# Electric Field Contributions

Figure 1: Two thin parallel wires

Figure 2: The cross section of a hollow sphere containing a smaller, hollow sphere

The electric fields for both figures are calculated using different principles. In figure 1, the electric field is the vector sum of each thin wire's electric field. In figure 2, the electric field between both shells is just the electric field of the inner shell. It is found using Gauss's law: Enclose the inner sphere with a gausssian sphere and the electric field is, effectively, the field of one point charge:

$$\Delta V = - \int_{Ra}^{Rb} (KQ/r^2) dr$$

Why doesn't figure two take into account the field produced by the outer shell?

The second field calculation uses Gauss's law, which does not work as you have stated, instead it says that the flux of electric field out of a closed surface equals the charge (divided by $\epsilon_0$ contained within. Only in a case of high symmetry will you be able to say that the electric field cuts the spherical surface perpendicularly and that the flux is $4\pi r^2 E$.