Chapter 41: Problem 9

Question: Start from Eqs. 5 and 9 (Capter 41) and show that ${\bf E}(x,t)$ and ${\bf B}(x,t)$, the electric and magnetic field components of a plane traveling electromagnetic wave, must satisfy the ``wave equations''

$${\partial^2\over\partial t^2}{\bf E}~=~c^2 {\partial^2\over\partial x^2}{\bf E}$$

$${\partial^2\over\partial t^2}{\bf B}~=~c^2 {\partial^2\over\partial x^2}{\bf B} \ \ \ .$$ (4)

Solution: We have, from Eq. 5,

$${\partial\over\partial x}{\bf E}~=~-{\partial\over\partial t}{\bf B}$$

$${\partial\over\partial x}\left[ {\partial\over\partial x}{\bf E}~=~-{\partial\over\partial t}{\bf B}\right]$$

$${\partial^2\over\partial x^2}{\bf E}~=~- {\partial^2\over\partial t\partial x}{\bf B} \ \ \ .$$ (5)

We have, from Eq. 9,

$${\partial\over\partial x}{\bf B}~=~-{1\over c^2}{\partial\over\partial t}{\bf E}$$

$${\partial\over\partial t}\left[ {\partial\over\partial x}{\bf B}~=~-{1\over c^2}{\partial\over\partial t}{\bf E}\right]$$

$${\partial^2\over\partial t\partial x}{\bf B}~=~-{1\over c^2}{\partial^2\over\partial t^2}{\bf E} \ \ \ .$$ (6)

Combining these relations we find

$${1\over c^2}{\partial^2\over\partial t^2}{\bf E}~=~ {\partial^2\over\partial x^2}{\bf E} \ \ \ ,$$ (7)

and similarly for the magnetic field.