Question

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

Answer 1

1

Answer 2

The torque on a magnetic dipole $\vec{m}$ in an external magnetic field $\vec{B}$ is given by $\vec{\tau} = \vec{m} \times \vec{B}$. The magnetic field $\vec{B}$ at a point P, at a distance $r$ from a short magnetic dipole $\vec{m_1}$ at O, is given by: $\vec{B} = \frac{\mu_0}{4\pi} \left( \frac{3(\vec{m_1} \cdot \vec{r})\vec{r}}{r^5} - \frac{\vec{m_1}}{r^3} \right)$ where $\vec{r}$ is the position vector from O to P.

Assuming $\vec{m_1}$ is along the y-axis and P is on the x-axis, $\vec{m_1} = m_1 \hat{j}$ and $\vec{r} = r \hat{i}$. The dot product $\vec{m_1} \cdot \vec{r} = 0$. Thus, the magnetic field at P is: $\vec{B} = -\frac{\mu_0 m_1}{4\pi r^3} \hat{j}$ If $\vec{m_2}$ is along the x-axis, $\vec{m_2} = m_2 \hat{i}$. The torque on $m_2$ is: $\vec{\tau_2} = \vec{m_2} \times \vec{B} = (m_2 \hat{i}) \times \left(-\frac{\mu_0 m_1}{4\pi r^3} \hat{j}\right) = -\frac{\mu_0 m_1 m_2}{4\pi r^3} (\hat{i} \times \hat{j}) = -\frac{\mu_0 m_1 m_2}{4\pi r^3} \hat{k}$ The magnitude of the torque is $|\vec{\tau_2}| = \frac{\mu_0 m_1 m_2}{4\pi r^3}$.

Given $m_1 = 1$ A $m^2$, $m_2 = 1$ A $m^2$, $r = 1$ m, and $\mu_0 = 4\pi \times 10^{-7}$ T m/A. $|\vec{\tau_2}| = \frac{(4\pi \times 10^{-7}) \times 1 \times 1}{4\pi \times 1^3} = 10^{-7} \text{ N m}$ The value to fill in the blank is 1.