Two short bar magnets of equal dipole moment M are fastened...

Two short bar magnets of equal dipole moment $M$ are fastened perpendicularly at their centres, figure. The magnitude $T_{N}$ of resultant of two magnetic field at a distance $d$ from the centre on the bisector of the $P$ right angle is (a) $\frac{\mu_{e}}{4 m} \frac{2 \sqrt{2} M}{d^{7}}$ (b) $\frac{\mu_{0}}{4 \pi} \frac{2 M}{d^{3}}$ (c) $\frac{\mu_{e}}{4 \pi} \frac{M}{d^{4}}$ (d) $\frac{\mu_{0}}{4 \pi} \frac{\sqrt{2} M}{d^{3}}$

Two short bar magnets of equal dipole moment $M$ are fastened perpendicularly at their centres, figure. The magnitude $T_{N}$ of resultant of two magnetic field at a distance $d$ from the centre on the bisector of the $P$ right angle is
(a) $\frac{\mu_{e}}{4 m} \frac{2 \sqrt{2} M}{d^{7}}$
(b) $\frac{\mu_{0}}{4 \pi} \frac{2 M}{d^{3}}$
(c) $\frac{\mu_{e}}{4 \pi} \frac{M}{d^{4}}$
(d) $\frac{\mu_{0}}{4 \pi} \frac{\sqrt{2} M}{d^{3}}$

Key Concepts

Magnetic Dipole Moment

This is a fundamental quantity that characterizes the strength and orientation of a magnetic source. It is analogous to an electric dipole moment in electrostatics and plays a crucial role in determining the magnetic field produced by magnets or current loops.

Magnetic Field of a Dipole

The magnetic field generated by a dipole decreases with distance, typically following an inverse cube law. This concept is essential when analyzing how the magnetic influence of a dipole changes in space, and forms the basis for more complex magnetic field arrangements.

Vector Addition of Magnetic Fields

Since magnetic fields are vector quantities, their magnitudes and directions must be considered when they are combined. Proper vector addition allows one to determine the resultant magnetic field in cases where multiple fields interact, such as when dipoles are arranged perpendicularly.

Distance Dependence in Dipole Field Calculations

When dealing with dipole fields, an understanding of the fall-off behavior with distance is essential. The field strength typically decreases as the inverse cube of the distance from the dipole, a property that influences both the magnitude and spatial distribution of the resultant field.

Superposition Principle

In electromagnetism, the superposition principle states that the total magnetic field at any point due to multiple sources is the vector sum of the fields produced by each individual source. This principle is critical when calculating the net field emerging from the combination of different dipoles.

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