Magnetic Moment of a Current Loop

The magnetic moment is defined as the magnetic strength and alignment of a magnet or other object that generates a magnetic field. Loops of electric current such as electromagnets, permanent magnets, particles such as electrons and many astronomical objects all have magnetic moments.

More precisely, the term magnetic moment refers typically to a system’s magnetic dipole moment, which is a part of the magnetic moment that can be represented by an equivalent magnetic dipole: magnetic north and south poles that are separated by a very small distance. For small magnets or long distances, the magnetic dipole component is sufficient. Higher-order terms (such as the magnetic quadrupole moment) may be required in addition to the dipole moment for extended objects.

Magnetic dipoles have magnetic fields that are proportional to their magnetic dipole moments. The dipole component of an object’s magnetic field is symmetric to the direction of its magnetic dipole moment and decreases as the inverse cube of the object’s distance.

Magnetic field on a current loop

If we examine the direction of the magnetic field produced by a current-carrying segment of wire, it reveals that all parts of the loop contribute to the magnetic field in the same direction inside the loop.

The electric current in a circular loop generates a magnetic field that is more concentrated in the centre of the loop than outside the loop. The stacking of multiple loops will concentrate the field even more into what is known as a solenoid.

Magnetic field on the axis of a circular current loop

In magnetics, there are two methods for calculating magnetic fields. The first is the Biot-Savart law and the second is the Ampere’s law. We use Biot-Savart law to calculate the magnetic field of a highly symmetric configuration carrying a steady current  whereas Ampere’s Circuital Law is used to find the magnetic field due to an infinitesimally small current-carrying wire at some point.

Biot-Savart Law

The study of the relationship between current and the magnetic field it produces is given by Biot-Savart’s law.

For example, let’s say there is a finite conductor XY that is carrying a current I.

Here the magnetic field dB due to an infinitesimal element (dl) of the conductor is to be determined at a point P at a distance of r from it.

Assuming that the angle between dl and the position vector ( r) is θ, according to Biot-Savart’s law, the magnitude of the magnetic field  dB is proportional to the current (I), the elemental length

|dl| is inversely proportional to the square of the distance (r).

Also, its direction is perpendicular to the plane, which contains dl and  r.

Therefore Biot- Savart Law can be written as,

|dB|=(μ₀/4π)(IdlsinΘ/r2)———–(1)

In this,

μ0 is the permeability of free space and is equal to 4π × 10-7TmA-1.

Consider the magnetic field produced by a circular coil along its axis. The evaluation necessitates summarising the effect of minute current elements (I.dl).

(I) represents the steady current, and the evaluation is performed in a vacuum or free space.

For instance, (I) denotes a circular loop carrying a constant current. The circular loop has a radius R and is located in the y-z plane, with its centre at the origin O. The x-axis is the loop’s axis.

x = the distance P from the loop’s centre O.

dl = a loop conducting element

|dB|=(μ₀/4π)(I|dl∗r|/r3)

where

r2= x2+ R2 ———–(2)

The displacement vector from the element to the axial point will be perpendicular to any element of the loop.

For example, the element dl present in the figure is in the y-z plane. The displacement vector r from dl to the axial point P is in the x-y plane.

Hence,

|dl∗r|=r.dl

Thus,

|dB|=(μ₀/4π)Idl/(x2+R2)

The direction of dB is as shown in the figure and is perpendicular to the plane formed by dl and r. It has an x-component dBx and a component perpendicular to the x-axis, dB⊥.

A null result is obtained when the components perpendicular to the x-axis are summed over, and they cancel out.

The dB component due to dl is cancelled by the contribution due to the diametrically opposite dl element. This is represented in the above figure.

Hence, only the x-component survives. The net contribution along the x-direction can be obtained by integrating dBx = dB cos θ over the loop.

cosΘ=R/(x2+R2)1/2 ———–(3)

From equation (2) and (3)

dBx=μ₀Idl/4πR(x2+R2)

The summation of elements dl over the loop yields 2πR, the circumference of the loop. Thus, the magnetic field at P due to the entire circular loop is,

B = μ₀IR2/2(x2+R2)3/2

Current loop as a magnetic dipole

Ampere observed that the distribution of magnetic pressure traces around a finite energizing solenoid became much like that produced through a bar magnet.

This may be visible from the reality that the compass needle indicates comparable deflection because it acts around those bodies.

After noting those similarities, Ampere confirmed that an easy current loop behaves like a bar magnet, arguing that every magnetic phenomenon is because of the stream of the modern. This is the ampere hypothesis.

Long straight wire current-carrying loop

B=μ₀I/2πr

Where,

I = current

r = shortest distance to the wire

μ0=4π×10-7 Tm/A is the permeability of free space.

Points to remember :

1. Right-hand rule  (RHR) specifies the direction of the magnetic field created by a long straight wire: Point the right hand’s thumb in the direction of the current and the fingers curl in the direction of the magnetic field loops it creates.
2. The magnetic field produced by current flowing along any path is the sum of the fields produced by segments along the path, resulting in Ampere’s law, a general relationship between the current and magnetic field.
3. Magnetic field strength (at the centre of the loop) is B=μ₀I/2R.

Where,

R = radius of the loop

B = μ₀nI/(2R) for a flat coil of N loops

• Right hand rule-2 expresses the direction of the loop.
• Solenoid is a long Coil and its magnetic field is B=μ₀nI

Where,

N= number of loops.

Inside field has a very uniform direction and magnitude.

Conclusion

As there are no magnetic monopoles, current-carrying loops behave like tiny magnets, which we call dipoles and the strength of that tiny magnet, which we call magnetic dipole strength, is given by the product of the number of loops times the current times the area.