Animated Solution for Physics - Magnetic Effects of Current: Two short magnetic dipoles $m_1$ and $m_2$ each having magnetic moment of $1{\text{ Am}}^2$ are placed at point $O$ and $P$, respectively. The distance between $OP$ is $1\text{ m}$. The torque experienced by the magnetic dipole $m_2$ due to the presence of $m_1$ is ...... $\times 10^{-7}\text{ Nm}$.

The interaction between magnetic dipoles is one of the most elegant and fundamental concepts in electromagnetism. It forms the basis for understanding everything from the behavior of compass needles to the complex magnetic properties of materials. In this problem, we are tasked with finding the torque exerted by one magnetic dipole on another. Let's embark on this journey step-by-step.

Two Dipoles in Space

Imagine you are looking at a two-dimensional plane. We have two short magnetic dipoles, which you can think of as tiny, powerful bar magnets. The first dipole, ${\mathbf{m}}_1$, is anchored at the origin $O$ and points straight up along the y-axis. The second dipole, ${\mathbf{m}}_2$, is located at a point $P$ on the x-axis, exactly $1\text{ m}$ away from the origin, and it points to the right along the positive x-axis.

The term short magnetic dipole is crucial here. It tells us that the physical size of the dipoles is negligible compared to the distance $r$ between them. This allows us to use the simplified, far-field approximations for their magnetic fields.

Magnetic Field at the Equator

To find the torque on ${\mathbf{m}}_2$, we must first understand the magnetic environment it is sitting in. This environment is created entirely by ${\mathbf{m}}_1$.

Look closely at the geometry: point $P$ lies on the x-axis, while ${\mathbf{m}}_1$ points along the y-axis. This means point $P$ is on the equatorial line of dipole ${\mathbf{m}}_1$. The magnetic field produced by a short dipole at an equatorial point is given by the formula:

But what about its direction? For an equatorial point, the magnetic field is always anti-parallel to the magnetic moment of the source dipole. Since ${\mathbf{m}}_1$ points upwards, the magnetic field ${\mathbf{B}}_1$ at point $P$ must point straight downwards.

Torque on a Dipole

Now, let's shift our focus to the second dipole, ${\mathbf{m}}_2$. It is immersed in the downward magnetic field ${\mathbf{B}}_1$. We know that a magnetic dipole in an external magnetic field experiences a torque that tries to align it with the field. This torque is given by the cross product:

The magnitude of this torque is:

where $\theta$ is the angle between the dipole moment and the magnetic field. In our setup, ${\mathbf{m}}_2$ points to the right (horizontal) and ${\mathbf{B}}_1$ points downwards (vertical). Therefore, the angle $\theta$ is exactly $90^{\circ }$. Since $\sin 90^{\circ }=1$, the torque magnitude simplifies beautifully to just $m_2{B}_1$.

Bringing It All Together

We are now ready to substitute our expression for $B_1$ into the torque equation:

Let's plug in the numerical values provided in the problem: - $m_1=1{\text{ Am}}^2$ - $m_2=1{\text{ Am}}^2$ - $r=1\text{ m}$ - $\frac{{\mu }_0}{4\pi }=10^{-7}\text{ T m/A}$

Substituting these into our equation yields:

The problem asks for the value that fills in the blank for $......\times 10^{-7}\text{ Nm}$. As we have calculated, the coefficient is exactly $1$.