Current Loop as a Magnetic Dipole

When an electric charge moves, it produces an electric current and creates a magnetic field around it. This magnetic field can exert forces on nearby moving charges and magnets. Stationary charges do not produce magnetism, while moving charges can cause attraction or repulsion.

Magnets and Magnetic fields

A moving charge creates a magnetic field, and the force experienced in a magnetic field is called the magnetic force. Charge is a fundamental property of matter that allows it to produce and experience electrical and magnetic effects. The region in space around a magnet where its magnetic influence is felt is called the magnetic field of the magnet.

If a point charge q moves with velocity v at position r in the presence of both an electric field E_r and a magnetic field B_r​, the total force on the charge is given by:

F = q [E_r + v × B_r] = E_Electric + F_magnetic

This formula, known as the Lorentz force, was stated by H. A. Lorentz based on the experiments of Ampere and others.

Characteristics of Magnets

• Like poles repel, and opposite poles attract.
• Poles cannot be separated; each piece of a broken magnet becomes a new magnet with both poles.
• When freely suspended, the north pole points to the geographical north and the south pole to the geographical south.
• The magnetic field is strongest at the poles and weakest at the middle.

Magnetic Dipole and Magnetic Dipole Moment

A magnetic dipole consists of two equal and opposite magnetic poles separated by a small distance. A current-carrying loop behaves like a magnetic dipole and produces a magnetic field similar to a bar magnet.

The magnetic dipole moment is a vector quantity that measures the strength and direction of a magnetic dipole. Its direction is from south to north, and its SI unit is A·m².

The torque on a magnetic dipole is:

Where,

• τ is the torque acting on the dipole
• m is the magnetic dipole moment
• B is the external magnetic field

A magnetic dipole consists of two equal and opposite magnetic poles (north and south), which together produce a magnetic field. Example: when a bar magnet is broken into smaller pieces, each piece behaves as an independent magnetic dipole, with its own north and south poles.

Magnetic Dipole Moment of a Circular Loop

When an electric current flows through a circular loop, it behaves like a magnetic dipole. This happens because the moving charges in the loop generate a magnetic field, similar to that produced by a bar magnet. The loop tends to align itself in an external magnetic field, just like a magnetic dipole.

The magnetic dipole moment of a current-carrying loop is defined as the product of the current flowing through the loop and the area enclosed by it. For a coil having multiple turns, the dipole moment increases proportionally with the number of turns.

μ = nIA

Where,

• n is the number of turns in the loop
• I is the current flowing in the loop
• A is the area enclosed by the loop

The SI unit of magnetic dipole moment is ampere–meter² (A·m²), while in the CGS system, it is expressed in erg per gauss. Here, erg represents the unit of energy and gauss represents the unit of magnetic flux density.

Characteristics

• Magnetic field lines around a current-carrying loop are circular near the wire.
• At the center of the loop, the field lines are nearly straight and parallel.
• One face of the loop acts as a north pole, while the opposite face acts as a south pole.
• The magnetic dipole moment is directed perpendicular to the plane of the loop, from south to north.

Current Loop as a Magnetic Dipole

A circular current-carrying loop behaves like a magnetic dipole because the flow of current in a closed loop produces a magnetic field similar to that of a bar magnet. The direction of this magnetic field is determined by the right-hand thumb rule. Due to this, the loop exhibits north and south pole behavior, making it equivalent to a magnetic dipole.

The magnetic dipole moment of a circular current loop is given by:

Where,

• n is the number of turns
• i is the current flowing in the loop
• r is the radius of the loop

Current Loop as a Magnetic Dipole Derivation

Consider a circular loop of radius R placed on a table, carrying a current i in the anti-clockwise direction. Let a point P be located on the axis of the loop at a distance l from its centre. The magnetic field at point P is given by:

For simplification, assume that the point P is very far from the loop (l >> R). Then, the magnetic field can be approximated as:

The area of the loop is:

Thus, the magnetic field can be expressed in terms of the magnetic dipole moment μ as:

where μ=iA is the magnetic dipole moment of the current loop.

This formula for the magnetic field of a current loop is analogous to the electric field of a dipole:

Here, is the electric dipole moment and r is the distance from the dipole.

Solved Problems

Question 1: Two wires of the same length are shaped into a square and a circle. If they carry the same current, then the ratio of their magnetic moments is:

Solution: The magnetic moment of a loop is given by

Let the side of the square be a and the radius of the circle be r. Since the wires have the same length

The area of the square loop:

Thus, the magnetic moment of the square loop:

The area of the circular loop:

Thus, the magnetic moment of the circular loop:

The ratio of magnetic moments:

Question 2: A circular coil of 300 turns and a diameter of 14 cm carries a current of 15 A. The magnitude of the magnetic moment associated with the loop is going to be

Solution: Given: 

Number of turns (N) = 300

Radius of coil (r) = 14/2 = 7cm = 7 × 10^-2 m

Current in coil (I) = 15A

Magnetic moment of a circular coil:

Substitute the values:

Question 3: A circular coil of 100 turns, each turn of radius 8 cm carries a current of 0.4 A. What is the magnitude of the magnetic field B at the center of the coil? 

Solution: Given:

Number of turns (N) = 100

radius of each turn (r) = 8cm = 8 × 10^-2m

Current flowing in the coil (I) = 0.4 A

permeability of free space (μ₀) = 4π × 10^-7 TmA^-1

The magnetic field at the center of a circular coil is given by:

Substitute the values:

Question 4: A long straight wire in the horizontal plane carries 50 A in the north-to-south direction. Give the magnitude and direction of B at a point 2.5 m east of the wire.

Solution: Given:

Current in the wire (I) = 50 A

Distance of a point from the wire (r) = 2.5 m

permeability of free space (μ₀) = 4π × 10^-7 TmA^-1

The magnetic field due to a long straight wire is:

Substitute the values:

Using the right-hand thumb rule, the magnetic field at a point east of the wire (current north-to-south) points into the page.

T, into the page

Unsolved Problems

Question 1: A circular coil of 50 turns has a radius of 10 cm. It carries a current of 2 A. Find its magnetic moment.

Question 2: A straight wire 1 m long carries a current of 5 A. It is placed in a uniform magnetic field of 0.1 T perpendicular to the wire. Find the force on the wire.

Question 3: Two long parallel wires, 1 m apart, carry currents of 6 A and 9 A in the same direction. Find the force per meter between them.

Question 4: Two wires of equal length are shaped into a square and an equilateral triangle. Both carry the same current. Find the ratio of their magnetic dipole moments.

Question 5: A circular coil of 300 turns and radius 8 cm carries a current of 0.5 A. Find the magnetic field at a point on its axis, 20 cm from its centre.