1
$\begingroup$

The 1972 book by L. Eyges's, The Classical Electromagnetic Field, on p. 184, in $\S$11.7, Integral Forms of The Potential, the statement

"We now turn to the problem of finding $\mathbf{A}$ and $\mathbf{\Phi}$ in terms of $\mathbf{J}$ and $\rho$. For this purpose, the Lorenz gauge is the more convenient one. In this gauge we have four equations in (11.33)."

appears. Equation 11.33 is stated on p. 182 as

$$ \nabla^2 \mathbf{A}- \frac{1}{c^2} \frac{\partial^2 \mathbf{A}}{\partial t^2} = - \frac{4 \pi \mathbf{J}}{c}, \\ \nabla^2 \phi - \frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} = - 4 \pi \rho $$

Why does the author claim that this is four equations when only two are clearly written?

$\endgroup$
5
$\begingroup$

The Equation,

$$ \nabla^2 \mathbf{A}- \frac{1}{c^2} \frac{\partial^2 \mathbf{A}}{\partial t^2} = - \frac{4 \pi \mathbf{J}}{c} $$ ,

is actually three seperate equations in three dimensions. In Cartesian coordinates this Equation expands to,

$$\nabla^2 A_x- \frac{1}{c^2} \frac{\partial^2 A_x}{\partial t^2} = - \frac{4 \pi J_x}{c}, \\ \nabla^2 A_y- \frac{1}{c^2} \frac{\partial^2 A_y}{\partial t^2} = - \frac{4 \pi J_y}{c}, \\ \nabla^2 A_z- \frac{1}{c^2} \frac{\partial^2 A_z}{\partial t^2} = - \frac{4 \pi J_z}{c}. $$

Therefore, given these three equations, and the equation for $\phi$, there are four total equations.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.