# 03. Analysis Tools 2

### Potential Difference

*The long, hollow plastic cylinder at the bottom has inner radius a, outer radius 2a, and uniform charge density r. Find the electric potential difference between the inner and outer radius.*

pic 1

An alternative method for calculating the electric potential at a point, or the electric potential difference between two points, is by using knowledge of the electric field. The following relation,

pic 2

states that the potential difference between two points can be determined by integrating the component of the electric field that lies along the path connecting the two points. This means that if you know the electric field in a region of space, you can easily (more or less) find the potential difference between any two points that lie in that region. This relation is particularly useful in conjunction with Gauss' Law for situations with cylindrical or spherical symmetry.

To find the potential difference between points a and b (i.e., what a voltmeter would read if connected across points a and b), we need the electric field in this region. Gauss' Law can be used to find that (review Gauss' Law if this step is a little fuzzy):

pic 3

This electric field points radially away from the center of the cylinder.

For simplicity, choose a path that directly connects a to 2a, i.e., a radial path.

pic 4

pic 5

Notice that this method directly calculates the *difference* in electric potential between two points, without actually determining the *value* of the electric potential at either point. Since electric potential is related to electric potential energy, this method allows to you to find the difference in energy between two points but not the actual value of the energy of an electric charge.

This should strike you as quite similar to the gravitational case. For gravitational potential energy, the choice of the zero-point is arbitrary and only energy differences lead to changes in kinetic energy. For electrical potential energy, the situation is identical. The zero-point of electric potential energy (and electric potential) is typically taken at infinity, although you can "zero" the potential at a more convenient point by "grounding" the system at that point. The physical act of grounding a point on an electrical device is mathematically equivalent to setting the potential equal to zero at that point.

### Electric Potential Energy

*In many applications, oppositely charged parallel plates (with small holes cut for the beam to pass through) are used to accelerate beams of charged particles. In this example, a proton is injected at v _{1} into the space between the plates. The potential difference between the plates is DV(the left plate is at higher potential). What is the velocity of the proton as it exits the device?*

pic 6

Since the electric potential changes as the proton moves from the left plate to the right plate, its potential energy changes. This change in potential energy results in a change in kinetic energy of the proton by energy conservation.

pic 7