Vector Potential: Proving Partial Derivative of A = E

[captain]

can anyone show me how to prove that the partial derivaive of vector potential A with respect t is equal to the electric field E?

 

[Pythagorean]

can anyone show me how to prove that the partial derivaive of vector potential A with respect t is equal to the electric field E?

Maxwell's equations... maybe something like the divergence theorem or one of those convenient integral identities in the front of Griffith's Electrodynamics.

Maybe you could describe the problem better. If you've given me all the information, try what I said above... maybe you'll have to take a derivative or something, but it should be fairly straightforward.

 

[m2003]

I am happy to assist in proving the relationship between the partial derivative of the vector potential A and the electric field E. Firstly, it is important to understand that the vector potential A is a mathematical quantity used in the study of electromagnetism, while the electric field E is a physical quantity that describes the electric force experienced by a charged particle.

To prove that the partial derivative of A with respect to time is equal to E, we can start by using the definition of the vector potential A, which is given by the cross product of the magnetic field B and a vector potential coefficient μ0. Mathematically, this can be represented as A = μ0B.

Next, we can take the partial derivative of A with respect to time, which is denoted as ∂A/∂t. Using the product rule of differentiation, we get ∂A/∂t = μ0(∂B/∂t).

Now, according to Faraday's law of induction, the time rate of change of the magnetic field (∂B/∂t) is equal to the negative of the electric field E. Therefore, we can substitute ∂B/∂t with -E in the above equation, giving us ∂A/∂t = -μ0E.

Finally, we can rearrange the equation to get A = -μ0∫E dt, where the integral represents the time dependence of the electric field. This shows that the vector potential A is directly proportional to the electric field E, and their partial derivatives are also equal.

In conclusion, by using the definition of the vector potential and Faraday's law of induction, we have proven that the partial derivative of A with respect to time is equal to the electric field E. This relationship is fundamental in understanding the behavior of electromagnetic fields and can be used in various applications in science and engineering. I hope this explanation was helpful in understanding the proof.