Derive an expression for the magnetic field intensity at any point on the axial line of a bar magnet. Also, write the condition for a short bar magnet.

Q: Derive an expression for the magnetic field intensity at any point on the axial line of a bar magnet. Also, write the condition for a short bar magnet.

Introduction to Magnetic Field of a Bar Magnet\nA bar magnet behaves like a magnetic dipole consisting of two equal and opposite magnetic poles (North and South) separated by a small distance. The magnetic field intensity at any point in space due to this bar magnet can be calculated using Coulomb's law for magnetism or by treating it as an equivalent solenoid. Derivation for Axial Line\nLet us consider a bar magnet of length $2l$ and pole strength $m$. Let $P$ be a point on its axial line at a distance $r$ from the center $O$ of the magnet. Magnetic field at point P due to North Pole ($N$): The distance of point $P$ from the North pole is $(r - l)$. The magnetic field $B1$ directed away from the North pole is given by: $$B1 = \frac{\mu0}{4\pi} \frac{m}{(r - l)^2}$$ Magnetic field at point P due to South Pole ($S$): The distance of point $P$ from the South pole is $(r + l)$. The magnetic field $B2$ directed towards the South pole is given by: $$B2 = \frac{\mu0}{4\pi} \frac{m}{(r + l)^2}$$ Resultant Magnetic Field ($B$): Since $B1$ and $B2$ act along the same straight line in opposite directions, the magnitude of the resultant magnetic field at point $P$ is: $$B = B1 - B2 = \frac{\mu0}{4\pi} m \left \frac{1}{(r - l)^2} - \frac{1}{(r + l)^2} \right$$ $$B = \frac{\mu0}{4\pi} m \left \frac{(r + l)^2 - (r - l)^2}{(r^2 - l^2)^2} \right$$ $$B = \frac{\mu0}{4\pi} \frac{m (4rl)}{(r^2 - l^2)^2}$$ We know that the magnetic dipole moment $M = m \times 2l$. Substituting this: $$B = \frac{\mu0}{4\pi} \frac{2Mr}{(r^2 - l^2)^2}$$ Special Case for a Short Bar Magnet\nIf the magnet is very small compared to the distance $r$ (i.e., $l \ll r$), then $l^2$ can be neglected in comparison to $r^2$: $$B = \frac{\mu0}{4\pi} \frac{2Mr}{r^4} = \frac{\mu0}{4\pi} \frac{2M}{r^3}$$ Conclusion\nThus, the magnetic field intensity at an axial point of a short bar magnet is inversely proportional to the cube of the distance from its center.

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Introduction to Magnetic Field of a Bar Magnet
A bar magnet behaves like a magnetic dipole consisting of two equal and opposite magnetic poles (North and South) separated by a small distance. The magnetic field intensity at any point in space due to this bar magnet can be calculated using Coulomb's law for magnetism or by treating it as an equivalent solenoid. Derivation for Axial Line
Let us consider a bar magnet of length $2l$ and pole strength $m$. Let $P$ be a point on its axial line at a distance $r$ from the center $O$ of the magnet. Magnetic field at point P due to North Pole ($N$): The distance of point $P$ from the North pole is $(r - l)$. The magnetic field $B1$ directed away from the North pole is given by: $$B1 = \frac{\mu0}{4\pi} \frac{m}{(r - l)^2}$$ Magnetic field at point P due to South Pole ($S$): The distance of point $P$ from the South pole is $(r + l)$. The magnetic field $B2$ directed towards the South pole is given by: $$B2 = \frac{\mu0}{4\pi} \frac{m}{(r + l)^2}$$ Resultant Magnetic Field ($B$): Since $B1$ and $B2$ act along the same straight line in opposite directions, the magnitude of the resultant magnetic field at point $P$ is: $$B = B1 - B2 = \frac{\mu0}{4\pi} m \left \frac{1}{(r - l)^2} - \frac{1}{(r + l)^2} ight$$ $$B = \frac{\mu0}{4\pi} m \left \frac{(r + l)^2 - (r - l)^2}{(r^2 - l^2)^2} ight$$ $$B = \frac{\mu0}{4\pi} \frac{m (4rl)}{(r^2 - l^2)^2}$$ We know that the magnetic dipole moment $M = m 	imes 2l$. Substituting this: $$B = \frac{\mu0}{4\pi} \frac{2Mr}{(r^2 - l^2)^2}$$ Special Case for a Short Bar Magnet
If the magnet is very small compared to the distance $r$ (i.e., $l \ll r$), then $l^2$ can be neglected in comparison to $r^2$: $$B = \frac{\mu0}{4\pi} \frac{2Mr}{r^4} = \frac{\mu0}{4\pi} \frac{2M}{r^3}$$ Conclusion
Thus, the magnetic field intensity at an axial point of a short bar magnet is inversely proportional to the cube of the distance from its center.

CBSE · Class 12 · Physics · Magnetism and MatterDerive an expression for the magnetic field intensity at any point on the axial line of a bar magnet. Also, write the condition for a short bar magnet.

Derivation for Axial Line\nLet us consider a bar magnet of length $2l$ and pole strength $m$. Let $P$ be a point on its axial line at a distance $r$ from the center $O$ of the magnet.

Special Case for a Short Bar Magnet\nIf the magnet is very small compared to the distance $r$ (i.e., $l \ll r$), then $l^2$ can be neglected in comparison to $r^2$:

Conclusion\nThus, the magnetic field intensity at an axial point of a short bar magnet is inversely proportional to the cube of the distance from its center.