Dipole in Uniform Magnetic field

Most people are familiar with magnets primarily as toys or as decorative items on a refrigerator door

Magnetism has a much broader range of applications

Considering two opposite poles (i.e., North and South poles) as a system is known as a magnetic dipole

Therefore, based on the definition we can say that a bar magnet forms a magnetic dipole

Let us see what happens when we place a Dipole in Magnetic field

When a bar magnet or magnetic dipole is placed in a uniform Magnetic field, each pole will experience a force

Due to this Force, the Dipole is bound to experience a Torque

Let us see the Force and Torque acting on a Magnetic Pole in a field

When an electric charge is placed in an Electric field, Force acts on the charge due to the Electric Field Intensity

Similarly, a Magnetic pole experiences a Force due to a Magnetic Field

The Force acting on the Magnetic Pole is given by $\overrightarrow{F}=m\overrightarrow{B}$

Where, $m$=The Magnetic pole strength $\overrightarrow{B}$=Magnetic induction

The pole strength acting on the North Pole is $+m$, so the Force acting on it will be $\overrightarrow{F}=m\overrightarrow{B}$

The direction of the Force acting on the North magnetic Pole will be in the direction of the Magnetic Field

The pole strength of the South magnetic Pole will be $-m$, so the Force acting on it will be $\overrightarrow{F}=-m\overrightarrow{B}$

The direction of the Force acting on the South magnetic Pole will be opposite to the direction of Magnetic Field

Now, let us consider a Dipole placed in a Uniform Magnetic Field $\overrightarrow{B}$ at an angle $\theta$

The Force acting on the dipole is zero since the forces acting on the individual poles are equal and opposite

The Torque acting on a dipole can be mathematically expressed as, $\tau =F\times$ perpendicular distance between North Pole N and South Pole P

The vertical distance between the North Pole and the South Pole is $NP$

In $△SPN$, $$\begin{gathered} \sin \theta =\frac{NP}{SN} \\ \therefore NP=SN\sin \theta =2l\sin \theta \end{gathered}$$

Replacing the value of $NP=2l\sin \theta$, $\tau =mB\times 2l\sin \theta$

Rearranging the terms, $$\begin{gathered} \tau =(m)(2l)B\sin \theta \\ \therefore \tau =MB\sin \theta \end{gathered}$$

Where, $M=m(2l)$ is the Magnetic Moment on the Dipole

The equation of Torque can be written in vector form as, $\overrightarrow{\tau }=\overrightarrow{M}\times \overrightarrow{B}$

In the equation $\tau =MB\sin \theta$, if $\theta =0^o$ or $\theta =180^o$, the Dipole is lying Parallel or Anti-Parallel to the Magnetic Field

In this case, the Torque acting on the Dipole will be Zero

This is because the Torques acting on the North Pole $mB$ and South Pole $-mB$ will be balanced

Now, if $\theta =90^o$, $\sin 90^o=1$

In this case, the Torque acting on the Dipole will be $\tau =MB$

Additionally, if $B=1T$, the Torque acting on the Dipole will be equal to Magnetic Moment $M$

The Torque acting on a Dipole placed in a Unit Magnetic field perpendicular to the direction of Magnetic field is known as the Magnetic Dipole Moment

Let us calculate the Work done on the Dipole

Work is done whenever a Torque is used to rotate a Dipole by an angle

Work done can be mathematically expressed as Work done = Torque x Angular Displacement in the direction of the Force

The Work done to rotate a bar magnet or Dipole by a small angle $\theta$ is given by, $dW=\tau d\theta$

Now, further Work done is required to move the Dipole from an initial angle ${\theta }_0$ to final angle $\theta$

This Work done is calculated by, $W=\int dW$

$$\begin{gathered} \int dW={\int }_{{\theta }_0}^{\theta }MB\sin \theta d\theta \\ \Rightarrow W=MB{\int }_{{\theta }_0}^{\theta }\sin \theta d\theta \end{gathered}$$

$$\begin{gathered} \Rightarrow W=MB{{\theta }_0}_{\theta }^{-cos\theta } \\ \Rightarrow W=MB[\cos {\theta }_0-\cos \theta ] \end{gathered}$$

If ${\theta }_0=0$, $W=MB[1-\cos \theta ]$

If ${\theta }_0=90^o$, $$\begin{gathered} W=-MB\cos \theta \\ \therefore W=-\overrightarrow{M}.\overrightarrow{B} \end{gathered}$$ Since, $\overrightarrow{A}.\overrightarrow{B}=AB\cos {\theta }_0$

This Work done will be stored in the form of Potential Energy. Therefore, $P.E=-MB\cos \theta$

Revision

A magnet in which opposite poles (i.e., North and South poles) are on opposite sides of the magnet is called a Dipole magnet

When a bar magnet or magnetic dipole is placed in a uniform Magnetic field, it will experience a torque

The Force acting on a Dipole multiplied by the perpendicular distance between the Poles is the Torque acting on the Dipole

The work done on the dipole if ${\theta }_0=90^o$, will be $W=-\overrightarrow{M}.\overrightarrow{B}$

The End