In this article, we shall study to derive an expression for magnetic induction and magnetic potential at any point in a magnetic field created by a bar magnet.

Magnetic Induction at Any Point Due to a Short Bar Magnet:

Consider a short magnetic dipole NS.  Let  be the magnetic moment of the dipole

M = m x 2l ………………(1)

The direction of magnetic induction is along the axis from S-pole to N-pole inside the magnet.

Consider a point ‘P’ near the dipole at distance ‘r’ from its centre O. i.e. OP = r Let ‘ θ’ be the angle between the line joining the point from the centre O and the axis of the dipole (angle between OP and SN). Resolving magnetic moment into two mutually perpendicular components, we have,  the component M Cosθ along OP and M Sinθ perpendicular to OP.

Now, the point P lies on the axis of M Cosθ. Hence, the magnetic induction at, the axis point of M Cos θ is given by

Also, the given point P lies on the equatorial-line of component M Sin θ. Hence, the magnetic induction at the equatorial point of M Sin θ is given by

Let B₁ and B₂ be represented by sides PQ and PT of completed parallelogram PQRT. The diagonal PR represents the resultant magnetic induction in magnitude and direction.

This is the magnitude of the resultant induction B at point P.

Let ∝ be the angle made by the resultant B with the direction of OP

This is the angle made by B with OP.  Hence, the total inclination of the resultant induction B with the axis of the dipole is  ( θ + ∝ )

Special cases:

Case 1: If P is a point on the axis of the dipole, then θ = 0° or θ = 180° and Cos θ = ± 1

Case – 2: If P is a point on the equator of the dipole, then θ = 90° and Cos θ = 0

Magnetic Potential at Any Point Due to a Short Bar Magnet:

The magnetic potential at a point in a magnetic field is defined as the work done in moving unit north pole from infinity to that point. It is denoted by ‘V’ and its S.I. unit is J/Am or Wb/m.

In free space, the magnetic potential at a point due to the magnetic pole of strength ‘m’ units and at a distance, r is given by

Consider a short magnetic dipole NS.  Let M be the magnetic moment of the dipole

M = m x 2l ………………(1)

The direction of M is along the axis from S-pole to N-pole.

Consider a point ‘P’ near the dipole at distance ‘r’ from its centre O. i.e. OP = r. Let ‘θ’ be the angle between the line joining the point from the centre O and the axis of the dipole (angle between OP and SN).

Now the magnetic potential due to the North Pole of a magnetic dipole is given by

The magnetic potential due to the North Pole of a magnetic dipole is given by

Since the magnetic potential is a scalar quantity, the resultant potential at a point P is given by

Case 1: If P is a point on the axis of the dipole, then θ = 0° or θ = 180° and Cos θ = ± 1

Case – 2: If P is a point on the equator of the dipole, then θ = 90° and Cos θ = 0

Hence V_equator = 0

Science > Physics > Magnetism > Magnetic Induction and Magnetic Potential at any Point

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In this article, we shall study magnetic induction at the point on the magnetic axis and magnetic equator of bar magnet.

Magnetic Induction at a Point on Axis of Bar Magnet:

The line passing through the poles of a bar magnet is called the axis of the magnet. Consider a bar magnet having pole strengths +m & -m and magnetic length to ‘2l’. The magnetic dipole moment vector is given by

M = m × 2l …….. (1)

Its direction is from the south pole to the north pole.

Consider point P on the axis of the magnet at a distance of ‘r’ from the centre of magnet O.

Consider the north pole. Magnetic induction at P due to the north pole is given by

The direction of magnetic induction is away from the north pole and along the axis of the magnet.

Consider the south pole. Magnetic induction at a point on the axis due to the south pole is given by

The direction of magnetic induction is towards the south pole along the axis of the magnet. Let B be the resultant magnetic induction at P

Then,    B = B₁ +  B₂ ………….(4)

This is an expression for magnetic induction at a point on the axis of a bar magnet.

For short bar magnet, l is very less than r. (l << r), hence l can be neglected. (i.e.  l = 0)

This is an expression for magnetic induction at a point on the axis of the short bar magnet.

Magnetic Induction at a Point on Equator of Bar Magnet:

The perpendicular bisector of the segment joining the north pole and south pole of a bar magnet is called equator of the magnet.

Consider a bar magnets having pole strength +m & -m & m.l. 2l the magnetic dipole movement vector is given by

M = m × 2l ………….. (1)

The direction of the magnetic dipole moment is from the south pole to north pole.

Let P be the point on the equator of a bar magnet at a distance of r from the centre of magnet O.

