The magnetic moment is a vector that relates the current that passes through a loop or closed loop with its area. Its modulus is equal to the product of the intensity of the current and the area, and its direction and sense are given by the right-hand rule, as shown in figure 1.
This definition is valid regardless of the shape of the loop. Regarding the unit of the magnetic moment, in the International System of SI units it is Ampere × m 2 .
In mathematical terms, denoting the magnetic moment vector with the Greek letter μ (in bold because it is a vector and thus it is distinguished from its magnitude), it is expressed as:
μ = IA n
Where I is the intensity of the current, A is the area enclosed by the loop and n is the unit vector (with a module equal to 1) that points in the direction perpendicular to the plane of the loop, and whose direction is given by the rule of the right thumb (see figure 1).
This rule is very simple: by curling the four fingers of the right hand so that they follow the current, the thumb indicates the direction and sense of n and therefore that of the magnetic moment.
The above equation is valid for a loop. If there are N turns as in a coil, the magnetic moment is multiplied by N:
μ = N IA n
Magnetic moment and magnetic field
It is easy to find expressions for the magnetic moment of turns with regular geometric shapes:
-Square loop of side ℓ: μ = Iℓ 2 n
– Rectangular loop with sides a and b : μ = Iab n
– Circular loop of radius R: μ = IÏ€R 2 n
Dipole magnetic field
The magnetic field produced by the loop or loop of current is very similar to that of a bar magnet and also that of the Earth.
Bar magnets are characterized by having a north pole and a south pole, where opposite poles attract and like poles repel. The field lines are closed, leaving the north pole and reaching the south pole.
Now, the magnetic poles are inseparable, which means that if you divide a bar magnet into two smaller magnets, they still have their own north and south poles. It is not possible to have the magnetic poles isolated, that is why the bar magnet is called a magnetic dipole .
The magnetic field of a circular loop of radius R, carrying a current I, is calculated using the Biot-Savart law. For the points belonging to its axis of symmetry (in this case the x-axis), the field is given by:
Relationship between the magnetic field and the magnetic moment of the dipole
Including the magnetic moment in the previous expression results:
In this way, the intensity of the magnetic field is proportional to the magnetic moment. Note that the field intensity decreases with the cube of the distance.
This approximation is applicable to any loop, as long as x is large compared to its dimensions.
And since the lines of this field are so similar to those of the bar magnet, the equation is a good model for this magnetic field and that of other systems whose lines are similar, such as:
-Moving charged particles like the electron.
-The atom.
-The Earth and other planets and satellites of the Solar System.
-Stars.
Effect of an external field on the loop
A very important characteristic of the magnetic moment is its link to the torque that the loop experiences in the presence of an external magnetic field.
An electric motor contains coils through which a current of changing direction passes and which, thanks to the external field, experience a spinning effect. This rotation causes an axis to move and electrical energy is converted to mechanical energy during the process.
Torque on a rectangular loop
Suppose, for ease of calculations, a rectangular loop with sides a and b , whose normal vector n , projecting from the screen, is initially perpendicular to a uniform magnetic field B , as in Figure 3. The sides of the loop experience given forces for:
F = I L x B
Where L is a vector of magnitude equal to the length of the segment and directed according to the current, I is its intensity and B is the field. The force is perpendicular to both L and the field, but not all sides experience force.
In the figure shown, there is no force on short sides 1 and 3 because they are parallel to the field, remember that the cross product between parallel vectors is zero. However, the long sides 2 and 4, which are perpendicular to B , experience the forces denoted as F 2 and F 4.
These forces form a pair : they have the same magnitude and direction, but opposite directions, therefore they are not capable of transferring the loop in the middle of the field. But they can make it rotate, since the torque  τ exerted by each force, with respect to the vertical axis that passes through the center of the loop, has the same direction and sense.
According to the definition of torque, where r is the position vector:
τ = r x F
Then:
τ 2 = τ 4 =  (a / 2) F (+ j )
The individual torques do not cancel, since they have the same direction and sense, so they add:
τ net = τ 2 + τ 4 = a F (+ j )
And being the magnitude of the force F = IbB, it results:
net τ = I⋅a⋅b⋅B (+ j )
The product a⋅b is the area A of the loop, so Iab is the magnitude of the magnetic moment μ . Therefore  τ net = μ⋅B (+ j )
It can be seen that, in general, the torque coincides with the vector product between the vectors μ and B :
net τ = μ x B
And although this expression was derived from a rectangular loop, it is valid for a flat loop of arbitrary shape.
The effect of the field on the loop is a torque that tends to align the magnetic moment with the field.
Potential energy of the magnetic dipole
To rotate the loop or dipole in the middle of the field, work must be done against the magnetic force, which changes the potential energy of the dipole. The variation of the energy ΔU, when the loop rotates from angle θ or to angle θ is given by the integral:
ΔU = -μB cos θ
Which in turn can be expressed as the scalar product between the vectors B and μ :
Au = – μ · B
The minimum potential energy in the dipole occurs when cos θ = 1, which means that μ and B are parallel, the energy is maximum if they are opposite (θ = π), and it is zero when they are perpendicular (θ = π / 2).
