## Sunday, March 23, 2008

### Concept Review Ch.36 Permanent Magnets

Model of magnetic charge

a. There are two types of magnetic charges, positive magnetic charge and negative magnetic charge.

A magentic charge when placed in a magnetic field experiences a force.

The force on a postive magnetic charge is along the field and the force on a negative magnetic charge is opposite the field.

b. A magnetic charge produces a magnetic field.
The field is radially outward if the charge if positive and is inward if it is negative.

C. A magnetic dipole is formed when a negative magnetic charge -m and a positive magentic charge +m are placed at a small separation d.

md is called magnetic dipole moment and its direction is from -m to +m.
The line joining -m to +m is called the axis of the dipole.

d. a magnetic dipole of dipole moment md can be replaced by a current loop or area carrying a current i.

md = iA

How natural magnets exist?

Matter is made of atoms and each atom contains electrons circulating around the nucleus. The circulation of electron around a nucleus can be assumed as small circular current loop. We already mentioned in the model of magnetism that a current loop is equivalent to a magnetic dipole. In permanent magnets, these small circular loops are arranged nearly parallel to each other along the length of the magnet. At any point inside the magnet, the net current is zeo because the currents from the adjacent loops cancel each other over any cross section or face. However on the surface, such cancellation of current does not take place and hence there is a net current. Due to such a surface current, a magnet in the form of a cylinder is equivalent to a closely wound, current carrying solenoid and hence produces a magnetic field similar to the solenoid.

Magnetic length of the magnent

Magnetic poles appear slightly inside the bar. The distance between the locations of the assumed poles is called the magnetic length of the magnet. the distance between the ends is called the geometrical length.

It is found from observation that magnetic length/geometrical length = 0.84.

Magnetic moment of a bar magnet

It is denoted by M. Magnetic length of a bar magnet is denoted by 2l.If m is the pole strength and 2l is magnetic length of a bar magnet, its magnetic moment is

M = 2ml

Torque on a bar magnet placed in a magnetic field

Г = MB sin θ

θ is the angle between the magnetic field assumed to be along Y axis and the bar magnet. B is the magnetic field and M is the magnetic moment of the bar magnet.

Potential energy of a bar magnet placed in a magnetic field

U(θ) = -MB Cos θ = -M.B

When potential energy is assumed to be zero when θ = 90° (Magnetic field assumed to be along Y axis)

Magnetic field due to a bar magnet

End on position
A position on the magnetic axis of a bar magnet is called an end-on position. The magnetic field at a point in end-on position at a distance d from the centre of the magnet is:

B = (µ0/4 π)[(2Md)/(d² – l²)²] …(6)

If d is very large compared to l, then

B = (μ0/4π)(2M/d³)

A position on a perpendicular bisector of the bar magnet is called broadside-on position. The magnetic field at a point in broadside-on position at a distance d from the centre of the magnet is:

B = (µ0/4 π)[m 2l/(d² + l²)3/2 ] …(8)
= (µ0/4 π)[M/(d² + l²)3/2 ]

If d is very large compared to l,

B = (μ0/4π)(M/d³)

Magnetic scalar potential

Change in potential is defined as
V(r2) – V(r1) = -r1r2 B.dr

Normaly, the potential at infinity ( a point far away from all sources of magnetic field) is taken to be zero.

Magnetic scalar potential due to a magnetic dipole

Magnetic scalar potential at a point P which is at a distance r from the mid point of the magnetic dipole, and the angle between the dipole axis and the line joining the mid point of the dipole to the point P is θ

V = (µ0/4 π)[Mcos θ /r²)
Where
M = 2ml =magnetic moment of the dipole

Magnetic field due to a dipole

Magnetic field at P =

0/4 π)[M /r²)√(1 +3 cos² θ)] ..

Terrestrial magnetism
(There are questions in past papers about the magnetic needle placed at various places on the earth)

Earth has a magnetic dipole of dipole moment about 8.0*10^23 /T located at its centre. The axis of this dipole makes an angle of about 11.5 with the earth’s axis of rotation.

The dipole axis cuts the earth’s surface at two points, one near the geographical north pole and the other near the geographical south pole. The first of these points is called geomagnetic north pole and the other is called geomagnetic south pole.

If a bar magnet is freely suspended at a point near the earth’s surface, it will stay along the magnetic field there, the north pole pointing towards the direction of magnetic field. At the geomagnetic poles, the magnetic field is vertical. If the bar magnet is suspended near the geomagnetic north pole, it will become vertical with its north pole towards the earth’s surface. Similarly, a bar magnet freely suspended near the geomagnetic south pole, will become vertical with its south pole pointing towards the earth’s surface. Hence at geomagnetic poles a freely suspended bar magnet becomes vertical and this property can be definition of geomagnetic poles.

