Working w/ Complex representation E-field

In summary, when dealing with a time varying E-field represented in complex notation, the potential can be obtained by taking the negative gradient of the E-field. This can be done without converting to the "real" representation of the E-field. The imaginary numbers in complex notation behave well even when dealing with time operators or in Maxwell's equations, resulting in the correct answer.
  • #1
Apteronotus
202
0
Often a time varying E-field is represented in complex format. I have a simple E-field (uniform in space) given by

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

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

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
 
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  • #2
Apteronotus said:
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 eiwt can be left in unmodified. It is a time-dependent factor; the gradient operation is a spatial one.
 
  • #3
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.
 

FAQ: Working w/ Complex representation E-field

1. What is a complex representation of an electric field?

A complex representation of an electric field is a mathematical representation that combines the magnitude and direction of an electric field into a single complex number. This is often used in physics and engineering to simplify calculations and analysis of electric fields.

2. How is a complex representation of an electric field different from a vector representation?

In a complex representation, the magnitude and direction of an electric field are combined into a single complex number, while in a vector representation, they are represented as separate components. Additionally, complex representations can be used in more advanced calculations, such as for AC circuits, while vector representations are typically used for simpler calculations.

3. What are the advantages of using a complex representation for electric fields?

One advantage of using a complex representation is that it simplifies calculations and analysis of electric fields, especially in AC circuits. It also allows for more efficient use of mathematical tools, such as complex analysis, in solving problems related to electric fields.

4. How do I convert between a complex representation and a vector representation of an electric field?

To convert from a complex representation to a vector representation, you can use the real and imaginary components of the complex number as the x and y components of the vector, respectively. To convert from a vector representation to a complex representation, you can use the Pythagorean theorem to find the magnitude of the vector and the inverse tangent function to find the direction of the vector.

5. Can a complex representation be used for other types of fields besides electric fields?

Yes, complex representations can be used for other types of fields, such as magnetic fields and gravitational fields. In these cases, the complex number represents the magnitude and direction of the field, just as it does for electric fields.

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