \sqrt{M^2 + M^2 + 2M^2\cos\theta} \] Step 3: Substitute given values.
\[ M_R = \sqrt{2M^2 + 2M^2 \cos 60^\circ} \] \[ \cos 60^\circ = \dfrac{1}{2} \] Step 4: Simplify.
\[ M_R = \sqrt{2M^2 + M^2} = \sqrt{3M^2} \] But from the orientation of poles in the figure, the effective angle between magnetic moments is $120^\circ$.
Step 5: Correct calculation using $120^\circ$.
\[ M_R = \sqrt{2M^2 + 2M^2\cos 120^\circ} \] \[ \cos 120^\circ = -\dfrac{1}{2} \] \[ M_R = \sqrt{2M^2 - M^2} = M \]