Maharashtra Board Class 12 Physics Chapter 11 Magnetic Materials PDF Download

11. Magnetic Materials

Can You Recall?

1. What are magnetic lines of force?

2. Why magnetic monopoles do not exist?

3. Which materials are used in making magnetic compass needle?

11.1 Introduction

You have studied about magnetic dipole and dipole moment due to a short bar magnet. Also you have studied about magnetic field due to a short bar magnet at any point in its vicinity. When such a small bar magnet is suspended freely, it remains along the geographic North South direction. (This property can be readily used in Navigation.)

In the present Chapter you will learn about the behaviour of a short bar magnet kept in two mutually perpendicular magnetic fields. You will also learn about different types of magnetism, viz. diamagnetism, paramagnetism and ferromagnetism with their properties and examples. At the end you will learn about the applications of magnetism such as permanent magnet, electromagnet, and magnetic shielding.

Teacher's Note

A magnet always points North-South when freely suspended. This is like how a compass needle works. You have seen this if you ever used a compass during a school trip.

Exam Trick

Remember: Free magnet = North-South direction. Just like birds know which way to fly, magnets know which way to point!

Points to Remember

Magnetic dipole moment is important in magnetism.
Bar magnets have North and South poles.
Magnetic field is strongest at the poles.
A free magnet always aligns North-South.
This property is used in compasses and navigation.

Activity

You have already studied in earlier classes that a short bar magnet suspended freely always aligns in North South direction (as shown in Fig. 11.1). Now if you try to forcefully move and bring it in the direction along East West and leave it free, you will observe that the magnet starts turning about the axis of suspension. Do you know from where does the torque which is necessary for rotational motion come from? (as studied in rotational dynamics a torque is necessary for rotational motion).

Try This

Now we will extend the above experiment further by bringing another short bar magnet near to the freely suspended magnet. Observe the change when the like and unlike poles of the two magnets are brought near each other. Draw conclusion. Does the suspended magnet rotate continuously or rotate through certain angle and remain stable?

11.2 Torque Acting On A Magnetic Dipole In A Uniform Magnetic Field

You have studied in the previous Chapter that the torque acting on a rectangular current carrying coil kept in a uniform magnetic field is given by

\[\tau = \vec{m} \times \vec{B}\]

\[\tau = mB \sin \theta\] --- (11.1)

where \(\theta\) is the angle between \(\vec{m}\) and \(\vec{B}\), the magnetic dipole moment and the external applied uniform magnetic field, respectively as shown in Fig. 11.2. The same can be observed when a small bar magnet is placed in a uniform magnetic field. The forces exerted on the poles of the bar magnet due to magnetic field are along different lines of action. These forces form a couple. As studied earlier, the couple produces pure rotational motion. Analogous to rectangular magnetic coil in uniform magnetic field, the bar magnet will follow the same Eq. (11.1).

Due to the torque the bar magnet will undergo rotational motion. Whenever a displacement (linear or angular) is taking place, work is being done. Such work is stored in the form of potential energy in the new position (refer to Chapter 8). When the electric dipole is kept in the electric field the energy stored is the electrostatic Potential energy.

Magnetic potential energy

\[U_m = -\int_0^{\theta} \tau(0) \, d\theta\] --- (11.2)

\[U_m = -\int_0^{\theta} mB\sin\theta \, d\theta\]

\[U_m = -mB \cos \theta\] --- (11.3)

Let us consider various positions of the magnet and find the potential energy in those positions.

Case 1- When \(\theta = 0°\), \(\cos 0 = 1\), \(U_m = -mB\). This is the position when \(\vec{m}\) and \(\vec{B}\) are parallel and bar magnet possess minimum potential energy and is in the most stable state.

Case 2- When \(\theta = 180°\), \(\cos 180° = -1\), \(U_m = mB\). This is the position when \(\vec{m}\) and \(\vec{B}\) are antiparallel and bar magnet possesses maximum potential energy and thus is in the most unstable state.

Case 3- When \(\theta = 90°\), \(\cos 90° = 0\), \(U_m = 0\). This is the position when bar magnet is aligned perpendicular to the direction of magnetic field.

The potential energy as function of \(\theta\) is shown in Fig. 11.3.

As discussed in the activity earlier, suppose the bar magnet is suspended using inextensible string. When we perform similar activity (Refer Fig. 11.1) we observe that the magnet rotates through angle and then becomes stationary. This happens because of the restoring torque as studied in Chapter 10 generated in the string opposite to the deflecting torque. In equilibrium both the torques balance.

\[\tau = I \frac{d^2\theta}{dt^2}\] --- (11.4)

where I is moment of inertia of bar magnet and \(\frac{d^2\theta}{dt^2}\) is the angular acceleration. As seen from Fig. (11.2), the direction of the torque acting on the magnet is clockwise. Now if one tries to rotate the magnet anticlockwise through an angle \(d\theta\), then there will be a restoring torque acting as given by the equation, in opposite direction.

