IITJEE 2014-Physics-School Handout-Magnetism and Matter

Hand-Out Chapter - 5 Physics: Magnetism and Matter 1.

2.

MAGNETIC FIELD LINES Magnetic Field: The space around a magnet (or a conductor carrying current) where its magnetic effect can be experienced. A magnetic field line is the path along which an isolated north pole will tend to move, if placed inside a magnetic field. The magnetic field lines give the visual representation of the magnetic field. Properties  The magnetic field lines form continuous closed loops.  The tangent to the field line at a point represents the direction of the net magnetic field at that point.  The larger the number of field lines crossing per unit area, the stronger is the magnitude of magnetic field.  Two magnetic field lines of the same field do not intersect, for if they did, the direction of the magnetic field would not be unique at the point of intersection. Bar Magnet as an Equivalent Solenoid The magnetic field lines of a bar magnet and a solenoid are similar. A bar magnet can be thought of as a large number of circulating currents similar to that of a solenoid. The strength of magnetic field along the axial line and equatorial line of a solenoid and a bar magnet have similar expressions. MAGNETIC FIELD DUE TO A MAGNETIC DIPOLE ALONG AXIAL LINE Consider a bar magnet NS, whose each pole is of strength m. Let 2l be the magnetic length of the magnet and O be its centre. Let p be a point on axial line of the magnet at a distance r from the centre of the magnet.  The magnetic field Baxial at point P due to the bar magnet will be the resultant of the magnetic fields B1 (due to N-pole     of the bar magnet) and B2 (due to S-pole of the bar magnet), i.e. Baxial  B1  B2  μ μ m m  0 (along px ) Now = B1  0 4π (NP)2 4π (r  l ) 2  μ μ m m And = B2  0  0 (along PS ) 2 4π (SP) 4π ( r  l ) 2   It follows that B1 is greater than B2 .

  Since B1 and B2 act along the same line but in opposite directions. 

Or

   Baxial  B1  B2 (along PX)

Baxial =

 1 μ0 1   m  2 4π  ( r  l ) (r  l ) 2 

 (r  l )2  (r  l )2  μ0   m = 4π   (r 2  l 2 )2  

=

 (4rl )  μ0 m 2 2 2  4π  ( r  l ) 

We have, m (2l) = M, magnitude of the magnetic dipole moment of the magnet. 

Baxial 

μ0 2 Mr  2 2 2 along PX 4π ( r  l )

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… (1)

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Hand-Out Chapter - 5 Physics: Magnetism and Matter In vector form

  μ 0 2M r Baxial  4π (r 2  l 2 )2 When the length of the bar magnet is very small,

l  r

i.e.,  Baxial 

μ 0 2M  along PX 4π r 3

… (2)

Similarly, magnetic field on equatorial line of a bar magnet can be derived which comes out to be   μ0 M Bequatorial   4π r 3   From equations (1) and (2), we find that Baxial  2Bequatorial 3.

DIPOLE IN UNIFORM MAGNETIC FIELD Torque on a Magnetic Dipole in a Uniform Magnetic Field: Let a magnet NS having   pole strength M and length 2l be placed in a uniform magnetic field of strength B making an angle  with the direction of magnetic field lines. The expression can be derive using the same same method as in electrostatic and moving charge in magnetism.    The expression of torque on the bar magnet is τ  M  B Magnitude of torque: τ  MBsin θ

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POTENTIAL ENERGY OF A BAR MAGNET PLACED IN A MAGNETIC FIELD The magnitude of torque acting on a magnetic dipole of moment M in a magnetic field of strength B is given by τ  M B sin θ This torque tends to align the magnet along the direction of the field. If the magnet is to be rotated against the action of this torque, then work has to be done. Suppose that the magnet is rotated through an infinitesimally small angle d  under the action of the constant torque t. Work Required dw  τ  dθ ; dw  M B sin dθ

If the magnet is rotated from initial position θ = θ 1 to final position θ = θ 2, then the total work done is given by

W



θ2

M B sin θ dθ

θ1

 MB 

θ2

sin θ dθ

θ1

 MB cos θ θ2 θ

1

 MBcos θ2  cos θ1  = MBcos θ1  cos θ2 

Suppose magnet is initially perpendicular to the direction of the magnetic field, i.e.  1 = 90°. Then potential energy of the magnet in any position making an angle  with the direction of the field can be obtained by setting 1  90 and 2   in equation U = MB (cos 90° – cos θ °) = – M B cos θ   U  M  B Physics/Class XII

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Hand-Out Chapter - 5 Physics: Magnetism and Matter Special Case

 = 0°, U= – MB cos 0° = – MB (min.) (iii) When  = 180°, U= – MB cos 180° = + MB (max.) (i)

5.

