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Let Q be a point in end on position (It is a position lying on the magnet axis of the bar magnet) at a distance x from the centre of the short bar magnet with magnetic length 2L.

We consider NS is a short bar magnet with magnetic length 2L and pole strength m such that \[\left( {L < < x} \right)\] as shown below.

Now, the magnetic field at point Q due to north-pole is:-

\[{B_N} = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{m}{{{{(x - L)}^2}}}\], with its direction is away from the magnet.

Similarly, the magnetic field at point Q due to south-pole is:-

\[{B_s} = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{m}{{{{(x + L)}^2}}}\], with its direction is towards the magnet.

The Resultant magnetic field at the point Q is:

\[{\vec B_R} = {\vec B_s} + {\vec B_N}\]

Since,\[{\vec B_N}\] and \[{\vec B_s}\] are opposite in direction as shown in the above diagram.

So, Magnitude of the resultant magnetic field is written as:

\[{B_R} = {B_N} - {B_s}\]…………… (i)

Substitute the values of \[{B_N}\]and \[{B_s}\]in the eqn (i), we get:

\[{B_R} = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{m}{{{{(x - L)}^2}}} - \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{m}{{{{(x + L)}^2}}}\]

\[ = \dfrac{{{\mu _0}m}}{{4\pi }}\left[ {\dfrac{1}{{{{(x - L)}^2}}} - \dfrac{1}{{{{(x + L)}^2}}}} \right]\]

\[ \Rightarrow {B_R} = \dfrac{{{\mu _0}m}}{{4\pi }}\left[ {\dfrac{{4Lx}}{{{{({x^2} - {L^2})}^2}}}} \right]\]

Hence, the expression of magnitude for the intensity of magnetic field in end on position is

\[ \Rightarrow {B_R} = \dfrac{{{\mu _0}m}}{{4\pi }}\dfrac{{4Lx}}{{{{({x^2} - {L^2})}^2}}}\]

We can also write it in the term of magnetic moment of the bar magnet as:

\[ \Rightarrow {B_R} = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{2Mx}}{{{{({x^2} - {L^2})}^2}}}\]………… (ii) [Where we take\[\left( {M = 2mL} \right)\]which is the magnetic moment of the bar magnet.]

Applying small-size approximation in eqn (ii)

Since, \[\left( {L < < x} \right)\] therefore we can ignore L in the equation (ii), we get

\[ \Rightarrow {B_R} = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{2Mx}}{{{{({x^2})}^2}}}\]

\[ = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{2M}}{{{x^3}}}\]

Hence, the Intensity of magnetic field for short size bar magnet at the end-on position is expressed as:

\[{B_R} = \dfrac{{{\mu _0}}}{{4\pi }}\dfrac{{2M}}{{{x^3}}}\]