Consider the north pole. Magnetic induction at P due to the north pole is given by

Consider the south pole. Magnetic induction at P due to the south pole is given by

Resolving the magnetic induction B₁ & B₂ along the axis of the magnet and the along the equator of the magnet. The components B₁sinθ and B₂sinθ are equal & opposite hence cancel each other. The component B₁ cos θ and B₂ cos θ are in the same direction where they reinforce (support) each other. Let B be the resultant magnetic induction at P then

This is an expression for Magnetic induction at a point on the equator of the bar magnet.

For short bar magnet (l << r). l is small so can be neglected. (l = 0)

This is an expression for magnetic induction at a point on the equator of a short bar magnet. Its direction is from the north pole to south pole.

Numerical Problems:

Example – 01:

Find the magnetic induction at a point distant 10 cm on the axis of a short bar magnet of moment 0.2 Am².

Given: Distance from centre = r = 10 cm = 0.1 m, Magnetic moment = 0.2 Am². Point on axis.

To Find: B =?

Solution:

Ans: The magnetic induction at the point is 4 x 10^-5 T

Example – 02:

Find the magnetic induction at a point distant 20 cm on the equator of a short bar magnet of moment 5 Am².

Given: Distance from centre = r = 20 cm = 0.2 m, Magnetic moment = 5 Am²., The point on the equator.

To Find: Magnetic induction= B =?

Solution:

Ans: The magnetic induction at the point is 6.25 x 10^-5 T

Example – 03:

Find the magnetic induction at a point 0.5 m from either pole on the equator of a bar magnet of moment 5 Am².

Given: Distance from either pole = 0.5 m, Magnetic moment = 5 Am².

To Find: B =?

Solution:

As the point is equidistant from either pole it is on the equator of the bar magnet

∴ r² + l² = (0.5)² = 0.25

Ans: The magnetic induction at the point is 4 x 10^-6 T

Example – 04:

Find magnetic induction due to a short bar magnet at a point distant 10 cm a) on its axis and b) on its equator, given that the magnetic dipole moment of the magnet is 0.25 Am².

Given: Distance from centre = r = 10 cm = 0.1 m, Magnetic moment = 0.25 Am².

To Find: B_axis =? B_equator =?

Solution:

Ans: The magnetic induction at the point on the axis is 5 x 10^-5 T and

that on the equator is 2.5 x 10^-5 T

Example – 05:

A bar magnet has pole strength 10 Am and a magnetic length of 5 cm. Find B at equidistant point of 10 cm from either pole.

Given: Distance from either pole = 10 cm = 0.1 m, pole strength = m = 10 Am, magnetic length = 2l = 5 cm = 0.05 m

To Find: B =?

Solution:

Magnetic Moment = M = m . 2l = 10 x 0.05 = 0.5 Am²

As the point is equidistant from either pole it is on the equator of the bar magnet

∴ r² + l² = (0.1)² = 0.01

Ans: The magnetic induction at the point is 5 x 10^-6 T

Example – 06:

Find magnetic induction due to a short bar magnet at a point distant 20 cm a) on its axis and b) on its equator, given that the magnetic dipole moment of the magnet is 0.5 Am².

Given: Distance from centre = r = 20 cm = 0.2 m, Magnetic moment = 0.5 Am².

To Find: B_axis =? B_equator =?

Solution:

Ans: The magnetic induction at the point on axis is 1.25 x 10^-5 T and

that on equator is 6.25 x 10^-6 T

Example – 07:

Find the magnetic induction at a point distant 8 cm on the equator from the centre of a short bar magnet of moment 0.2 JT^-1.

Given: Distance from centre = r = 8 cm = 0.08 m, Magnetic moment = 0.2 JT^-1, Point on equator.

To Find: B =?

Solution:

Ans: The magnetic induction at the point is 3.91 x 10^-5 T

Example – 08:

Find the magnetic induction at a point distant 10 cm from the centre of the magnet on the axis of a short bar magnet of moment 0.24 JT^-1.

Given: Distance from centre = r = 10 cm = 0.1 m, Magnetic moment = 0.24 JT^-1. Point on axis.

To Find: B =?

Solution:

Ans: The magnetic induction at the point is 4.8 x 10^-5 T

Example – 09:

Find the magnetic induction due to a bar magnet of magnetic induction 0.5 Am² at a point on its axis at a distance of 15 cm from the nearest pole. The magnetic length of magnet is 10 cm.

Given: Magnetic moment = M = 0.5 Am², distance from nearest pole = 15 cm = 0.15 m, magnetic length = 2 l = 10 cm, l = 5 cm = 0.05 m, distance of point from centre = r = 0.15 + 0.05 = 0.20 m,  μ_o/4π = 10^-7 Wb/Am.

To Find: B_axis =?

Solution:

Ans: The magnetic induction at the point is 1.422 x 10^-5 T

Example – 10:

The magnetic induction at a point on the equator of a magnetic dipole at a distance of 10 cm from its centre is 5 x 10^-5 Wb/m². Calculate the magnetic moment of the magnet.

