Vector to scalar potential, transformation of fields

In summary, the author asked a question about a specific expression for V in Griffith's Electrodynamics and how it relates to the inverse Lorentz transformations. The answer provided explains the Lorentz transformation matrix and its inverse, and how to use them to obtain the desired result for V. The author thanked the respondent for the explanation.
  • #1
OhNoYaDidn't
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Hey guys. So, as i was going through Griffith's Electrodynamics, and i came across this problem:

Screen_Shot_2016_09_04_at_18_36_08.png

In the solutions:
sol.png


How to they actually get to that expression for V = (V(bar)+vAx(bar) )Ɣ? I understand everything after that, but this just made me very confused. How do they get this from the inverse Lorentz transformations?

Thank you.
 

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  • #2
The Lorentz transformation that goes from ##S## to ##\bar S## is given by the matrix:
$$\Lambda(S\rightarrow \bar S)=\begin{pmatrix}\gamma& -\beta\gamma & 0 & 0\\ -\beta\gamma &\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\
\end{pmatrix} $$
The inverse is simply:

$$\Lambda(\bar S\rightarrow S)=\begin{pmatrix}\gamma& \beta\gamma & 0 & 0\\ \beta\gamma &\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\
\end{pmatrix} $$
You can easily verify this by multiplying these two in any order and using ##\gamma=1/\sqrt{1-\beta^2}##. Basically it's the transformation with the velocity in the opposite direction, so ##\beta\rightarrow -\beta##, while ##\gamma## doesn't change since it's quadratic. If you take into account that
$$
\bar A^\mu=\begin{pmatrix}\bar V/c\\ \bar A_x \\ \bar A_y \\ \bar A_z\\
\end{pmatrix} $$

and ##A=\Lambda(\bar S \rightarrow S)\bar A##, you should be able to get the desired result.
 
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Likes vanhees71 and OhNoYaDidn't
  • #3
Thank you so much, it all makes sense now, even how to apply those!
 

1. What is a vector potential?

A vector potential is a mathematical concept used in vector calculus to represent a vector field. It is defined as a vector function whose curl (a measure of the rotation of a vector field) is equal to the original vector field. It is often used in electromagnetism to simplify calculations and describe the magnetic field.

2. What is a scalar potential?

A scalar potential is a mathematical concept used in vector calculus to represent a scalar field. It is defined as a scalar function whose gradient (a measure of the rate of change of a scalar field) is equal to the original scalar field. It is often used in electromagnetism to describe the electric field.

3. How does one transform a vector field into a scalar field?

To transform a vector field into a scalar field, one must take the curl of the vector field and set it equal to zero. This will result in a scalar potential that describes the original vector field. This process is known as the transformation of fields.

4. What is the significance of vector to scalar potential transformation?

The transformation of vector fields to scalar fields is important in many areas of physics, particularly in electromagnetism. It allows for simpler and more efficient calculations, and also provides a better understanding of the underlying physical principles behind vector fields.

5. Can the transformation of fields be applied to any type of vector field?

Yes, the transformation of fields can be applied to any type of vector field. However, it is most commonly used in electromagnetic theory, where it allows for a simpler and more intuitive description of the electric and magnetic fields.

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