The Neutral Point in a Magnetic Field

A bar magnet placed on a horizontal table experiences the influence of both the horizontal component of Earth's magnetic field and its own magnetic moment. The neutral point refers to the location where the magnetic field due to the magnet and the Earth's magnetic field cancel each other out, resulting in no net magnetic field at that point.

Magnetic Moment of the Bar Magnet

The magnetic moment of a bar magnet is a measure of its strength and is given by the product of its pole strength and the distance between its poles. It is denoted by the symbol 'm' and has the unit A.m² (ampere meter squared). In this case, the magnetic moment of the bar magnet is given as 1.44 A.m².

Earth's Magnetic Field

Earth has its own magnetic field, which can be approximated as a dipole field. The horizontal component of Earth's magnetic field is the component that is parallel to the surface of the Earth. In this case, the magnitude of the horizontal component is given as 18 μT (microtesla).

Neutral Point Calculation

To determine the distance of the neutral point from the magnet, we need to consider the balance of magnetic forces acting on the magnet. At the neutral point, the magnetic field due to the magnet and the Earth's magnetic field must cancel each other out.

The magnetic field due to a bar magnet at a point on its axial line is given by the formula:

B = (μ₀/4π) * (2m/d³)

Where:
B is the magnetic field at the point
μ₀ is the permeability of free space (4π × 10⁻⁷ T.m/A)
m is the magnetic moment of the bar magnet
d is the distance of the point from the center of the magnet

To find the neutral point, we equate the magnetic field due to the magnet to the Earth's magnetic field and solve for d.

(μ₀/4π) * (2m/d³) = B_earth

Substituting the given values:
(4π × 10⁻⁷ T.m/A) * (2 * 1.44 A.m² / d³) = 18 × 10⁻⁶ T

Simplifying the equation:
2.88 × 10⁻⁷ / d³ = 18 × 10⁻⁶ / (4π × 10⁻⁷)

Further simplification:
d³ = (4π × 10⁻⁷ * 18 × 10⁻⁶) / (2.88 × 10⁻⁷)

d³ = 28.274

Taking the cube root of both sides:
d ≈ 3.04 cm

Therefore, the neutral point is approximately 3.04 cm away from the magnet.