6.

When

(ii)

When

 = 90°, U= – MB cos 90° = 0

  Thus, dipole possesses minimum potential energy, when M and B are like parallel and maximum potential energy when   M and B anti parallel. GAUSS’S LAW IN MAGNETISM An isolated magnetic pole does not exist. In other words, a surface may enclose a magnetic dipole i.e. a pair of equal and opposite magnetic poles so that the net pole strength enclosed by the surface is zero.   Therefore, the magnetic analogue of Gauss’s law in electrostatic may be stated as B  ds  0 , i.e., surface integral of  magnetic field B over a closed surface is always zero. It is called Gauss’s law in magnetism. 1. If a number of magnetic field lines are leaving a closed surface, an equal number of field lines must also be entering the surface. 2. Isolated magnetic poles do not exist, i.e., magnetic poles exist in pairs of equal strengths. THE EARTH’S MAGNETISM  A vertical plane passing through the geographic axis is called the geographic meridian.

 A vertical plane passing through the magnetic axis of the earth is called magnetic meridian. Elements of Earth’s Magnetism The physical quantities, which determine the intensity of earth’s total magnetic field completely (both in magnitude and direction), are called magnetic elements. They are: (i) Magnetic Declination: Declination at a place is the angle between the geographic meridian and the magnetic meridian. It is denoted by . In the figure, ABCD and AB´C´D represent magnetic and geographic meridians respectively. BAB  θ represents the magnetic declination. (ii) Magnetic Inclination or Dip: Dip at a place is defined as the angle made by the direction of earth’s total magnetic field with the horizontal direction. It is denoted by . In the figure, BAP is the angle of dip. (iii) Horizontal Component of Earth’s Magnetic Field: It is the component of earth’s magnetic field along the horizontal direction. It is denoted by BH. From point P (refer to the figure), if we drop PL perpendicular to AB and PM perpendicular to AD. Then AL and AM represent horizontal component (BH) and vertical component (BV) of earth’s magnetic field. In right ALP, cos δ 

Physics/Class XII

AL BH  AP B

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Hand-Out Chapter - 5 Physics: Magnetism and Matter Or

BH  B cos δ Also sin δ 

… (1) LP AM BV   AP AP B

BV  B sin δ

… (2)

Squaring and adding the equations (1) and (2) , we have B2H  B2V  B2 cos 2 δ  B 2 sin 2 δ  B 2

B  B2H +B2V

 B

… (3)

Dividing equation (2) by (1), we have Bsin δ B V  B cos δ B H

 7.

tan δ 

BV BH

… (4)

MAGNETISATION AND MAGNETIC INTENSITY (i)

Magnetic Permeability : It is the ability of a material to permit the passage of magnetic lines of force thorugh it i.e. the degree or extent to which magnetic field can penetrate or permeate a material is called relative magnetic permeability of the material. It is represented by μ r Relative magnetic permeability of a material is defined as the ratio fo the number of lines of magnetic induction per unit area (i.e., flux density B) in that material to the number of magnetic lines per unit area that would be present, if the medium were replaced by vaccum. (i.e., flux density B0) μr 

i. e.

B B0

μ r has no dimension. Its value for vacuum is one. Relative magnetic permeability of a material may also be defined as the ratio of magnetic permeability of the material μ  and magnetic permeability of free space μ 0  

μr 

μ μ0

or

μ  μ rμ 0

We know that μ 0  4π 10 –7 weber / amp - metre (Wb A–1 m–1) or henry / metre (Hm–1) 

S.I. units of permeability μ  are Hm–1 = Wb A –1m –1 = Tm2 A–1m–1 = Tm A–1

 (ii) Magnetising force or magnetic Intensity H

 

The degree to which a magnetic field can magnetise a material is represented in terms of magnetising force or  magnetic intensity H .

 

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Hand-Out Chapter - 5 Physics: Magnetism and Matter The magnetic induction of the field produced in the material to the toroidal solenoid is

B = μn I The product n I is called the magnetising force or magnetic intensity H. i.e. H  n I , so that B  μ H If inside the toroidal solenoid, there is free space, then magnetic induction B0  μ0 H

 (iii) Intensity of Magnetisation I .



It represents the extent to which a specimen is magnetised, when placed in a magnetising field. Quantitatively, the intensity of magnetisation of a magnetic material is defined as the magnetis moment per unit volume of the material.   Magnetic moment M I  .........(i) volume V If a = uniform area of cross -section of the magnetised specimen (a rectangular bar) 2 l = magnetic length of the specimen. m = strength of each pole of the specimen ,  from (i), I=

m  2l m  a  2l a

Hence intensity of magnetisation of a magnetic material is also defined as the pole strength per unit area of acrosssection of the material. magnetic moment Volume  These area S.I. unit of I .

