Electric potential as an integral over the charge distribution + uniformly charged ring potential.
00:00 How do we compute electric potential when we don't know the electric field around a charge distribution? Either we have to compute the electric field first, or we can compute the potential directly by integrating over the charge distribution! In this video, we learn how to break a continuous charge distribution into charge increments and add up the potential contributions of all the charge increments by expressing the electric potential as an integral over the charge distribution. We apply this physical integration for potential to the case of the uniformly charged ring potential, and we compute the electric potential of the ring at an observation point lying on the symmetry axis of the ring.
00:40 Electric potential of a point charge. We quickly review the potential for a point charge starting from the expression for the potential energy between two point charges. Since the potential is the potential energy per unit charge U/q_0, we divide the test charge out of the expression for potential energy and get the potential due to a point charge.
01:56 Electric potential for a collection of point charges. Now we generalize the formula for potential to the electric potential of a collection of point charges. Starting from the electric potential energy of a test charge near a collection of point charges, we divide the test charge out and arrive at a formula for the electric potential near a group of point charges.
03:21 Electric potential for a continuous distribution of charge. A continuous distribution of charge is really a collection of infinitely many infinitesimal point charges! We visualize a point charge in a continuous distribution as a dq a distance r from the observation point. This dq makes a contribution of dq/(4pi*epsilon_0*r) to the potential at the observation point. Now we have to sum these infinitely many infinitesimal dq's, and that's an integral! So we arrive at a formula for the electric potential due to a distribution of charge as integral (dq/(4pi*epsilon_0*r). This is the formula we've been looking for!
04:56 Example: electric potential for a uniformly charged ring. We swing the mighty hammer of physical integration by computing the potential for a uniformly charged ring using our new formula. We visualize dq and write it in terms of the linear charge density as dq=lambda*R*d(theta), and we write the distance from dq to the observation point using the pythagorean theorem. Plugging into the integral, we find that everything is a constant except for d(theta) and we quickly integrate to find the potential at a distance of d from the uniformly charged ring on its symmetry axis.
