Magnetic Induction at a Point on the Axis and the Equator

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

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