Electric fields and charges

You may assume that the electric field is entirely perpendicular to the central axis. The electric field, $\v E$, is the gradient of the electric potential, $\psi$.

$$\v E = \nabla \psi$$

The relationship between the charge in the conductor and the electric field is Gauss' theorem (also known as the divergence theorem):

$$\iint_S \v E \cdot \widehat{n} dA = \frac{Q}{\epsilon}$$

where S is a Gaussian surface enclosing the conductor, Q is the charge enclosed in the surface and $\epsilon$is the dielectric constant for the material..

The electric potential, $\psi$, satisfies the Poisson equation

$$\nabla^2 \psi = \frac{\rho}{\epsilon}$$

where $\rho$ is the charge density and $\epsilon$ is the dielectric constant of the material. The problem has a cylindrical geometry, so you will need to review what $\nabla^2$ is in cylindrical coordinates. Since the sensor has a very low load, you must constrain the problem so that there is zero potential difference between the conductor and the shielding.