Derive an expression for magnetic field induction on the equatorial line of a bar magnet.

To derive an expression for the magnetic field induction along the equatorial line of a bar magnet, we first need to understand the fundamental properties of a bar magnet and the nature of its magnetic field. A bar magnet can be modeled as a magnetic dipole, which has a north and a south pole. The magnetic field produced by this dipole can be analyzed using the concept of magnetic field lines and the dipole moment.

Understanding the Magnetic Dipole Moment

The magnetic dipole moment (m) of a bar magnet is defined as:

• m = m * l,

where m is the pole strength and l is the length of the magnet. The direction of the dipole moment is from the south pole to the north pole.

Magnetic Field Due to a Dipole

The magnetic field (B) at a point in space due to a magnetic dipole can be expressed in spherical coordinates. However, for our purpose, we will focus on the equatorial line, which is the line perpendicular to the axis of the dipole and equidistant from both poles.

For a magnetic dipole, the magnetic field at a distance r from the center of the dipole along the equatorial line is given by the formula:

• B = (μ₀ / 4π) * (2m / r³),

where μ₀ is the permeability of free space, and r is the distance from the center of the dipole to the point where the magnetic field is being calculated.

Deriving the Expression

To derive this expression, we start with the general formula for the magnetic field due to a dipole:

• B = (μ₀ / 4π) * [3(m·r)r - m r²] / r⁵,

In this equation, r is the unit vector pointing from the dipole to the point of interest. For points along the equatorial line, the angle between the dipole moment and the position vector is 90 degrees, making the dot product zero. Thus, the equation simplifies significantly.

On the equatorial line, the magnetic field can be simplified to:

• B = (μ₀ / 4π) * (2m / r³).

Key Points to Remember

Here are some important aspects to keep in mind:

• The magnetic field is strongest close to the magnet and decreases with the cube of the distance.
• The direction of the magnetic field on the equatorial line is opposite to that of the dipole moment.
• This expression is valid in the region far from the magnet, where the dipole approximation holds true.

In summary, the magnetic field induction on the equatorial line of a bar magnet can be expressed as B = (μ₀ / 4π) * (2m / r³). This formula highlights the relationship between the magnetic field strength and the distance from the magnet, illustrating the fundamental principles of magnetism in a clear and concise manner.