As



I=

I=

Amp.metre 2  A m –1 metre3

(iv) Magnetic Susceptibility  χ m  . It is a property which determines how easily a specimen can be magnetised. Quantitatively, susceptibility of a magnetic material is defined as the ratio of the intensity of magnetisation (I) induced in the material to the magnetising force (H) applied. susceptibility is represented by χ m Thus (v)

χm 

I H

Magnetic Induction: When a piece of magnetic material is placed inside a magnetic filed (B0), it gets magnetised and produce its own magnetic field. Magnetic induction denotes the number of magnetic lines of induction (magnetic field lines inside the material) crossing per unit area normally through the magnetic substance. It is denoted by B.    The magnetic induction B is the sum of the magnetic field B0 and the induced field Bi    B  B0  Bi Relation between magnetic permeability and susceptibility     By definition: B0  μ 0 H and Bi = μ 0 I    Then, B  μ 0 (H  I)

B  μ 0 H  I Physics/Class XII

... (i) 5

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Hand-Out Chapter - 5 Physics: Magnetism and Matter Magnetic induction is also known as magnetic flux density or simply magnetic field. Dividing equation (i) by H, Using μ 

 B I  μ 0 1     H H

B I and χ m  , H H

μ  μ 0 I  χ m  μ  1 χ m μ0

r  1   m (as μ  μ r , relative permeability of the magnetic substance) μ0

8.

MAGNETIC PROPERTIES OF MATERIALS Different materials show different magnetic behaviour e.g. iron, steel, cobalt and nickel are strongly magnetised whereas aluminium, copper, gold, mercury etc, are weakly magnetised in an applied magnetic field. Curie’s law Curie’s Law can be stated as, magnetic susceptibility of a material varies inversely with temperature (in Kelvin). Curie discovered experimentally that intensity of magnetisation I of a magnetic material is directly proportional to magnetic field intensity B and inversely proportationl to the temperature T (in kelvin) i.e.,

or

B T

or

I

I 1  H T

or

χm 

I

H T C T

where C is called Curie constant. Increase in magnetisation (I) with decrease in temperature has a limit when the material becomes saturated. Figure I shows a curve between (I) and (B / T). Once saturation is achieved further decrease in temperature does not bring change in magnetisation. Curie temperature for iron is about 1000 K, for cobalt it is about 1400 K and for nickel it is about 600 K. Classification / Categoes of Magnetic Material. Faraday divided the materials in three classes according to their magnetic behaviour : (a) Ferromagnetic Materials : The materials which are strongly magnetised in the direction of the applied magnetic field are known as Ferromagnetic Materials. Iron, steel, nickel, cobalt and alloy’s like alnico (aluminium + nickel + cobalt) are ferromagnetic materials. Ferromagnetic substances can be easily magnetised. Ferromagnetic effect is noticed even in the presence of weak magnetic field. with the rise in temperature it becomes comparitively less easier to magnetise the ferromagnetic substance. (b) Paramagnetic Materials : The materials which are weakly magnetised in the direction of applied magnetic field are known as Paramagnetic Materials. Aluminium, chromium, manganese platium, antimony, sodium, copper chloride, salt solutions of iron and nickel, liquid oxygen, cross glass etc are paramagnetic materials. Paramagnetic materials. Aluminium, chromium, managanese platinum, antimony, sodium, copper chloride, salt solutions of iron and nickel, liquid oxygen, cross glass etc, are paramagnetic materials. Paramagnetic materials tend to lose their magnetic behaviour the rise in temperature. Paramagnetic materials can not be easily magnetised.

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Hand-Out Chapter - 5 Physics: Magnetism and Matter (c) Diamagnetic Material : the materials which are weakly magnetised in a direction apposite to the direction of applied magnetic field are known as diamangetic Materials, Gold, silver, zinc, lead, bismuth, mercury, marble, glass, quartz, water, alcohol, air, helium, argon, hydrogen, salts like sodium chloride etc are diamagnetic materials.It is very difficult to magnetise a diamagnetic material. They require very strong magnetic field to show magnetic properties. Their magnetic behaviour normally does not depend upon change in temperature. Properties of Magnetic Materials A comparative chart of properties of ferromagnetic, paramagnetic and diamagnetic materials is given below: Ferromagnetic Materials (a) They are strongly attracted by a magnet. (b) A freely suspended ferremagnetic rod quikly sets itself along the direction of external magnetic field as shown in figure 2 (A).