Thus we write the restoring torque

\[\tau_s = -mB \sin \theta\] --- (11.5)

From the two equations we get

\[I \frac{d^2\theta}{dt^2} = -mB \sin \theta\]

When the angular displacement \(\theta\) is very small, \(\sin \theta \approx \theta\) and the above equation can be written as

\[I \frac{d^2\theta}{dt^2} = -mB\theta\]

\[\frac{d^2\theta}{dt^2} = -\frac{mB}{I}\theta\] --- (11.6)

The above equation has angular acceleration on the left hand side and angular displacement \(\theta\) on the right hand side of equation with m, B and I being held constant.

Teacher's Note

When you push a swing and let it go, it swings back and forth. A magnet in a magnetic field moves the same way - this is called oscillation.

Exam Trick

Remember: Torque makes magnet rotate. Like a fan that spins when you turn it on.

Points to Remember

Torque \(\tau = mB \sin \theta\) causes rotation.
Magnetic potential energy is \(U_m = -mB \cos \theta\).
Parallel position has lowest energy (most stable).
Antiparallel position has highest energy (unstable).
Perpendicular position has zero potential energy.

This is the angular simple harmonic motion (S.H.M) (Chapter 5) analogous to linear S.H.M. governed by the equation

\[\frac{d^2x}{dt^2} = -\omega^2 x\]

Here \(\omega^2 = \frac{mB}{I}\)

\[\therefore \omega = \sqrt{\frac{mB}{I}}\] --- (11.7)

The time period of angular oscillations of the bar magnet will be

\[T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{I}{mB}}\] --- (11.8)

Vibration Magnetometer

Vibration Magnetometer is used for the comparison of magnetic moments and magnetic field. This device works on the principle, that whenever a freely suspended magnet in a uniform magnetic field, is disturbed from its equilibrium positions, it starts vibrating about the mean position. It can be used to determine horizontal component of Earth's magnetic field.

Teacher's Note

A vibrating magnet is like a pendulum that swings back and forth. Scientists use this to measure Earth's magnetic field strength.

Exam Trick

Remember: Vibration Magnetometer = magnet swinging = measuring magnetic field. Like a swing measuring how hard you push it.

Points to Remember

Time period \(T = 2\pi \sqrt{\frac{I}{mB}}\).
Vibrating magnet undergoes simple harmonic motion.
Used to compare magnetic moments.
Can measure Earth's magnetic field.
Works by counting oscillations.

Location Of Magnetic Poles Of A Current Carrying Loop

You have studied that a current carrying conductor produces magnetic field and if you bend the conductor in the form of loop, this loop, behaves like a bar magnet (as discussed in Chapter 10).

Observe And Discuss

What is a North pole or South pole of a bar magnet? For understanding this, you just have to draw a circular loop on a plane glass plate and show the direction of current (say clockwise direction). Now place on it a wire loop having clockwise current flowing through it. According to right hand rule, the top surface will behave as a South pole. Now just turn the glass and see the same loop through other surface of glass. You will find that the direction of current is in anticlockwise direction (in reality there is no change in the direction of current) and hence the loop side surface behaves like North pole.

Example 11.1

A bar magnet of moment of inertia of 500 g cm\(^2\) makes 10 oscillations per minute in a horizontal plane. What is its magnetic moment, if the horizontal component of earth's magnetic field is 0.36 gauss?

Given: Moment of Inertia I = 500 g cm\(^2\), Frequency n = 10 oscillation per minute = 10/60 oscillations per second, Time period T = 6 sec, B\(_H\) = 0.36 gauss

Solution: From Eq. (11.8),

\[m = \frac{4\pi^2 I}{T^2 B}\]

\[m = \frac{4 \times (3.14)^2 \times 500 \times 10^{-7}}{36 \times 0.36 \times 10^{-4}}\]

m = 1.52 Amp m\(^2\)

Teacher's Note

This formula helps scientists measure how strong a magnet is. The more times it swings, the stronger it is.

Exam Trick

Remember: More oscillations = stronger magnet. Count the swings to know magnet strength!

Points to Remember

Time period depends on moment of inertia.
Time period depends on magnetic moment.
Time period decreases with stronger field.
Frequency increases with stronger magnet.
Used in labs to measure magnetic properties.

11.3 Origin Of Magnetism In Materials

In order to understand magnetism in materials we have to use the basic concepts such as magnetic poles and magnetic dipole moment. In XIth Std., you have studied about the magnetic property of a short bar magnet such as its magnetic field along its axial or equatorial direction. What makes some material behave like a magnet while others don't? To understand it one must consider the building blocks of any material i.e., atoms. In an atom, negatively charged electrons are revolving about the nucleus (consisting of protons and neutrons). The details about it will be studied in Chapter 15. You have studied in the periodic table in Chemistry in XIth Std. that chemical properties are dominated by the electrons orbiting in the outermost orbit of the atom. This also applies to magnetic properties as described below.

Teacher's Note

Electrons moving around atoms create tiny magnets. Like a spinning top creates a spinning effect, electrons create magnetic effects.

Exam Trick

Remember: Electrons moving = tiny magnets. All magnetism comes from moving electrons!

Points to Remember

Atoms have electrons moving around nucleus.
Moving electrons create magnetic dipole moment.
Magnetic properties come from electrons.
Outermost electrons are most important.
Both orbital and spin motion matter.

This is a preview of the first 3 pages. To get the complete book, click below.