Bar Magnet

A bar magnet is a rectangular-shaped permanent magnet that produces a magnetic field around it, with clearly defined north and south poles. It is one of the most common types of magnets used in physics experiments, educational demonstrations, and real-world applications such as electric motors, compasses, and magnetic separators. The magnetic field of a bar magnet is strongest at its poles and weakest at the center, forming a distinctive magnetic pattern that helps explain the basic principles of magnetism.

1.0Definition of Bar Magnet(Magnetic Dipole)

Bar Magnet

A bar magnet is typically made from iron, steel, or other ferromagnetic materials or composites that possess permanent magnetic properties. It has two distinct poles — the North Pole and the South Pole — where the magnetic strength is concentrated.

2.0Properties of Bar Magnet

1. When a bar magnet is suspended freely near the surface of the earth, it tends to always align in the nearly North-South direction. One end of the bar magnet always pointing towards the geographic North is called the North pole of the magnet. The other end that always points towards the geographic South is called the South pole of the magnet.

2. Like poles repel and unlike poles attract each other.

3. The North and South poles of a bar magnet cannot be separated. The magnetic monopoles do not exist.

4. Bar magnets are made up of ferromagnetic materials, like iron, cobalt, nickel etc.

5. The magnetic field lines outside a bar magnet move from north pole to south pole.

6. The magnetic field lines inside a bar magnet move from south pole to north pole.

3.0Characteristics of Magnetic Field Lines

1. The tangent to a magnetic field line at a point gives the direction of the net magnetic field acting at that point.

2. Magnetic field lines are always closed paths, whatever may be the source of the magnetic field. Hence,there is no beginning or end point for a magnetic field line. There are no sources or sinks for magnetic field lines in a magnetic field. That is, there is no isolated magnetic pole.

3. The magnetic field lines can never intersect with each other.

4. The number of magnetic field lines across unit area in a region of space is a measure of the strength of the magnetic field in that region.

4.0Magnetic Dipole

A magnetic dipole consists of a pair of magnetic poles of equal and opposite strength separated by a small distance.

Example: Magnetic needle, Bar Magnet, Current carrying coil/solenoid etc.

Dipole Moment of a bar magnet

$\text{Dipole Moment}M\left(\vec{M}\right)$

• Vector quantity
• Direction: South → North
• Unit: $A-m^2$

The magnetic dipole moment of a bar magnet is a vector quantity equal to the product of its pole strength (m) and effective length (l), directed from the south pole to the north pole.

$$\begin{gathered} \vec{M}=m\vec{l} \\ Wherem:\text{Pole Strength}(Unit-A-m) \\ l:\text{Effective distance between North pole and South Pole} \\ M:\text{Dipole Moment of Bar Magnet} \end{gathered}$$

Note:

The number of field lines emerging from each pole indicate the pole strength of the pole.

Pole strength (m) of each pole $\propto$ area of cross section for a given bar magnet.

As magnetic moment is a vector, in case of two magnets having magnetic moments $M_1$ and $M_2$ with angle  between them, the resulting magnetic moment.

${\left[M=\sqrt{M_1^2+M_2^2+2M_1{M}_2\cos \theta }\right]}^{1/2}\text{with}\tan \varphi =\left[\frac{M_2\sin \theta }{M_1+M_2\cos \theta }\right]$

5.0Magnetic Moment of Current Carrying Coil (Loop)

The current carrying coil (or loop) behaves like a magnetic dipole. The face of the coil in which current appears to flow anticlockwise acts as north pole while face of coil in which current appears to flow clockwise acts as south pole.

A current carrying coil acts as a Magnetic Dipole.

$$\begin{gathered} \text{For a coil of turns ‘N’, geometrical area ‘A’, carries a current ‘I’} \\ \text{Magnetic Moment, }\vec{M}=NI\vec{A} \\ \text{Where}\vec{A}\text{is an area vector perpendicular to the plane of the coil and along its axis.} \\ {\text{SI unit : A – m}}^2\text{or J/T} \\ \text{Direction of}\vec{M}\text{can be found out by right hand thumb rule} \\ \text{Curling fingers - In the direction of current} \\ \text{Thumb – Gives the direction of}\vec{M} \\ \text{For a current carrying coil, its magnetic moment and magnetic field vectors both are parallel axial} \end{gathered}$$

6.0Torque and Force on Magnetic Dipole

Torque on a Magnetic Dipole in Uniform Magnetic Field: When a bar magnet is placed in a uniform magnetic field the two poles experience a force. They are equal in magnitude and opposite in direction and do not have the same line of action. They constitute a couple of forces which produce a torque. The torque tries to rotate the magnet so as to align it parallel to direction of field

(a) Bar magnet

Consider a bar magnet of magnetic moment $\vec{M}$ held in a uniform magnetic field $\vec{B}$ making an angle with the field. Then the magnet experiences a torque given by

$$\begin{gathered} \tau =\text{Force}\times \text{perpendicular distance between force couple} \\ \tau =F(l\sin \theta ) \\ \tau =(mB)(l\sin \theta ) \\ \tau =(ml)(B\sin \theta ) \\ \tau =MB\sin \theta ,\text{where }M=ml \\ \vec{\tau }=\vec{m}\times \vec{B} \\ \tau =MB\sin \theta \\ (1)\theta =90^{\circ }\Rightarrow \tau =MB(\text{Maximum}) \\ (2)\theta =0^{\circ }\text{or}180^{\circ }\Rightarrow \tau =0(\text{Minimum}) \end{gathered}$$

