What will be the value of ψ

Question

Given that $\nabla^2\phi$=0,the solution of $\nabla^2ψ$=k$(\nabla\phi)$.$(\nabla\phi)$
first i started with the given condition, since i know that the divergence of the curl is zero so i can write the condition as$$\nabla.(\nabla\phi)=0$$
which implies that $(\nabla\phi)$=$(\nabla*A)$ then i put this into the above given equation and got $$\nabla^2ψ=k(\nabla*A).(\nabla*A)$$
after this i got stuck now how to proceed?

Answer 1

here we will first look at $\nabla^2\phi=0$ we can write this as $$\nabla.(\nabla\phi)=0$$
so we can write $\phi=$ $\alpha x+\beta y+\gamma z$ and now if we substitute $\nabla\phi=0$ in the given equation then we can write $$\nabla^2ψ = k ( \alpha+\beta+\gamma)^2$$
now let us assume that $ψ$=$\frac{k\phi^2}{2}$ is the solution. Then on substituting this solution in the given equation we get the above result hence the solution is $$ψ=\frac{k\phi^2}{2}$$