Chapter 12: 53 (page 568)

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

Expert verified

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

Step by step solution

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

$\frac{δ F^{μ ν}}{δ X^{ν}} = μ_{0} J^{μ}$ …… (1)

It is known that:

$\frac{δ}{δ X^{v}} = δ_{v}$

Substitute $δ_{v}$ for $\frac{δ}{δ X^{v}}$ in equation (1).

$δ_{v} F^{μ ν} = μ_{0} J^{μ}$ …… (2)

Differentiate the equation (2).

$δ_{v} δ_{μ} F^{μ ν} = μ_{0} δ_{μ} J^{μ}$

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

$δ_{μ} δ_{v} = δ_{ν} δ_{μ} (s y m m e t r i c) F^{μ ν} = - F^{μ ν} (A n t i - s y m m e t r i c)$

Since it is known that:

$δ_{μ} δ_{v} F^{μ ν} = 0$

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

$δ_{μ} δ_{v} F^{μ ν} = δ_{v} δ_{μ} F^{μ ν} = δ_{μ} δ_{v} (- F^{μ ν}) = - δ_{μ} δ_{v} F^{μ ν}$

Now, as the above quantity is equal to minus itself, it must be zero. Hence,

$\frac{δ J^{μ}}{δ X^{μ}} = 0$

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

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