Paramagnetic Materials

Diamagnetic Materials

(a) They are weakly attracted by a magnet.

(a) They are weakly repelled by a magnet.

(b) A freely suspended paramagnetic rod slowly sets itself along the direction of external magnetic field as shown in figure 2 B.

(b) A freely suspended diamagnetic rod slowly sets itself at right angle to the direction of external magnetic field as shown in figure 2 C.

Figure 2 (A) (c) When they are placed in a magnetic field, the lines of force prefer to pass through them.

Figure 2 (B) Figure 2 (C) (c) When they are placed in a magnetic (c) When they are placed in a magnetic field, most of the lines of force field, the lines of force do not prefer to prefer to pass through them. pass through them.

Figure 3 (A) This behaviour indicates that (i) Field within the sample is much more than the magnetic intersity (figure 3A) i.e. perme-

Figure 3 (B)

ability (μ) is much more than

This behaviour indicates that

B unity. (B > > H or > > 1 or H μ > > 1).

(ii) Flux density (B) inside a ferromagnetic material is much larger than in air. (iii) The sample gets strongly magnetised in the direction of magnetising field. (iv) Intensity of magnetisation (I) has large positive value. Physics/Class XII

(i) Field within the sample is more than the magnetic intersity (figure 3B) i.e. permeability (μ) is more than unity (B > H or

B of μ > 1). H

(ii) Flux density (B) inside a paramagnetic material is larger than in air.

Figure 3(C) (i) Field within the sample is decreased to a very small value (figure 3C) i.e. permeability ( μ ) is always less than unity (B < H or

B < 1 of μ < 1) H

(ii) Flux density (B) inside a diamagnetic material is less than in air.

(iii) The sample gets weakly magnetised (iii) The sample gets weakly magnetised in in the direction of magnetising the direction opposite to the direction field. of magnetising field. 7

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Hand-Out Chapter - 5 Physics: Magnetism and Matter (v) Susceptibility has a large positive (iv) Intensity of Magnetisation (I) has small positive value. 1 value χ m  : I is large + ve so H (v) Susceptibility has a small positive χ m is large + ve). 1 value (χ m  : I is small + ve so H (d) They obey Curie’s law. At a certain temperature i.e. Curie χ m is small + ve). point, ferromagnetic properties disappear and material starts (d) They obey curie’s law. They are behaving as paramagnetic. badly affected with the rise in

(iv) Intensity of magnetisation (I) has small negative value. (v) Susceptibility has a small negative 1 : I is small – ve so χ m H is small – ve).

value (χ m 

(d) They do not obey Curie’s law. Normally their magnetic properties do not change with temperature*

temperature. Due to rise in tempera(e) Liquids and gases do not show ture they lose magnetic property. ferromagnetism. If a finely (e) If a diamagnetic liquid in a watch glass powered ferromagnetic material in is placed on closely spaced magnetic a watch glass is placed on closely (e) If a paramagnetic liquid in a watch poles and then widely placed magnet glass is placed on closely spaced spaced magnetic poles and then poles, the effect is observed as shown magnetic poles and then widely widely placed magnet poles, the in figure. In the first case there is a placed magnet poles, the effect is effect is observed as shown in depression in the middle but in the observed a shown in figure. In the figure. second case middle but in the second case

Figure 4 (A) It shown that such materials move from weaker to stronger magnetic field. (f) When a sample of ferromagnetic material in a very finely powered from is placed in U-tube and magnetic field is applied to one limb, the level rises in that limb (Figure 5A)

Figure 5 (A)

Physics/Class XII

Figure 4(B) there is a depression in the middle. It shows that such materials move from weaker to stronger magnetic field. (f) When a sample of paramagnetic liquid is put in a U-tube and magnetic field is applied to one limb, the level rises in that” limb i.e. from weaker to stronger magnetic field.

Figure 5 (B)

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Figure 4 (C) there is a rise in the middle. it shows that such material move from stronger to weaker magnetic field. (f) When a sample of diamagnetic liquid is put in a U-tube and magnetic field is applied to one limb the level falls in that limb i.e., from stronger to weaker magnetic field.

Figure 5 (C)

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Hand-Out Chapter - 5 Physics: Magnetism and Matter 9.

