Question

Suppose there did exist magnetic monopoles. How would you modifyMaxwell's equations and the force law to accommodate them? If you think thereare several plausible options, list them, and suggest how you might decide experimentally which one is right.

Short answer

The correctly modified Maxwell’s equation is$F=\frac{{\alpha }_0}{4\pi }\frac{q_{m1}{q}_{m2}}{r^2}\widehat{r}$ .

Step-by-step solution

Determine the Maxwell’s equation

Write the Maxwell’s equation.

$\begin{array}{l}\nabla \cdot \mathbf{E}\mathbf{=}\frac{\rho }{{\mathbf{Î}}_0}\left(Gausslaw\right)\\ \nabla \times E=0\\ \nabla \cdot B=0\end{array}$

Write the amperes law

$\nabla \cdot \mathbf{B}\mathbf{=}{\mathbf{\mu }}_0\mathbf{J}$

Determine the right Maxwell’s equation 

As the magnetic monopole exists then there will be no change in Ampere’s law and gauss law.

Actually $\nabla \cdot B=0$implies there will be no magnetic monopoles.Then if magnetic monopoles exist, then

$\nabla \cdot B={\alpha }_0{\rho }_m$

Here${\rho }_m$,is the density of magnetic change and${\alpha }_0$is the same constant.

Rewrite the Maxwell’s equation as

$\nabla \times E={\beta }_0{J}_m$

Here, $J_m$is the magnetic current density and${\beta }_0$is another constant.

Thus, magnetic charge is conserved. Hence, ${\rho }_m$and$J_m$satisfy continuity equation and is written as

$\nabla \cdot J_m+\frac{\partial {\rho }_m}{\partial t}=0$

Write the expression for force on magnetic monopole.

$F=q_m[B+(\upsilon \times E)]$

Consider both the equation directionally not correct.

Here, $E$has same units as$\upsilon B$ .

Hence, divide $\upsilon \times E$with dimensions of velocity squared and rewrite the equation as

$F=q_e[E+(\upsilon \times B)]+q_m[B-\frac{1}{c^2}(\upsilon \times E)]$

Now, write the expression for magnetic field along the lines ofCoulomb’s law.

.

$F=\frac{{\alpha }_0}{4\pi }\frac{q_{m1}{q}_{m2}}{r^2}\widehat{r}$