Two short magnetic dipoles $$m_1$$ and $$m_2$$ each having magnetic moment of 1 A m$$^2$$ are placed at point O and P respectively. The distance between OP is 1 m. The torque experienced by the magnetic dipole $$m_2$$ due to the presence of $$m_1$$ is _________ $$\times 10^{-7}$$ N m

We need to determine the torque experienced by the magnetic dipole $$m_2$$ due to the magnetic field produced by the magnetic dipole $$m_1$$.

1. Identify the Given Parameters

From the problem parameters , we have two short magnetic dipoles with the following values:

• Magnetic moment of the first dipole ($$m_1$$) = $$1\text{ A m}^2$$
• Magnetic moment of the second dipole ($$m_2$$) = $$1\text{ A m}^2$$
• Distance between the points O and P ($$r$$) = $$1\text{ m}$$

Based on standard dipole configurations for calculating torque, the dipole $$m_1$$ at point O creates an axial magnetic field at the position of dipole $$m_2$$ at point P, where $$m_2$$ is oriented perpendicular to this field.

2. Calculate the Magnetic Field ($$B_1$$) Due to $$m_1$$

The magnetic field produced by a short magnetic dipole at a point on its axial line at distance $$r$$ is given by the formula:

$$B_1 = \frac{\mu_0}{4\pi} \frac{2m_1}{r^3}$$

Substituting the standard value of $$\frac{\mu_0}{4\pi} = 10^{-7}\text{ T m A}^{-1}$$ and the given problem constants:

$$B_1 = 10^{-7} \times \frac{2 \times 1}{(1)^3} = 2 \times 10^{-7}\text{ T}$$

3. Determine the Torque experienced by $$m_2$$

The torque ($$\tau$$) acting on a magnetic dipole placed in a uniform external magnetic field is given by the vector cross product formula:

$$\tau = m_2 \cdot B_1 \cdot \sin(\theta)$$

Since the dipole $$m_2$$ is oriented perpendicular ($$\theta = 90^\circ$$) to the axial magnetic field line produced by $$m_1$$, the value of $$\sin(90^\circ) = 1$$. Substituting these values gives:

$$\tau = 1 \times (2 \times 10^{-7}) \times 1 = 2 \times 10^{-7}\text{ N m}$$

4. Match with the Requested Format

The problem asks for the torque in the expression format of $$\text{Value} \times 10^{-7}\text{ N m}$$:

$$\tau = 2 \times 10^{-7}\text{ N m} \implies \text{Value} = 2$$

Conclusion

The value of the torque experienced by the magnetic dipole is 2.