(b) Coil or Loop 

If a loop carries a current of magnitude I and magnetic moment M /vec{M} held in a uniform magnetic field B /vec{B} making an angle \theta,then the loop experiences a torque given by

$$\begin{gathered} \vec{\tau }=\vec{M}\times \vec{B} \\ \tau =MB\sin \theta \\ \vec{\tau }=I(\vec{A}\times \vec{B}) \\ \text{If loop have N turns then} \\ \vec{\tau }=NI(\vec{A}\times \vec{B}) \\ \tau =NIAB\sin \theta \\ (1)\theta =90^{\circ }\Rightarrow \tau =NIAB(\text{Maximum}) \\ (2)\theta =0^{\circ }\text{or}180^{\circ }\Rightarrow \tau =0(\text{Minimum}) \end{gathered}$$

Note:

• Torque on a dipole is an axial vector and it is directed along the axis of rotation of the dipole.
• Tendency of torque on dipole is try to align the $\vec{M}$ in the direction of $\vec{B}$ or tries to makes the axis of dipole parallel to $\vec{B}$ or makes the plane of coil (or loop) perpendicular to $\vec{B}.$
• Dipole in uniform magnetic field
• $$\begin{gathered} F_{\text{net}}=0(\text{no translatory motion}) \\ \tau \text{may or may not be zero}(\text{decides by }\theta ) \end{gathered}$$
• Dipole in non-uniform magnetic field
• $$\begin{gathered} F_{\text{net}}\text{may or may not be zero} \\ \tau \text{may or may not be zero (decides by }\theta ) \end{gathered}$$

7.0Potential Energy of Magnetic Dipole

Work Done in Rotating the Coil in Uniform Magnetic Field

When a bar magnet or coil of dipole moment Mis kept in a uniform magnetic field B It experiences a torque which tries to align it in the direction of the external magnetic field Work has to be done in rotation of the magnet or coil.

The work done in rotating dipole by small angle d is dW=d

The work done in rotating the magnet or coil from =1to =2 \theta=\theta_{1} to \theta=\theta_{2} is given by

$$\begin{gathered} W={\int }_{{\theta }_1}^{{\theta }_2}\tau d\theta \\ W={\int }_{{\theta }_1}^{{\theta }_2}MB\sin \theta d\theta \\ W=MB{\left[-\cos \theta \right]}_{{\theta }_1}^{{\theta }_2} \\ W=-MB\left(\cos {\theta }_2-\cos {\theta }_1\right) \\ W=MB\left(\cos {\theta }_1-\cos {\theta }_2\right) \end{gathered}$$

Potential Energy of the Coil in Uniform Magnetic Field

The potential energy of a dipole is defined as work done in rotating the dipole from a direction perpendicular to the given direction.

$$\begin{gathered} \text{The work done in rotating the dipole from}{\theta }_1=90^{\circ }to{\theta }_2=0^{\circ } \\ W=MB(\cos 90^{\circ }-\cos \theta ) \\ U=-MB\cos \theta \\ \text{In vector form} \\ U=-\vec{M}\cdot \vec{B} \end{gathered}$$

Important Note

Case-1	Case-2	Case-3
When $\vec{M}$ and $\vec{B}$ are parallel $$\begin{gathered} \theta =0^{\circ } \\ \tau =0 \\ {U}_{\min }=-MB \end{gathered}$$	When $\vec{M}\text{and}\vec{B}$ are perpendicular $$\begin{gathered} \theta =90^{\circ } \\ \tau =MB \\ U=0 \end{gathered}$$	When $\vec{M}\text{and}\vec{B}$ are anti-parallel $$\begin{gathered} \theta =180^{\circ } \\ \tau =0 \\ {U}_{\max }=MB \end{gathered}$$
The dipole has minimum potential energy and it is in stable equilibrium.	No Equilibrium as τ ≠ 0	The dipole has maximum potential energy and it is in unstable equilibrium.

8.0Field of a Bar Magnet

Magnetic field at large distances due to bar magnet of magnetic moment  can be obtained from the equation for electric field due to an electric dipole of dipole moment.

The magnetic field at a large distance from a bar magnet with magnetic moment $\vec{M}$ can be calculated using the formula for the electric field of a dipole with moment $\vec{p}$

9.0Angular SHM of Magnetic Dipole

When a dipole is suspended in a uniform magnetic field it will align itself parallel to the field. Now if it is given a small angular displacement with respect to its equilibrium position. The restoring torque acts on it :-

$$\begin{gathered} \tau =-MB\sin \theta \\ \text{For small angle,}\sin \theta \approx \theta \\ \tau =-MB\theta \\ \text{For angular SHM,} \\ \tau =-I{\omega }^2\theta \\ -I{\omega }^2\theta =-MB\theta \\ \omega =\sqrt{\frac{MB}{I}}=\frac{2\pi }{T} \\ T=2\pi \sqrt{\frac{I}{MB}} \\ \text{where I : Moment of Inertia}=\frac{m{l}^2}{12} \\ (m:\text{ mass of magnet},l:\text{ length of magnet}) \end{gathered}$$