Chapter 12: Q12.53P (page 568)

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

Short Answer

Expert verified

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

Step by step solution

Expression for Maxwell’s equation:

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

$\frac{\partial F^{μv}}{\partial x^{v}} = μ_{0} J^{μ}$ …… (1)

It is known that:

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

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

$\partial_{v} F^{μ v} = μ_{0} J^{μ}$ …… (2)

Determine the continuity equation from Maxwell’s equation:

Differentiate the equation (2).

$\partial_{v} \partial_{μ} F^{μ v} = μ_{0} \partial_{μ} J^{μ}$

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

$\begin{matrix} \partial_{v} \partial_{μ} = \partial_{μ} \partial_{v} (Symmetric) \\ F^{μ v} = - F^{μ v} (Anti-symmetric) \end{matrix}$

Since it is known that:

$\partial_{μ} \partial_{v} F^{μ v} = 0$

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

$\begin{matrix} \partial_{μ} \partial_{v} F^{μ v} = \partial_{v} \partial_{μ} F^{μ v} \\ = \partial_{μ} \partial_{v} (- F^{μ v}) \\ = - \partial_{μ} \partial_{v} F^{μ v} \end{matrix}$

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

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

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

Short Answer

Step by step solution

Expression for Maxwell’s equation:

Determine the continuity equation from Maxwell’s equation:

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