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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 ∂Jμ∂xμ=0.

Step by step solution

01

Expression for Maxwell’s equation: 

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

∂Fμv∂xv=μ0Jμ …… (1)

It is known that:

∂∂xv=∂v

Substitute ∂vfor ∂∂xvin equation (1).

∂vFμv=μ0Jμ …… (2)

02

Determine the continuity equation from Maxwell’s equation:

Differentiate the equation (2).

∂v∂μFμv=μ0∂μJμ

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

∂v∂μ=∂μ∂v   (Symmetric)Fμv=−Fμv   (Anti-symmetric)

Since it is known that:

∂μ∂vFμv=0

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

∂μ∂vFμv=∂v∂μFμv=∂μ∂v(−Fμv)=−∂μ∂vFμv

Now, as the above quantity is equal to minus itself, it must be zero. Hence,∂Jμ∂xμ=0

Therefore, the continuity equation is obtained as∂Jμ∂xμ=0 .

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