# Fourier components of vector potential

• dkin
In summary, the conversation is discussing the complex nature of the Fourier coefficients in the equation defining the vector potential. While the text indicates that the bar indicates complex conjugates, there is confusion about whether the coefficients themselves are complex or real. It is determined that the coefficients will be complex regardless due to the complex nature of the \mathbf{e}^{(1)}(\mathbf{k}) and \mathbf{e}^{(-1)}(\mathbf{k}) vectors.
dkin
Hi all,

I was reading the http://en.wikipedia.org/wiki/Quanti...d#Electromagnetic_field_and_vector_potential" and I am a little bit confused.

In this equation defining the vector potential

$\mathbf{A}(\mathbf{r}, t) = \sum_\mathbf{k}\sum_{\mu=-1,1} \left( \mathbf{e}^{(\mu)}(\mathbf{k}) a^{(\mu)}_\mathbf{k}(t) \, e^{i\mathbf{k}\cdot\mathbf{r}} + \bar{\mathbf{e}}^{(\mu)}(\mathbf{k}) \bar{a}^{(\mu)}_\mathbf{k}(t) \, e^{-i\mathbf{k}\cdot\mathbf{r}} \right)$

are the $a^{(\mu)}_\mathbf{k}(t)$ and $\bar{a}^{(\mu)}_\mathbf{k}(t)$ complex or real?

On my first reading I assumed complex as the text says the bar indicates complex conjugates but the vector potential is later defined as a linear addition with complex $\mathbf{e}^{(1)}(\mathbf{k})$ and $\mathbf{e}^{(-1)}(\mathbf{k})$ basis vectors so why would these vectors have complex multipliers? They are also not in bold font which would indicate scalers.

Is the bar in this case simply a label for the scaler?

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Fourier coefficients of a function are generally complex, because they are the Fourier transform of the function.

atyy said:
Fourier coefficients of a function are generally complex, because they are the Fourier transform of the function.

Thanks for the reply, I think the Fourier coefficients will be complex regardless as the $\mathbf{e}^{(1)}(\mathbf{k})$ and $\mathbf{e}^{(-1)}(\mathbf{k})$ vectors are complex therefore $\mathbf{e}^{(1)}(\mathbf{k}) a^{(1)}_\mathbf{k}(t)$ should be complex regardless of whether $a^{(1)}_\mathbf{k}(t)$ is.. I think.

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## 1. What is the Fourier transform of the vector potential?

The Fourier transform of the vector potential is a mathematical operation that decomposes the vector potential into its constituent frequencies. It is commonly used in electromagnetic theory to analyze the behavior of electromagnetic fields.

## 2. How are Fourier components of vector potential used in physics?

Fourier components of vector potential are used in many areas of physics, including electromagnetism, quantum mechanics, and signal processing. They allow for the analysis and representation of complex vector fields in terms of simpler harmonic components.

## 3. Can Fourier components of vector potential be visualized?

Yes, Fourier components of vector potential can be visualized using various techniques, such as plotting the amplitude and phase of each component as a function of frequency. This allows for a better understanding of the behavior of the vector potential and its effects on other physical quantities.

## 4. What is the relationship between Fourier components of vector potential and Maxwell's equations?

The Fourier components of vector potential are a fundamental part of Maxwell's equations, particularly in the study of electromagnetic fields. They can be used to solve for the electric and magnetic fields in a given system, and to analyze the behavior of electromagnetic waves.

## 5. Are there any applications of Fourier components of vector potential in engineering?

Yes, Fourier components of vector potential have many applications in engineering, particularly in the design and analysis of electromagnetic devices. They are also used in signal processing and image reconstruction techniques, such as magnetic resonance imaging (MRI).

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