Show that the given electric field is a plane wave

[Blanchdog]

Homework Statement

Show that each wavefront of the electric field forms a plane

Relevant Equations

$E(r, t) = E_0 \text{cos}(k(\hat{u} \cdot r - c t) + \phi)$

A wavefront is defined as a surface in space where the argument of the cosine has a constant value. So I set the argument of the cosine to an arbitrary constant s.

$k(\hat{u} \cdot r - c t) + \phi = s$

The positional information is is in r, so I rearrange the equation to be

$\hat{u} \cdot r = \frac s k + ct + \phi = \text{const}$
$u_x x + u_y y + u_z z = \text{const}$

And that's where I'm stuck.

 

[ergospherical]

You made a small error in the rearrangement, but once corrected the last line is almost right; that's the general equation of a plane of normal $\hat{n}$, but $\hat{u} \cdot r$ depends on $t$ so isn't constant in time (that's why the plane translates).

 

[Blanchdog]

You made a small error in the rearrangement, but once corrected the last line is almost right; that's the general equation of a plane of normal $\hat{n}$, but $\hat{u} \cdot r$ depends on $t$ so isn't constant in time (that's why the plane translates).

Whoops you're right, I wasn't very careful with my minus signs since I knew it was all going to be wrapped up into a constant anyway.

It looks like my equation is in the form of the scalar equation of a plane... is it really that simple? I have the equation of a plane so it is a plane wave?

 

[ergospherical]

It looks like my equation is in the form of the scalar equation of a plane... is it really that simple? I have the equation of a plane so it is a plane wave?

Yeah, pretty much