Given: Distance from centre = r = 10 cm = 0.1 m, Magnetic moment = B = 5 x 10^-5 Wb/m^2, Point on equator.

To Find: Magnetic moment = M =?

Solution:

Ans: The magnetic moment is 0.5 Am²

Example – 11:

The magnetic induction at a point on the axis at a distance 20 cm from the centre of the magnet is 1.5 x 10^-5 Wb/m². Find the magnetic induction at a point on the equator at the same distance from the centre.

Given: distance of point from centre of magnet = r = 20 cm for both the points. Magnetic induction = B_axis = 1.5 x 10^-5 Wb/m²,

To Find: B_equator =?

Solution:

Ans: Magnetic induction at a point on equator is 7.5 x 10^-6 Wb/m².

Example – 12:

The strength of magnetic field at a point on axis of a magnetic dipole at a distance of 10 cm from its centre is 4 x 10^-5 Wb/m². Calculate the magnetic moment of the magnet.

Given: Distance from centre = r = 10 cm = 0.1 m, Magnetic moment = B = 4 x 10^-5 Wb/m^2, Point on axis.

To Find: M =?

Solution:

Ans: The magnetic moment is 0.2 Am²

Example – 13:

The strength of magnetic field at point P on the axis of short bar magnet is equal to the magnetic induction at point Q on the equatorial line. Find the ratio of the distances from the centre of magnet.

Given: B_axis = B_equator,

To Find: ratio of distances r_P:r_Q = ?

Solution:

Ans: The required ratio of distances is 2^1/3: 1

Example – 14:

Earth’s magnetic field may be imagined to be due to a magnetic dipole located at the centre of the earth. If the magnetic field at a point on the magnetic equator is 3 x 10^-5 Wb/m². What is the magnetic moment of such a magnet? What is the value of earth’s magnetic field at the north pole of the earth? Radius of the earth = 6400 Km.

Given: B = 3 x 10^-5 Wb/m², Point on Equator, distance from centre = r = 6400 km = 6.4 x 10⁶ m

To Find: Magnetic moment = M =? Magnetic field at the north pole =?

Solution:

Ans: The magnetic moment of magnet is 7.86 x 10²² Am²

and magnetic induction at north pole is 6 x 10^-5 T.

Example – 15:

A magnet has magnetic length 0.10 m and pole strength 12 Am. Find the magnitude of the magnetic field B at a point on its axis at a distance of 0.02 m from centre.

Given: Magnetic length = 2l = 0.10 m, l = 0.05 m, pole strength = 12 Am, distance from the centre = r = 0.02 m, μ_o/4π = 10^-7 Wb/Am.

To Find: B_axis =?

Solution:

Ans: Magnetic induction on the axis is 3.4 x 10^-5 T.

Example – 16:

Two short magnets A and B of magnetic moments M₁ = 2.7 Am² and M₂ = 3.2 Am² respectively are kept as shown. Find the resultant magnetic field due to the magnets at P. r₁ = 30 cm and r₂ = 40 cm.

Given: M₁ = 2.7 Am², M₂ = 3.2 Am², r₁ = 30 cm = 0.3 m, and r₂ = 40 cm = 0.4 m.

Science > Physics > Magnetism > Magnetic Induction at a Point on the Axis and the Equator

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The branch of physics which deals with the study of earth’s magnetic field is called terrestrial magnetism or geomagnetism. The magnetism on the surface of the earth can be approximately represented by assuming a huge imaginary bar magnet in the interior of the earth. The source of magnetism of the earth is due to the electric current in the magma of the earth at the depth of about 3000 km.

Evidence of Geomagnetism:

The earth is a giant magnet. It is evident from the following examples.

A freely suspended magnetic needle always rests in the north-south direction:

The earth itself is a giant magnet. Its magnetic North pole is near the geographical South pole. Its magnetic South pole is near the geographical North pole. Now, unlike poles of a magnet always attract each other and like poles of the magnet always repel each other. Thus the north pole of the suspended magnet gets attracted towards the magnetic south pole of the earth (geographical north). Similarly, the south pole of the suspended magnet gets attracted towards the magnetic north pole of the earth (geographical south).

Thus the magnet when suspended in the air such that it is free to rotate about a transverse axis passing through its centre, always comes to rest in the north-south direction. Which confirms the existence of magnetic field by the earth.

An iron rod buried inside earth along N-S direction becomes a magnet:

As the magnet is buried inside the earth in the north-south direction, it is acted upon by the earth’s magnetic field. Due to which magnetism is induced in the iron rod and it behaves like a magnet. Which confirms the existence of the magnetic field by the earth.

