Magnetic Induction and Magnetic Potential due to a Bar Magnet

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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.

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

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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

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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)

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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.

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

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



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  • Resolving the magnetic induction B& 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 hence they reinforce (support) each other. Let B be the resultant magnetic induction at P then

Magnetic Induction 09

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)

Magnetic Induction 10



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:

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  • 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

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  • 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

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  • 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.

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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



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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

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Case – 2

  • If P is a point on the equator of the dipole, then θ = 90° and Cos θ = 0

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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

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

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  • 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

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

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

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Science > Physics > Magnetism >You are Here
Physics Chemistry Biology Mathematics

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