In the assumed magnetic dipole inside the earth, the south pole of the dipole will be towards the geomagnetic north pole and the north pole will be towards the geomagnetic south pole. That is why north pole of the freely suspended bar magnets points towards geomagnetic north pole, at that place. Similarly south pole of the freely suspended bar magnet will be towards the south pole at that place.

Earth’s magnetic field changes in both in direction and magnitude as time passes. Noticeable changes occur over 10 years long periods. Even reversals in direction take place. In the past 7.6*10^7 years 171 reversals have taken place. The latest reversal is believed to have taken place around 10,000 years ago.

Elements of the Earth’s magnetic field
a. declination b. inclination or dip and c. horizontal component of the field

Declination

A plane passing through the geographical poles (that is axis of rotation of the earth) and a given point P on the earth’s surface is called the geographical meridian at the point P. The plane passing through the geomagnetic poles (dipole axis of the earth) and the point P is called the magnetic meridian at the point P.

The angle between the magnetic meridian and the geographical meridian at a point is called the declination at that point.

Navigators use a magnetic compass needle to locate direction. The needle stays in equilibrium when it is in magnetic meridian. For finding the true north, navigators have use declination and arrive at the direction of true north.

Inclination or dip

The angle made by the earth’s magnetic field with the horizontal direction in the magnetic meridian, is called the inclination of dip at that point.

Magnetic meridian contains the dipole axis of earth.

In the magnetic northern hemisphere, the vertical component of the earth’s magnetic field points downwards, The north pole of a freely suspended magnet, therefore dips in northern hemisphere. The freely suspended magnet will not be horizontal.

The knowledge of declination and inclination completely specifies the direction of the earth’s magnetic field with respect to earth’s axis of rotation.

Horizontal component of earth’s magnetic field

It is the horizontal component of the earth’s magnetic field in the magnetic meridian at a point. If B is magnetic field and δ is dip at a point

Horizontal component BH = B cos δ

We can express it as B = BH/ cos δ

Determination of dip at a place

Dip at a place is measured by an apparatus known as dip circle.

Neutral point

If the horizontal component of the magnetic field due to a magnet is equal and opposite to the earth’s horizontal magnetic field at a point, the net horizontal field is zero at such a point. If a compass needle is placed at such a point, it can stay in any position. Such a point where horizontal component of earth’s magnetic field is cancelled by the horizontal field of a magnet is called neutral point.

Tangent Galvanometer

Tangent galvanometer is used to measure current. In this instrument a magnetic needed is placed at the centre of circular coil mounted in a vertical plane. When current i is passed through this apparatus, the magnetic field produced at the centre of the coil is

B = µ0in/2r

Where i = current through the coil
N = no of turns in the coil.
r = radius of the coil..

As the field is perpendicular to the plane of the coil, its direction is horizontal and perpendicular to the magnetic meridian and hence perpendicular to the horizontal component BH of the earth’s magnetic field. Hence the resultant horizontal field is

Br = √(B² + BH²)
The resultant will make angle θ with BH

tan θ = B/ BH

Hence B = BH tan θ = µ0in/2r

i = 2r BH tan θ/µ0n = Ktan θ

where K = 2r BH0n is a constant for the give galvanometer at a given place and it is called reduction factor of the galvanometer.

The reduction factor is measured by passed known current and measuring the deflection (θ) of the needle in the galvanometer.

Sensitivity of the galvanometer

Good sensitivity means that the change in deflection is large for a given fractional change in current.

We have i = K tan θ, we get dθ = ½ sin 2θ (di/i)

For good sensitivity, θ = 45°.
Hence when θ = 45°, the galvanometer is most sensitive.

Tangent Law of perpendicular fieds

When the compass needle is palced in the earth’s magnetic field it stays along the horizontal component BH of the field. The magnetic forces m BH and -m BH on the poles will be in a straight line and do not produce any torque. But if an external horizontal magnetic field B perpendicular to BH is introced, the needle deflects from its position. This concept is used in the tangent galvanometer.

The resulting force is

Br = √(B² + BH²)
The resultant will make angle θ with BH

tan θ = B/ BH

Hence B = BH tan θ

This is known as the tangent law of perpendicular fields.

Moving Coil galvanometer

In the magnetic field produced by two poles of a strong permanent magnet, a soft iron core over which rectangular insulated wire is wound is placed. The soft iron core is fixed to a torsion head through a fine strip of phosphor bronze on bottom side also it fixed through a spring made of phosphor bronze. When the current is passed through coil there is a torque on it and deflection takes place. The torque is equated to the torque produced in the phosphor bronze strip due to its twisting and current value is determined.

