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**Magnetic Induction:**

**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 south pole to north pole.

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

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

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

- Consider south pole. Magnetic induction at a point on the axis due to 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 magnetic dipole moment is from 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 north pole. Magnetic induction at P due to north pole is given by

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

- Resolving the magnetic induction B
_{1 }& B_{2}along the axis of the magnet and the along the equator of the magnet.The components B_{1}sinθ and B_{2}sinθ are equal & opposite hence cancel each other. The component B_{1}cos θ and B_{2}cos θ are in the same direction hence 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 north pole to south pole.

**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 represent 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:**

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

**Expression for a Potential at Any Point Due to a Short Magnetic Dipole:**

- Consider a short magnetic dipole NS. Let 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 cntre O and the axis of the dipole (angle between OP and SN).

Now the magnetic potential due to north pole of magnetic dipole is given by

The magnetic potential due to north pole of magnetic dipole is given by

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

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