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When there are a few point charges, we can just sum each electric field created by each point charge to get the total E field. In this class, we will also look at continuous charge distribution where the sum will be done via an integral.

We have not done such integrals in Phys 211 but we could have done so in the section about center of mass and moment of inertia. Before we tackle integrals involving the electric field, let us see how to do integral for rigid bodies. This is a review and extension of what we did in Phys 211.

The center of mass of a continuous distribution of mass is

x_{cm} = \frac{1}{M}\int_{over \; M} x dm and similarly for y and z coordinate.

Two important things to note.

• The integral must cover the whole object. You must think carefully how to set up the right limit of integrations.
• The infinitesimal element of mass dm must be expressed in terms of coordinates. This is a crucial part of this type of problem.

Similarly the moment of inertia is

Make sure you look at examples 12.2, 12.5 and 12.6. (as usual, if you have time, try them out without looking at solutions). If stuck, see my problem solving video below.

Problem solving video.

Find the moment of inertia of a thin plate through an axis through the center and through the plane.

The worksheet for this problem is available here