Moving coil galvanometer

A rectangular insulated conductor coil of several turns is wound over a soft iron core. This core with the coil is attached to a torsion head at the upper end through a strip of phosphor bronze and at the lower end to a spring made of phosphor bronze. The coil with the core is suspended between two pole pieces of a strong permanent magnet. When current is passed through the coil, as the coil is in the magnetic field B of the permanent magnet, a torque niAB acts on the coil and due to this torque the core with the coil deflected.. A small mirror is fixed to the phosphor bronze strip and a lamp scale arrangement is used to measure the deflection.

niAB = kθ

i = kθ/nAB

The constant k/nAB is called the galvanometer constant. The constant is found by passing a known current and measuring the deflection.

The sensitivity of a moving coil galvanometer is defined as θ/i. For large sensitivity, the filed B should be large. Soft iron core increases magnetic field.

Deflection magnetometer.

The instrument has a small compass needle pivoted at the centre of a graduated circular scale. This arrangement is kept in a wooden frame having two long arms having metre scales. We can find M/ BH ratio using this apparatus. M is the magnetic moment of a magnet and BH is the horizontal component of the earth’s magnetic field. This measurement can be made in Tan A position or Tan B position of the magnetometer.

Tan A position

The arms of the magnetometer are kept along the magnetic east-west direction. This direction is perpendicular to the direction of the needle. The magnet is kept on one of the arms parallel to its length. Hence the compass needle is in end-on position of the bar magnet. (End on position is a position on the magnetic axis of a bar magnet)

The magnetic field due to the bar magnet at the site of the needle is
B = (µ0/4 π)[(2Md)/(d² – l²)²]

Where l = length of the bar magnet and d = distance of the centre of the bar magnet from the centre of the compass needle.

From tangent law B = BH tan θ

Hence M/ BHt = (4π/µ0) [d² – l²)²/2d] tan θ

Tan-B position

In this position, the arms of the magnetometer are kept in the magnetic north-south direction. The bar magnet is placed on one of the arms symmetrically at right angles to the wooden arm.

In this position the compass centre point is in broadside-on position of the bar magnet.
Therefore the magnetic field due to the bar magnet at the compass centre point due to the magnet is

B = (µ0/4 π)[M/(d² + l²)3/2 ]

Using the tangent law B = BH tan θ

Hence M/ BH = (4π/µ0)[(d² + l²)3/2 ]tan θ

Applications of the deflection magnetometer

1. Comparison of the magnetic moments two magnets. If the magnetic moments of the two magnets are M1 and M2, then we can find M1/ BH and M2/ BH and then can find M1/M2.

Null method of finding M1/M2

In this method, the two magnets are placed at the opposite ends of the wooden legs of the deflection magnetometer and the distances of the magnets are adjusted on wooden arms so that deflection of the needle is zero. That means magnetic fields produced by the magnets cancel each other.

In Tan A position
B = (µ0/4 π)[(2M1d1)/(d1² – l1²)²] = (µ0/4 π)[(2M2d2)/(d2² – l2²)²]

M1/M2 = (d2/d1)[(d1² – l1²)²/)/(d2² – l2²)²]

In tan B position

B = (µ0/4 π)[M1/(d1² + l1²)3/2 ] = (µ0/4 π)[M2/(d2² + l2²)3/2 ]

M1/M2 = (d1² + l1²)3/2 /(d2² + l2²)3/2

2. The horizontal components of earth’s magnetic field at two places can be done. M/ BH1 and M/ BH2 help in this.

3. Verification of inverse law. The magnetic field due to a magnetic pole is inversely proportional to the distance of the point from the pole.

From the M/ BH expression we can find an expression for cot θ
And a graph is plotted between [d² – l²)²/2d] and cot θ for Tan A position readings. We get a straight line

Similarly a graph is plotted between [(d² + l²)3/2 ] and cot θ for readings in Tan B position. We get a straight line.

Oscillation magnetometer

Using oscillation magnetometer, the oscillations of bar magnet and the taken are determined.

T = time per one osciallation = 2 π√(I/MBH)

That gives MBH = 4π²I/T²

I moment of inertia of the bar magnet can be determined from I = W(a²+b²)/12 where W = mass of the magnet, a = geometric length, b = breadth of the magnet.

Determination of M and BH

We can find M/ BH using a deflection magnetometer and M BH using an oscillation magnetometer.

If M/ BH = X abnd M BH= Y

Then M² = XY

And M = √(XY)

BH = √(Y/X)

Gauss’s law for Magnetism

Magnetic field due to magnetic charge is

B = (µ0/4 π)(m/r²)

From the above expression we can derive Gauss’s law for magnetism as

∮B.ds = µ0minside

where ∮B.ds is the magnetic flux and minside is the “net pole strength inside the closed surface. But net magnetic stranegh enclosed by any closer surface is zero.

Hence ∮B.ds = 0