Manipulation of Electromagnetic fields using curls clarification

[gedanken6]

Homework Statement

Question 1. General Plane Waves.
We may represent a general electromagnetic plane wave by (real part of the complex exponentials)
$E=E_{0}*e^(i*<b>k</b>*<b>r</b>-iwt) B=B_{0}*e^(i*<b>k</b>*<b>r</b>-iwt)$

Show that Faraday's Law becomes iwB0=-ik x Eo

Homework Equations

dB/dt=- curl of E

The Attempt at a Solution

I transformed dB/dt into iwB0e^(ikr-iwt).

Attempted to perform curl of E on Eoe^(ikr-iwt) but not sure how to work with the r. I thought you might use curl in spherical polars but then you have theta and phi components and I think there's only x, y and z

 

[tiny-tim]

hi gedanken6!

$E=E_{0}*e^(i*<b>k</b>*<b>r</b>-iwt) B=B_{0}*e^(i*<b>k</b>*<b>r</b>-iwt)$
…
… not sure how to work with the r. I thought you might use curl in spherical polars …

just because there's an r, that doesn't mean it stands for "radius"!

k*r simply means k_xi + k_yj + k_zk

(erm … different k)

 

[vanhees71]

No, it's the dot product:
$$\vec{k}\cdot \vec{r}=k_x x+k_y y+k_z z.$$

 

[tiny-tim]

oops!

thanks, vanhees71! ​

 

[rude man]

Start with faraday's law, use Stokes' theorem to produce Maxwell's law for del x E, then just use the given plane wave equations to show the validity of the given relationship between E₀ and B₀, which is just a matter of some differential calculus.