Electromagnetism electric fields

[james walshe]

An Electric field has the following form in free space;

E=Re{Eo exp[j(ωt-κy)$\hat{i}$

I am confused as to why a unit i vector is in the expression for an electric field oscillating in the Y direction? and also what does the Re mean ? i was reading somewhere about it being to do with rectangular coordinates.

 

[vanhees71]

Re is the real part of a complex expression, and $\mathrm{j}$ is used by electrical engineers as the imaginary unit, i.e., $\mathrm{j}^2=-1$.

A electromagnetic wave in free space must be transverse (both, the electric and magnetic components of the electromagnetic field are vectors perpendicular to the direction of the wave propagation). So what you have here is a plane wave
$$\vec{E}(t,\vec{x})=\left [\mathrm{Re} \; E_0 \cos(\omega t-k y)-\mathrm{Im} \; E_0 \sin(\omega t-k y) \right ] \vec{i}.$$
This is a wave propagating in $y$ direction (if $k>0$) and oscillating in $x$ direction).

That the wave must be transverse follows from Maxwell's equations for free fields in vacuo. Among them you have
$$\vec{\nabla} \cdot \vec{E}=0,$$
and this is fulfilled for the above wave.

 

[james walshe]

Hi Vanhees71 thank you for the swift reply.
Did you use the Euler relationships to turn the exponential into the sine and cosine functions? i.e [e^jx=cosx+jsinx] and hence the imaginary part can be ignored? or can the expression be left in the exponential form.