# Finding constant in electrostatic equation given only charge

So in a region of free space with zero charge, all I'm given is that the electrostatic potential V is given by:

$$V(x,y,z) = kz(x^2) y + 2y - 6y(z^3)$$ [Volts]

To find the constant k, would I have to use the assumption that there is zero electric field and that the corresponding partial derivatives dV/dx, dV/dy, and dV/dz also equal to zero?

• Between the plates of a charged capacitor in a vacuum there is no charge. And yet the field is not zero. You should try to catch another fish, mon ami. – Sredni Vashtar Mar 1 at 1:18

Why electric field is zero? It is a gradient of potential

$$E=-grad(V)$$

Now use Maxwell's equation for electric field taking into account that there is now electric charge in your space

$$\mathrm{div}\,E = 0$$

Then combine the two equations

$$\mathrm{div}\,\mathrm{grad}\, V=\Delta V=0$$

Where $$\\Delta\$$ is a Laplace operator.

Apply the operator to your equation for V and you should be able to solve it for k.