Electric Field of a Ring of Charge

Описание: In this video we find the electric field of a ring of charge. This is the second example of finding the electric field of a continuous charge distribution.

In this problem, the ring of charge, or hoop of charge, has a radius R and a linear charge density. We'll look at a small point on the ring and its small amount of charge, delta Q. Looking at the ring carefully, we notice that for each point on the ring, there is another point on the opposite side. These charges will have horizontal components of their electric fields in opposite directions. The z-components will be in the same direction, straight up along the positive z-axis.

Each point of charge can be represented as a point charge, so we can use the equation for an electric field for a point charge. Since each point on the ring is the same distance from the point P, each point will contribute the same to the electric field. This will make problem easier than our previous example of a line charge.

We will use cosine of the angle given to find the z-component of electric field. Rewriting our distance 'r' and cosine of theta in terms of z, the perpendicular distance from the ring to P, and R, the radius of the ring, we find that everything is a constant. Also, the only thing we need to sum is our small charges. Summing up all of the small charges just leaves us with the total charge on the ring, and we're able to get our result without writing an integral.

Technically, we are integrating the charge and we sum them up. We can solve this problem a different way, this time using an integral and the linear charge density. Each small charge dQ can be written as the linear charge density times the length ds which represents a small arc length. Since the radius is constant, each small region of the arc can be written as R times the small angle d-phi.

Be careful not to confuse the angle phi of the loop, with the angle our distance r from the ring to P makes with the z-axis.

Now we can account for the charge by integrating over the coordinate phi of the loop. Writing everything again in terms of z and R, we find the only term that needs to be integrated is d-phi. Since the ring is a complete circle, we integrate from 0 to 2pi. Doing this gives us a result that looks different than the one we obtained previously. But if we substitute the total charge divided by the length, which in this case is the circumference of the ring, we get the same result.

We can plot the z-component of the electric field as a function of z and see if it agrees with what we expect. At the center of the ring, there is no z-component of the field and the x- and y-components still cancel, so the field is zero. Far away from the ring, we expect the field strength to drop off. Plotting the field we see this agrees with our logic and find the maximum field strength is just short of one radius length above the ring.

In this problem, we saw that there were additional symmetries that made this problem easier than the line of charge. We should always look for symmetries like this when possible. We also saw there were multiple ways to solve the problem and the results may look different, though they are equivalent. When finding examples online or in other textbooks, we often find a result that may look different than our or what we would expect. It's a good practice to try and understand the path to these other results and verify by substitutions that they indeed agree with our answer. Otherwise we should find the mistakes!