Question

Figure 40-22 shows three points at which a spin-up electron can be placed in a non-uniform magnetic field (there is a gradient along the z-axis).

(a) Rank the three points according to the energy Uof the electron’s intrinsic magnetic dipole moment $\overrightarrow{{\mathbf{\mu }}_{\mathbf{s}}}$, most positive first.

(b) What is the direction of the force on the electron due to the magnetic field if the spin-up electron is at point 2?

Short answer

a) The ranking of the three points according to the energy of the electron’s intrinsic magnetic dipole moment $\overrightarrow{{\mu }_s}$is1<2<3.

b) The direction of the force on the electron due to the magnetic field if the spin-up electron is at point 2 is towards leftwards (for increasing magnetic field).

Step-by-step solution

The given data

Three points are placed in a non-uniform magnetic field where a spin-up electron is placed.

Understanding the concept of dipole moment and magnetic field

Using the concept of potential energy, we can get the rank of the points. Thus, the potential energy of a system at a dipole point depends on its dipole moment, but the intrinsic dipole moment of an electron is negative. Again, the magnetic field is determined by the closely packed magnetic field lines near the point. Similarly, the direction of the force considering the dipole moment and the field is always in the direction of the increasing field.

Formulae:

• The potential energy of a system of an electron due to dipole moment,

$U=\overrightarrow{{\nu }_s}.\overrightarrow{B}$ …(i)

• The force on an electron due to a magnetic field,

$F={\mu }_B\frac{dB}{dz}$ ...(ii)

a) Calculation of the ranking of the points due to energy

We know that the intrinsic magnetic dipole moment of an electron is negative. Thus, the energy equation (i) becomes:

$$\begin{gathered} U=-{\mu }_{s}B\cos 0° \\ =-{\mu }_{s}B \end{gathered}$$

Again, from the given figure, we can see that the ranking of the magnetic field due to field lines of the points can be given as:1>2>3

But, from the energy expression, we can see that the value of energy is most negative at point 1 due to the increasing magnetic field.

Hence, the ranking of the points according to their energies, the most positive first is 1<2<3.

b) Calculation of the direction of the force at point 2

From equation (ii), we can get that the force is towards the increasing direction of the magnetic field:

$F\alpha \frac{dB}{dz}$

Again, for a spin-up electron, the force value is:

$F=-{\mu }_B\frac{dB}{dz}$

Considering the spin-up electron and the increasing magnetic field in the direction of the points, we can consider this line as south to the north pole. Thus, the deflection would be sideward, and for spin-up, it would be leftward in direction.

Hence, the direction of the force for the electron at point 2 is towards leftwards that I towards increasing field as the deflection is in an upward direction due to the spin-up electron.