Existence of neutral point in the magnetic field created by a magnet:

When a magnet is placed in a horizontal plane with its north pole facing geographical north and its south pole facing geographical south and magnetic lines of force created by the magnet are plotted.  We obtain two neutral points one on either side of the magnet on its broadside position as shown

When a magnet is placed in a horizontal plane with its north pole facing geographical south and its south pole facing geographical north and magnetic lines of force created by the magnet are plotted.  We obtain two neutral points one on either side of the magnet on its end on the position as shown

At these points, the magnetic field is zero. If we keep a magnetic needle at these points, it shows any direction. At these points, the magnetic field created by a magnet is balanced by the horizontal component of the earth’s magnetic field. Thus neutral points are the points where the magnetic field of a magnet is equal in magnitude but opposite in direction to the earth’s magnetic field. The existence of neutral points confirms the existence of the magnetic field by the earth.

A magnetic needle rests with its geometrical axis making different angles at different places on the earth:

If a magnetic needle is suspended in such a way that it is free to rotate in a vertical plane and it is taken around the earth through the earth’s geographical poles and following phenomena are observed:

• We obtain two points on the earth at which the magnetic needle is vertical i.e. it is perpendicular to the earth’s surface. The two places at which the magnetic needle becomes vertical are called magnetic poles. At magnetic poles, the magnetic lines of force by earth are perpendicular to the surface of the earth.
• We obtain two points on the earth at which the magnetic needle is horizontal i.e. it is parallel to the earth’s surface. The line joining these two places is called the magnetic equator. At the magnetic equator, the magnetic lines of force by earth are parallel to the surface of the earth.
• At all other places, the needle comes to rest making different angles with the horizontal. At these places, the magnetic lines of force by earth are inclined to the surface of the earth.

These situations confirm the existence of the magnetic field by the earth.

Earth as a Magnet (Geomagnetism):

The earth itself is a giant magnet. Its magnetic North pole is near the geographical South pole. Its magnetic South pole is near the geographical North pole. The magnetic axis of the earth is inclined at 20° to its geographical axis. North pole of the earth’s magnet is in Antarctica while its south pole is in Northern Canada. The magnetic equator passes through India, near Trivandrum, Kerala.

The magnetic field at the surface of the earth ranges from 3 × 10^-5 T at the equator to 6 × 10^-5 T at the poles. Note that the magnetic field on the axis is twice at that on the equator. The strength of the magnetic field of the earth is of the order of 1 oersted or 1 gauss (10^-4 T).

The strength of the magnetic field on the earth changes from place to place and w.r.t. time. When the change in the strength of the magnetic field is considerable or erratic, it is called the magnetic storm. It is not related to the climactic storm. It is related to the variation of the electric current in the earth’s atmosphere. There is a correlation between magnetic storms and sunspots.

Terminology of Geomagnetism:

• A straight line passing through the geographical poles is called the geographical axis or polar axis of the earth. It is also the axis of rotation of the earth.
• A vertical plane at a place passing trough geographical north and south poles of the earth is called the geographic meridian at that place.
• The great circle on the surface of the earth, in a plane perpendicular to the geographical axis, is called the geographic equator. All the points on the geographic equator are at equal distances from the geographical poles.
• A straight line passing through the magnetic poles is called the magnetic axis of the earth.
• A vertical plane at a place passing through magnetic north and south poles of the earth is called the magnetic meridian at that place.
• The great circle on the surface of the earth, in a plane perpendicular to the magnetic axis, is called the magnetic equator. All the points on the magnetic equator are at equal distances from the magnetic poles.
• The angle between the magnetic meridian and the geographic meridian at a place is called the declination or variation at that place. There is a periodic variation in the value of declination at a place.

At a given place the Earth’s magnetic field can be resolved into horizontal and vertical components. The angle made by the Earth’s magnetic field with its horizontal component is called the angle of dip or angle of inclination. The value of the angle of dip changes from place to place. It depends on the strength of the magnetic field of the earth at that place. Its value is zero at the magnetic equator and 90° at Earth’s magnetic poles. The angle of dip at a place is measured by a device called dip circle.

As the magnetic needle is moved towards the north pole or south pole, the angle of dip increases. In the northern hemisphere, the north pole of the magnetic needle dips whereas in the southern hemisphere the north pole  of the magnetic needle dips

The lines drawn up on the map through places having the same declination are called isogonic lines.

The line drawn through places that have zero declination is known as an agonic line.

The line joining all the places on the earth having the same angle of dip or inclination is called the isoclinic line.

The line drawn through places that have zero angles of dip or inclination is known as the clinic line and it is the magnetic equator.

The earth’s magnetic field creates a spectacular, visual effect called ‘aurora borealis’ in the north and ‘aurora australis’ in the south.  The solar winds consisting of the stream of electrons and protons are trapped at magnetic poles, above the atmosphere. When moving down these winds ionises molecules of the atmosphere, giving rise to the aurora effect.

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