Question

Figure 32-24 shows three loop models of an electron orbiting counterclockwise within a magnetic field. The fields are non-uniform for models 1 and 2 and uniform for model 3. For each model, are (a) the magnetic dipole moment of the loop and (b) the magnetic force on the loop directed up, directed down, or zero?

Short answer

1. The magnetic dipole moment of the loop for all the models will be downward perpendicular to the plane of the circular loop.
2. The magnetic force on the loop for models 1 and 2 is zero and model three is directed up.

Step-by-step solution

Given data

Figure 32-24 with three loop models of an electron orbiting counter-clockwise within a given magnetic field is given.

Understanding the concept of direction of the magnetic dipole moment and the force

The concept of magnetic dipole moment defines the amount of the current flowing through a given area. Using the Fleming’s left-hand thumb rule, when we curl the fingers of our hand in the direction of the effective current, then the thumb indicates the direction of the dipole moment.

The magnetic force gives the amount of attraction or repulsion that arises between the charged particles due to their motion. The direction of this force is given by using the right-hand rule for a negative charge as the given particle is an electron.

Formulae:

The magnetic dipole moment of a particle, $\mu =iA..........\left(1\right)$

The magnetic force of a particle, $\stackrel{\rightharpoonup }{F}=q\left(\stackrel{\rightharpoonup }{v}\times \stackrel{\rightharpoonup }{B}\right)orF=qvB\sin \theta$

where, $\theta$ is the magnetic field, q is the charge, v is the particle velocity, and $\theta$is the angle between particle velocity and magnetic field.

a) Calculation of the magnetic dipole moment for all the given models

As the direction of magnetic dipole moment depends on planar area of the current loop considering equation (1), thus, the direction of the moment is only given by the direction of motion of the effective current using thumb rule.

Here, the electron is in counterclockwise direction. Thus, the effective current will be opposite of electron’s direction that is clockwise. Now, using right-hand thumb rule, the direction of the magnetic file will be downward perpendicular to the plane of the circular loop for all the three given models.

b) Calculation of the direction of the magnetic force

For model 1 and model 2,

Here, the magnetic field is non-uniform and is directed downward to the plane with increasing intensity for both the models. Thus, every dipole present there is an equal and opposite force directed and thus they cancel each other resulting in a zero.

For model 3,

As the magnetic field is uniform and the electron is moving counter clockwise that is perpendicular to the field direction, considering equation (1), it can be said that the force on it is direction inward and perpendicular to both the particle and magnetic field direction.

Hence, the force is zero for model 1 and 2 while is directed upward.