Question

Obtain the continuity equation (Eq. 12.126) directly from Maxwell’s equations (Eq. 12.127).

Short answer

The continuity equation is obtained as $\frac{\partial {\mathrm{J}}^{\mathrm{\mu }}}{\partial {\mathrm{x}}^{\mathrm{\mu }}}=0$.

Step-by-step solution

Expression for Maxwell’s equation: 

Using equation 12.127, write the expression for Maxwell’s equation.

$\frac{\partial {\mathrm{F}}^{\mathrm{\mu v}}}{\partial {\mathrm{x}}^{\mathrm{v}}}={\mathrm{\mu }}_0{\mathrm{J}}^{\mathrm{\mu }}$ …… (1)

It is known that:

$\frac{\partial }{\partial x^v}={\partial }_v$

Substitute ${\partial }_v$for $\frac{\partial }{\partial x^v}$in equation (1).

${\partial }_v{F}^{\mu v}={\mu }_0{J}^{\mu }$ …… (2)

Determine the continuity equation from Maxwell’s equation:

Differentiate the equation (2).

${\partial }_v{\partial }_{\mu }{F}^{\mu v}={\mu }_0{\partial }_{\mu }{J}^{\mu }$

From the above equation, observe the symmetric and anti-symmetric combination.

$\begin{array}{c}{\partial }_v{\partial }_{\mu }={\partial }_{\mu }{\partial }_v\left(\text{Symmetric}\right)\\ F^{\mu v}=-F^{\mu v}\left(\text{Anti-symmetric}\right)\end{array}$

Since it is known that:

${\partial }_{\mu }{\partial }_v{F}^{\mu v}=0$

As the above indices are summed from 0 to 3, the term $\mu$and v can be pronounced as the same. Hence,

$\begin{array}{c}{\partial }_{\mu }{\partial }_v{F}^{\mu v}={\partial }_v{\partial }_{\mu }{F}^{\mu v}\\ ={\partial }_{\mu }{\partial }_v(-F^{\mu v})\\ =-{\partial }_{\mu }{\partial }_v{F}^{\mu v}\end{array}$

Now, as the above quantity is equal to minus itself, it must be zero. Hence,$\frac{\partial J^{\mu }}{\partial x^{\mu }}=0$

Therefore, the continuity equation is obtained as$\frac{\partial J^{\mu }}{\partial x^{\mu }}=0$ .