Question

Imagine two charged magnetic dipoles (charge q, dipole moment m), constrained to move on the z axis (same as Problem 6.23(a), but without gravity). Electrically they repel, but magnetically (if both m's point in the z direction) they attract.

(a) Find the equilibrium separation distance.

(b) What is the equilibrium separation for two electrons in this orientation. [Answer: 4.72x10^-13m.]

(c) Does there exist, then, a stable bound state of two electrons?

Short answer

(a) The equilibrium separation distance is $\frac{\sqrt{6}m}{qc}$.

(b) The equilibrium separation distance is $4.73\times {10}^{-13}m$.

(c) There is no exist stable bound state for two electrons.

Step-by-step solution

Write the given data from the question.

The charge is q.

The dipole moment is m.

Two charges move on the z axis.

Calculate the equilibrium separate distance.

(a)

The magnetic field due one charge is given by,

$B_1=\frac{{\mu }_{0}m}{4{\mathrm{\pi z}}^3}\hat{z}$

Here z is the distance.

The magnetic field due another charge is given by,

$B_2=\frac{{\mu }_{0}m}{4{\mathrm{\pi z}}^3}\hat{z}$

The total magnetic field due to both the charge is given by,

$B=B_1+B_2$

Substitute $\frac{{\mu }_{0}m}{4{\mathrm{\pi z}}^3}\hat{z}$ for B₁ and B₂ into above equation.

$$\begin{gathered} B=\frac{{\mu }_{0}m}{4{\mathrm{\pi z}}^3}\hat{z}+\frac{{\mu }_{0}m}{4{\mathrm{\pi z}}^3}\hat{z} \\ B=2\frac{{\mu }_{0}m}{4{\mathrm{\pi z}}^3}\hat{z} \\ B=\frac{{\mu }_{0}m}{4{\mathrm{\pi z}}^3}\hat{z} \end{gathered}$$

The force on the dipole is given by,

$F=\nabla (m·B)$

Substitute $\frac{{\mu }_{0}m}{2{\mathrm{\pi z}}^3}\hat{z}$ for B into above equation.

$$\begin{gathered} F=\nabla \left(m·\frac{{\mu }_{0}m}{2{\mathrm{\pi z}}^3}\hat{z}\right) \\ F=\frac{\partial }{\partial z}\left(m·\frac{{\mu }_{0}m}{2{\mathrm{\pi z}}^3}\hat{z}\right) \\ F=\frac{3m^2{\mu }_0}{2{\mathrm{\pi z}}^4}\hat{z} \end{gathered}$$

The expression for the Coulomb’s for between the 2 electrons is given by,

role="math" localid="1657716216750" $F_e=\frac{q^2}{4{\mathrm{\pi \epsilon }}_0{\mathrm{z}}^2}\hat{z}$

The net force on the charge is given by,

F + F_e= 0

substitute $\frac{q^2}{4{\mathrm{\pi \epsilon }}_0{\mathrm{z}}^2}\hat{z}$ for F_e and role="math" localid="1657716398268" $-\frac{3m^2{\mu }_0}{2{\mathrm{\pi z}}^4}\hat{z}$ for F inti above equation.

$\begin{array}{rcl}-\frac{3m^2{\mu }_0}{2{\mathrm{\pi z}}^4}\hat{z}+\frac{q^2}{4{\mathrm{\pi \epsilon }}_0{\mathrm{z}}^2}\hat{z}& =& 0\\ \frac{q^2}{4{\mathrm{\pi \epsilon }}_0{\mathrm{z}}^2}\hat{z}& =& -\frac{3m^2{\mu }_0}{2{\mathrm{\pi z}}^4}\hat{z}\\ \frac{q^2}{2{\epsilon }_0}& =& -\frac{3m^2{\mu }_0}{z^2}\\ z^2& =& -\frac{3m^2{\epsilon }_0{\mu }_0}{q^2}\end{array}$

Solve further as,

$$\begin{gathered} z^2=\frac{6m^2}{q^2\left({\displaystyle \frac{1}{{\epsilon }_0{\mu }_0}}\right)} \\ z=\sqrt{\frac{6m^2}{q^2\left(\frac{1}{{\epsilon }_0{\mu }_0}\right)}} \\ z=\frac{\sqrt{6}m}{q\left(\frac{1}{\sqrt{{\epsilon }_0{\mu }_0}}\right)} \end{gathered}$$

Substitute c for $\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}$into above equation.

role="math" localid="1657716909912" $z=\frac{\sqrt{6}m}{qc}$

Hence the equilibrium separation distance is $\frac{\sqrt{6}m}{qc}$.

Calculate the equilibrium separation distance.

(b)

The charge on electron,$q=1.6\times {10}^{-19}C$

The speed of the light,$c=3\times {10}^{8}m/s$

The dipole moment,$m=9.28\times {10}^{-24}J/T$

The separation equilibrium distance,

$z=\frac{\sqrt{6}m}{qc}$

Substitute $1.6\times {10}^{-19}C$ for q, $3\times {10}^{8}m/s$ for c and $9.28\times {10}^{-24}J/T$for m into above equation.

role="math" localid="1657717399234" $$\begin{gathered} z=\frac{\sqrt{6}\times 9.28\times {10}^{-24}}{1.6\times {10}^{-19}\times 3\times {10}^8} \\ z=\frac{22.731\times {10}^{-24}}{4.8\times {10}^{-11}} \\ z=4.73\times {10}^{-13}m \end{gathered}$$

Hence the equilibrium separation distance is $4.73\times {10}^{-13}m$.

Determine the existence of the bound state for electron.

(c)

Since the equilibrium point is unstable, therefore, the existence of the stable bound for two electrons does not exist.

Hence there is no existing stable bound state for two electrons.