Principle of Superposition (revisited) The presence of other charges does not change the force exerted by point charges. One can obtain the total force by adding or superimposing the forces exerted by each particle separately. Suppose we have a number N of charges scattered in some region. We want to calculate the force that all of these charges exert on some test charge.

We introduce the charge density or charge per unit volume How do we calculate the total force acting on the test charge ?

We chop the blob up into little chunks of volume ; each chunk contains charge. Suppose there are N chunks, and we label each of them with some index. Let be the unit vector pointing from th chunk to the test charge; let be the distance between chunk and test charge. The total force acting on the test charge is This is approximation!

The approximation becomes exact if we let the number of chunks go to infinity and the volume of each chunk go to zero – the sum then becomes an integral: If the charge is smeared over a surface, then we integrate a surface charge density over the area of the surface A: If the charge is smeared over a line, then we integrate a line charge density over the area of the length:

Another example on force due to a uniform line charge A rod of length L has a total charge Q smeared uniformly over it. A test charge q is a distance a away from the rod’s midpoint. What is the force that the rod exerts on the test charge?

Reading Quiz

The electric field y x has the same direction as

Michael Faraday “The best experimentalist in the history of science”

Electric field lines These are fictitious lines we sketch which point in the direction of the electric field. 1) The direction of at any point is tangent to the line of force at that point. 2) The density of lines of force in any region is proportional to the magnitude of in that region Lines never cross.

How to calculate ? 1) Put a “test charge” at some point and do not allow it to move any other charges 2) Calculate the electric force on and divide by to obtain The force that N charges exert on a test charge :

We also calculated the force that a blob of charged material with charge density exerts on a test charge: We wrote the similar formulas if the charge is smeared out over a surface with surface density, or over a line with line density. In all of these cases, the force ends up proportional to the test charge. We might factor it out. This is the electric field!

(N point charges) (Charge continuum) Given an electric field, we can calculate force exerted on some point charge :

Example 1: Electric field of a point charge is directly radially away from or toward the charge. Example 2: Electric field of a dipole

Have a great day! Hw: All Chapter 2 problems and exercises Reading: Chapter 2