Help understanding proof involving Maxwell equation

[U.Renko]

help understanding "proof" involving Maxwell equation

I'm currently taking a course on mathematical methods for physics.
(Like always I'm a bit confused about where exactly I should post these questions, should it go to the homework forum? )


anyway as I was reading the lecture notes I found this demonstration that if $\vec{J} = 0$ in Maxwell-Ampere's law satisfies the wave equation: $\left( \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2 \right)\vec B = \vec 0$ :

There is one small step I'm not sure why/how he did it.

he took the curl of $\nabla \times\vec B$, that is, $\nabla\times\left(\nabla\times\vec B \right) = \nabla\left(\nabla\cdot\vec B\right) - \nabla^2\vec B$

then since $\nabla\cdot\vec B = 0$ and $\nabla\times\vec B = \mu_0\epsilon_0\frac{\partial\vec E}{\partial t}$ he says $\nabla\times\left(\nabla\times\vec B \right) = \nabla\left(\mu_0\epsilon_0\frac{\partial\vec E}{\partial t}\right) = \mu_0\epsilon_0\left(\nabla\times\frac{\partial\vec E}{\partial t}\right)$

so far I understand everything, but now he says:
$\mu_0\epsilon_0\left(\nabla\times\frac{\partial\vec E}{\partial t}\right) = \mu_0\epsilon_0\frac{\partial\left(\nabla\times\vec E\right)}{\partial t}$ and it is this very last step that I'm not sure if it's allowed. And if is, why it is?

the rest of the demonstration follows easily, supposing this last step is correct.

 

[Matterwave]

That step just says that time derivatives commute with the curl. You should try this out for yourself and see that it is so. (Basically it's true simply because partial derivatives commute and the basis vectors are not time-dependent.)

Just write out the curl of E, take the time derivative. Then write out the curl of the time derivative of E and they are identical.

 

[Simon Bridge]

You missed one:
$$ \left( \frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2 \right)\vec B = 0\\
\implies \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\vec B =\nabla^2\vec B\\
\implies \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\vec B = \nabla(\nabla\cdot\vec B)-\nabla\times (\nabla\times \vec B)$$

Note:
$$\nabla\cdot\vec B = 0\\
\nabla\times \vec B = \mu_0\epsilon_0\frac{\partial}{\partial t}\vec E\\
\nabla\times \vec E = -\frac{\partial}{\partial t}\vec B$$

https://www.google.co.nz/search?q=m...0j5j0j69i60.3357j0j1&sourceid=chrome&ie=UTF-8

 

[UltrafastPED]

By Cairault's theorem [see http://calculus.subwiki.org/wiki/Clairaut's_theorem_on_equality_of_mixed_partials] continuous partial derivatives commute.

 

[sardar]

Hello,

I am happy to help you understand the proof involving the Maxwell equation. In this proof, we are trying to show that if the current density \vec{J} is equal to zero, then the magnetic field \vec{B} satisfies the wave equation \left( \frac{1}{c^2}\frac{\partial^2}{\partial t^2} - \nabla^2 \right)\vec B = \vec 0.

The step that you are confused about is where the author takes the curl of \nabla \times \vec{B} and then rewrites it as \nabla \times \left(\nabla \times \vec{B}\right) = \nabla \left(\mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}\right). This step is allowed because of a mathematical identity known as the vector identity. This identity states that the curl of the curl of a vector field is equal to the gradient of the divergence of that vector field minus the Laplacian of that vector field. In this case, the vector field is \vec{B}, so the identity becomes:

\nabla \times \left(\nabla \times \vec{B}\right) = \nabla \left(\nabla \cdot \vec{B}\right) - \nabla^2 \vec{B}

Now, since we know that \nabla \cdot \vec{B} is equal to zero (as stated in the proof), we can simplify the above equation to:

\nabla \times \left(\nabla \times \vec{B}\right) = -\nabla^2 \vec{B}

Finally, we can use the fact that \nabla \times \vec{B} is equal to \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t} (as stated in the proof) to rewrite the above equation as:

\nabla \times \left(\nabla \times \vec{B}\right) = \mu_0 \epsilon_0 \frac{\partial}{\partial t}\left(\nabla \times \vec{E}\right)

I hope this explanation helps you understand the last step of the proof. It is important to note