HYSTERESIS The lag of intensity of magnetisation behind the magnetising field during the process of magnetisation and demagnetisation of a ferromagnetic material is called Hysteresis.   The graph plotted with B against H for a ferromagnetic substance is referred to as hysteresis loop. It is shown below. In the figure, B increases non-linearly with H along OPA. If H is reduced to zero, B does not fall back to zero along the same curve, but decreases along path ABC, i.e. it is non retraceable. The lagging of B behind H is called hysteresis. Also, when H becomes zero, B does not do so. It has a certain value Br equivalent to OB (in the figure). Retentivity is a measure of this remaining field. Further, if H is increased in the reverse direction, the value of B reduces and becomes zero when H has value Hc = OC. This value of magnetic field is called coercive force or coercivity of the specimen. B B2 BH = B    has the dimensions of energy per unit volume. Therefore, the area within the BH loop represents    0  r

energy dissipated per unit volume in the material. Retentivity: The measure of the magnetic field remaining in the specimen when the magnetising field is removed. Coercivity: The value of reverse magnetising field required so as to reduce residual magnetism to zero. Both coercivity and retentivity depend on the nature of the material. 10. PERMANENT MAGNETS AND ELECTROMAGNETS Permanent Magnets 

Permanent magnets are the ferromagnetic substances which retain magnetism for a long time at room temperature.



One method to make a permanent magnet is to continuously run one end of a magnet on a fixed steel rod always in one direction. The steel rod then acquires permanent magnetism.



Another method is to use electric current. When a current is passed through a solenoid containing a steel rod, then the rod acquires permanent magnetism.



Steel or hard iron or hard alloys of iron like alnico are considered best for making permanent magnets.

Electromagnets 

Soft iron has large permeability and small retentivity and hence is suitable for making electromagnets.



When a current is passed through a solenoid wound around a rod of soft iron, magnetic field inside the iron rod increases many times making it an electromagnet.



On switching off the current, magnetic field more or less vanishes due to small retentivity of soft iron.



The soft iron rod will act as a magnet only as long as there is current in the solenoid. Electric bells, loud-speakers and telephone receivers use electromagnets. Huge electromagnets are used in cranes to lift heavy things made of iron.

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Hand-Out Chapter - 5 Physics: Magnetism and Matter

1.

1. 2.

1. 2.

1. 2.

1.

1. 2.

1. 2. 3.

1. 2. 3.

2013 Which of the following substances are diamagnetic? Bi, Al, Na, Cu, Ca and Ni.

(1 Mark)

2012 The horizontal component of the earth’s magnetic field at a place is B and angle of dip is 60°. What is the value of vertical component of earth’s magnetic field at equator? (1 Marks) –5 The susceptibility of a magnetic material is –2.6 × 10 . Identify the type of magnetic material and state its two properties. (2 Marks) 2011 The susceptibility of a magnetic material is – 4.2×10–6. Name the type of magnetic materials it represents. (1 Marks) A magnetic needle free to rotate in a vertical plane parallel to the magnetic meridian has its north tip down at 60° with the horizontal. The horizontal component of the earth’s magnetic field at the place is known to be 0.4 G. Determine the magnitude of the earth’s magnetic field at the place. [Ans. 0.69G ] (2 Marks) 2010 (i) Write two characteristics of a material used for making permanent magnets. (ii) Why is the core of an electromagnet made of ferromagnetic materials? (2 Marks) Draw magnetic field line when a (i) diamagnetic, (ii) paramagnetic substance is placed in an external magnetic field. Which magnetic property distinguishes this behaviour of the field line due to the substances? (2 Marks) 2008 Define magnetic susceptibility of a material. Name two elements, one having positive susceptibility and the other having negative susceptibility. What does negative susceptibility signify. (2 Marks) 2007 Why should the material used for making permanent magnets have high coercivity. (a) Distinguish the magnetic properties of dia, para- and ferro-magnetic substances in terms of (i) susceptibility, (ii) magnetic permeability and (iii) Give one example of each of these materials. (b) Draw the field lines due to an external magnetic field near a (i) diamagnetic substance, (ii) paramagnetic substance.

(1 Mark) coercivity.

(5 Marks)

2006 Steel is preferred for making permanent magnets whereas soft iron is preferred for making electromagnets. Give one reason. (1 Mark) Why does a paramagnetic substance display greater magnetisation for the same magnetising field when cooled. How does a diamagnetic substance respond to similar temperature changes. (2 Marks) Three identical specimens of magnetic materials; nickel, antimony, aluminium are kept in a non-uniform magnetic field. Draw the modification in the field lines in each case. Justify your answer. (3 Marks) 2005 What is the value of the horizontal component of the earth’s magnetic field at magnetic poles. Define the terms ‘Magnetic Dip’ and ‘Magnetic Declination’ with the help of relevant diagrams. Write two characteristic properties to distinguish between diamagnetic and paramagnetic materials.

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