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 Discussion Response Post to: Bar magnets and magnetic field
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edison (9095)

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Magnetic Field Lines

The compass needle always lies along the direction of the field. The figure below shows the lines or pattern of the field, when the compass needle is placed at several places. These lines do not really, tell us the effect that magnet has on the other.

These field lines are developed to visualize the effect of the magnetic field. If we imagine a number of small compass needles around a magnet, each compass needle experiences a torque to the field of the magnet. The path along which this compass needles are aligned is known as magnetic lines of force.

The arrangement of iron fillings surrounding a bar magnet. The pattern mimics magnetic field lines. They suggest that the bar magnet is a magnet is a magnetic dipole.

Properties of Magnetic Lines of Force

• Magnetic lines of force form closed continuous curves.
• Outside the body of the magnet, their direction is from north to south pole.
• The tangent to these lines at any point gives the direction of the magnetic field at that point.
• No two lines can intersect each other.
• The lines of force contract longitudinally and dilate laterally.

• Crowding of magnetic lines of force represents stronger magnetic field and vice-versa.

Note:

The following diagram depicts the magnetic lines of force between two north pole, two south pole; North-South pole.

Bar Magnet as an Equivalent Solenoid

Inserting an iron core increases the strength of the field

On comparing the two-field pattern, the current carrying solenoid from outside resembles a bar magnet. Inside the solenoid there is a strong magnetic field, which can magnetise a specimen. Solenoid is hollow from inside whereas the bar magnet is solid.

Here we note the close similarity between the magnetic field lines due to a solenoid.

A more efficient way to make a magnet is to place the steel rod inside a solenoid and run a current. Then the magnetic field of a solenoid magnetises the rod as well.

If we hold the compass needle in various directions at each point and for each direction measure the torque exerted on it by the field.

If the magnetic dipole moment of the needle is 'm', the torque is given by

The similarity in the behavior of electric and magnetic dipoles in electric and magnetic dipoles in electric and magnetic fields respectively can be made use of to obtain an expression for the potential energy of a magnetic dipole in a given magnetic field. The potential energy of an electric dipole of moment p at that point r in an electric field E is p.E(r) and the torque is p x E (r).

By a similar argument, it follows that a magnetic dipole of moment m situated at the point r in the field B has the potential energy m.B.

In the case of a solenoid, the direction of m is related to the sense of flow of current. The solenoid behaves like a bar magnet. Like a bar magnet the stable orientation of the solenoid corresponds to m parallel to B, the unstable equilibrium corresponds to m anti parallel to B.

Magnetic Field Intensity due to a Bar Magnet Along the Axis

Consider a point P located on the axial line of a short bar magnet of magnetic length '2l' and pole strength m. Let us find at a point p which is at a distance d from the centre of the magnet. Magnetic flux density at p due to N-pole is

Magnetic flux density at p due to s-pole is

Net magnetic flux density at P is

Since the bar magnet is short d >> l, l2 can be neglected as compared to d2.

Magnetic Field Intensity on the Equatorial Line of Bar Magnet (Magnetic Dipole)

Let P be a point on the equatorial line of a short bar magnet where flux density due to the magnet is to be found out.

SN = 2l,

Magnetic flux density at p due to N-pole is

Magnetic flux density at p due to s-pole is

B1 = B2

The resultant magnetic flux density at P is

Since the bar magnet is short d >> l, l2 can be neglected as compared to d2.

Note:

a) In the above derivation, we have assumed a unit north pole to be placed at P and that the magnetic field intensity is force experienced by that unit pole. This is the coulombian magnetic force.

Where m1 is pole strength of North Pole of a bar magnet and l represents the unit north pole at P.

b) The magnetic field due to a short magnet at any point on axial line of magnet is twice the magnetic field at a point on the equatorial line of the magnet at the same distance.

Torque on a Bar Magnet in a Uniform Magnetic Field

The behavior of a bar magnet in a uniform magnetic field is similar to that of electric dipole in a uniform electric field. The above figure.

Let 2l be the magnetic length of a magnet and m be the strength of each pole.

These two forces, being equal and opposite having different lines of action constitute a couple. This couple tends to rotate the magnet so as to align it along .

A paradox is an argument that starts with apparently acceptable assumptions and leads by apparently valid deductions to an apparent contradiction. Since logic admits no contradictions, either the apparently acceptable assumptions are not acceptable, or the apparently valid deductions are not valid, or the apparent contradiction is not a contradiction. A paradox moves us to reexamine the argument until we find out what is wrong.
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