Working w/ Complex representation E-field

[Apteronotus]

Often a time varying E-field is represented in complex format. I have a simple E-field (uniform in space) given by

\vec{E}(t)=E_o\cos(\omega t)\cdot\hat{k}
or equivalently, the real part of

\vec{E}(t)=E_o e^{\omega t}\cdot\hat{k}.

We know the potential is the negative gradient of the E-field.
If we want to calculate the potential of a field represented in the complex notation, do we need first convert to the "real" representation of and then take the gradient?

In other words how do we deal with all the calculations when the field is represented in the complex format.

Thanks

 

[Philip Wood]

We know the potential is the negative gradient of the E-field.

You have this the wrong way round. The E-field component in a particular direction is minus the gradient in that direction of the potential.

The answer to your question – suitably modified! – is that e^iwt can be left in unmodified. It is a time-dependent factor; the gradient operation is a spatial one.

 

[chrisbaird]

Even when you come across time operators, such as in Maxwell's equations, you can stick with complex notation. The imaginary numbers behave very nicely and you still get the right answer in the end.