4.10 TORQUE ON CURRENT LOOP, MAGNETIC DIPOLE
4.10.1 Torque on a rectangular current loop in a uniform magnetic field
We now show that a rectangular loop carrying a steady current
(a) A rectangular current-carrying coil in uniform magnetic field. The magnetic moment
(b) The couple acting on the coil.
We first consider the simple case when the rectangular loop is placed such that the uniform magnetic field
The field exerts no force on the two arms AD and BC of the loop. It is perpendicular to the arm AB of the loop and exerts a force
Similarly, it exerts a force
Thus, the net force on the loop is zero. There is a torque on the loop due to the pair of forces
(a) The area vector of the loop
(b) Top view of the loop. The forces
We next consider the case when the plane of the loop, is not along the magnetic field, but makes an angle with it. We take the angle between the field and the normal to the coil to be angle
The forces on the arms
But they are not collinear! This results in a couple as before. The torque is, however, less than the earlier case when plane of loop was along the magnetic field. This is because the perpendicular distance between the forces of the couple has decreased. Figure 4.22(b) is a view of the arrangement from the
As
where the direction of the area vector
This is analogous to the electrostatic case (Electric dipole of dipole moment pe in an electric field
From Eq. (4.29), we see that the torque
If the loop has
Example 4.11
A
The coil is placed in a vertical plane and is free to rotate about a horizontal axis which coincides with its diameter. A uniform magnetic field of
-
From Eq.(4.16)
Here, ; , and . Hence, The direction is given by the right-hand thumb rule. -
The magnetic moment is given by Eq. (4.30),
The direction is once again given by the right-hand thumb rule. -
Initially, . Thus, initial torque .
Finally, (or 90º).
Thus, final torque . -
From Newton’s second law,
where is the moment of inertia of the coil. From chain rule, Using this, Integrating from to
Example 4.12
- A current-carrying circular loop lies on a smooth horizontal plane. Can a uniform magnetic field be set up in such a manner that the loop turns around itself (i.e., turns about the vertical axis).
- A current-carrying circular loop is located in a uniform external magnetic field. If the loop is free to turn, what is its orientation of stable equilibrium? Show that in this orientation, the flux of the total field (external field + field produced by the loop) is maximum.
- A loop of irregular shape carrying current is located in an external magnetic field. If the wire is flexible, why does it change to a circular shape?
-
No, because that would require
to be in the vertical direction. But , and since of the horizontal loop is in the vertical direction, would be in the plane of the loop for any . -
Orientation of stable equilibrium is one where the area vector
of the loop is in the direction of external magnetic field. In this orientation, the magnetic field produced by the loop is in the same direction as external field, both normal to the plane of the loop, thus giving rise to maximum flux of the total field. - It assumes circular shape with its plane normal to the field to maximise flux, since for a given perimeter, a circle encloses greater area than any other shape.
4.10.2 Circular current loop as a magnetic dipole
In this section, we shall consider the elementary magnetic element: the current loop. We shall show that the magnetic field (at large distances) due to current in a circular current loop is very similar in behaviour to the electric field of an electric dipole. In Section 4.6, we have evaluated the magnetic field on the axis of a circular loop, of a radius
The expression of Eq. [4.31(a)] is very similar to an expression obtained earlier for the electric field of a dipole. The similarity may be seen if we substitute,
It can be shown that the above analogy can be carried further. We had found in Chapter 1 that the electric field on the perpendicular bisector of the dipole is given by [See Eq.(1.21)],
The results given by Eqs. [4.31(a)] and [4.31(b)] become exact for a point magnetic dipole.
The results obtained above can be shown to apply to any planar loop: a planar current loop is equivalent to a magnetic dipole of dipole moment
We have shown that a current loop
(i) produces a magnetic field (see Fig. 4.12) and behaves like a magnetic dipole at large distances, and
(ii) is subject to torque like a magnetic needle. This led Ampere to suggest that all magnetism is due to circulating currents. This seems to be partly true and no magnetic monopoles have been seen so far. However, elementary particles such as an electron or a proton also carry an intrinsic magnetic moment, not accounted by circulating currents.
4.10.3 The magnetic dipole moment of a revolving electron
In the Bohr model of hydrogen-like atoms, the negatively charged electron is revolving with uniform speed around a centrally placed positively charged (
In Chapter 12 we shall read about the Bohr model of the hydrogen atom. You may perhaps have heard of this model which was proposed by the Danish physicist Niels Bohr in 1911 and was a stepping stone to a new kind of mechanics, namely, quantum mechanics. In the Bohr model, the electron (a negatively charged particle) revolves around a positively charged nucleus much as a planet revolves around the sun. The force in the former case is electrostatic (Coulomb force) while it is gravitational for the planet-Sun case. We show this Bohr picture of the electron in Fig. 4.23.
The electron of charge (
and
Substituting in Eq. (4.32), we have
There will be a magnetic moment, usually denoted by
The direction of this magnetic moment is into the plane of the paper in Fig. 4.23. [This follows from the right-hand rule discussed earlier and the fact that the negatively charged electron is moving anticlockwise, leading to a clockwise current.] Multiplying and dividing the right-hand side of the above expression by the electron mass
Here, l is the magnitude of the angular momentum of the electron about the central nucleus (“orbital” angular momentum). Vectorially,
The negative sign indicates that the angular momentum of the electron is opposite in direction to the magnetic moment. Instead of electron with charge (
is called the gyromagnetic ratio and is a constant. Its value is
The fact that even at an atomic level there is a magnetic moment, confirms Ampere’s bold hypothesis of atomic magnetic moments. This according to Ampere, would help one to explain the magnetic properties of materials. Can one assign a value to this atomic dipole moment? The answer is Yes. One can do so within the Bohr model. Bohr hypothesised that the angular momentum assumes a discrete set of values, namely,
where
where the subscript ‘min’ stands for minimum. This value is called the Bohr magneton.
Any charge in uniform circular motion would have an associated magnetic moment given by an expression similar to Eq. (4.34). This dipole moment is labelled as the orbital magnetic moment. Hence, the subscript ‘