Introduction to capacitance, parallel plate capacitor derivation of C=epsilon_0*A/d, and example.
00:00 Introduction to capacitance: we introduce the idea of capacitance for a pair of conductors and how conductors in a capacitor end up with charges of equal magnitude and opposite sign. Next, we introduce the parallel plate capacitor and the definition of capacitance. The strategy for computing the capacitance of a pair of conductors is to charge them with a magnitude of Q, then compute the potential difference between them, V, and apply the definition of capacitance C=Q/V to get the capacitance.
02:04 Electric field between parallel plates: starting from the electric field due to an infinite sheet of charge (derived previously using Gauss' Law here: • Electric field of an infinite sheet c... ), we find that the electric field outside the plates of a parallel plate capacitor is zero, provided the sheets are large compared to the separation distance so that the infinite sheet approximation is valid. Between the plates of the capacitor, the electric fields generated by each sheet add vectorially giving us a field strength of E=sigma/epsilon_0 between the plates, where sigma is the area charge density and epsilon_0 is the permittivity of free space.
04:33 Path integral to find the potential difference between the plates: now we set ourselves up to compute the potential difference on the capacitor given a charge of Q on the plates. We call the potential zero at the negative plate and V at the positive plate and draw a path from the positive to negative plate so that the path is parallel to the electric field inside the capacitor. The potential difference is given by the path integral V=integral_a^b E*dl. The dot product in the path integral is trivial because E and dl point in the same direction. E is constant so it can be pulled out of the integral, and the resulting path integral is just the distance between the plates. Finally, sigma can be replaced with Q/A and we arrive at an expression for the potential difference between the plates. Now we apply the definition of capacitance C=Q/V and simplify the result to derive the capacitance for the parallel plate capacitor: epsilon_0*A/d.
07:12 Example using C=epsilon_0*A/d. In our example, we are given the side length for a parallel plate capacitor constructed from two square plates with a given separation distance, and we are given the voltage of the battery connected to the plates. We compute the capacitance using the formula for parallel plate capacitance, then we use Q=CV to compute the charge on the plates.
