I was studying for my differential equations exam and came across this problem:
Given the equations:
$$\begin{gather} \vec{B} = \nabla \times(\hat{z}\Psi) \\ \frac{\partial \vec{E}}{\partial t} = \nabla \times \vec{B} \\ \vec{E} = \frac{\partial(\hat{z}\Psi)}{\partial t}
\end{gather}$$
with all function independent of $z$ and $\hat{z}$ being the unit vector in the $z$ direction. It asks to find which the partial differential equation which satisfies $\Psi(x,y,t)$.
The solution says that $\frac{\partial^2 \Psi}{\partial x^2} + \frac{\partial^2 \Psi}{\partial y^2} =\frac{\partial^2 \Psi}{\partial t^2}$. I can see the obvious relations between the equations but I am not confident in working with the $\nabla$ operator. Could someone show me how this equation is derived?
The next parts of the question asks, given that $\Psi(x,t)$ is independent of $y$ what is the form of the differential equation. I know that I have to look at the eigenvalues of the equation which are $-1$ and $1$ thus it is parabolic. Is this correct? The final bit asks to solve using separation of variables, which should not be an issue.
Best Answer
I assume you can derive this: $$\frac{\partial^2(\hat{z}\Psi)}{\partial t^2}= \nabla \times \left(\nabla \times(\hat{z}\Psi)\right).$$ From here you just need to use the definition of curl: for a given function $F$, the curl is obtained by computing the following determinant: $$\nabla \times F = \begin{vmatrix} \hat i & \hat j & \hat k \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ F_x & F_y & F_z \end{vmatrix}.$$ In your case, you have to apply it twice. Note that $\hat{z}\Psi = (0, 0, \Psi_z)$, so the computation is quite easier.