W = q E (z2 - z1).
If q is all alone between these plates, it will increase its kinetic energy by the amount of work done.
In working with electric circuits, we will rarely need to think explicitly in terms of electric fields produced by point charges. Instead we will make use of the ideas of electric potential and electric current. Let's assume the electric field is uniform, as can be produced between two large, closely spaced charged plates (a capacitor). Then E is a constant, and we'll assume it is in the z direction. The work done by the electric force on a charge q in this field is found by recalling the definition of work in terms of forces and displacements. For an infinitesimal displacement dz of the charge q, the work done by the electric force on it is dW = Fdz = qEdz. If the charge is moved from z1 to z2, the total work done is found by integrating dW, which in this situation is trivial:
W = q E (z2 - z1).
If q is all alone between these plates, it will increase its kinetic energy by the amount of work done.
An alternate way of accounting for work in this situation is to invent the electric potential energy, U, and equate the work done to a decrease in the potential energy. Then the increase in kinetic energy of the charged particle can be equated to the decrease in electric potential energy.
Note that the force experienced by q can be related to the derivative of the potential energy with respect to position:
Fz = - dU/dz.
Recall that as a result, only differences in potential energy are important.
The value of U can be adjusted up or down everywhere by an arbitrary additive constant without changing the forces that act or the resulting motion of the charge.
In working with electrostatic systems and with electric circuits, the behavior of the system is more often characterized by the electric potential, V, instead of electric potential energy U, which depends on the amount of charge, q, placed in the electric field. The difference in electric potential between two locations is given in terms of the difference in potential energy and the charge:
V2 - V1 = (U2 - U1) / q.
This is independent of q in the end, since U itself is proportional to q.
The difference in electric potential is more commonly referred to as the voltage
between two points. The electric potential indicates the amount of potential energy that one unit (i.e. one coulomb) of charge will have when it is placed at a particular location in a region with an electric field. So it is often referred to as the energy per unit charge. It carries dimensions of energy/charge, which in S.I. units will be joules/coulombs. A point in space where the potential (or voltage) is such that 1 C of charge will have 1 J of potential energy is said to be at a potential (or voltage) of 1 volt.
In the case of the charged parallel plates, the difference in potential
between points 1 and 2,
V2 - V1 = - E (z2 - z1),
so that the potential decreases as one moves from the left plate to the right plate. A positive charge q placed between the plates is accelerated by the electric force it experiences toward the right, toward the region of lower potential. So we can conclude that positive charges tend to move toward locations of lower voltage.
Negative charges placed between the plates experience a force in the opposite direction, toward the positively charged plate. For these charges, the lowest potential energy is toward the left (a consequence of a negative q), but this still corresponds to a region of higher potential or voltage.
The potential between the plates is independent of what kind of test charge q we choose to use. So we can conclude that negative charges tend to move
to locations of higher potential or voltage. Both positive and negative charges
still tend to move toward locations of lower potential energy, as required by the relation between the force and the derivative of the potential energy. It is important to understand what electric potential measures, and how and why charges respond to differences in potential or voltage. Also note that the electric field can be found directly from the electric potential:
Ez = -dV/dz.
A simple analogy with gravitational forces and gravitational potential energy is a useful tool in thinking about the ideas of voltage and, shortly, electric current. Imagine skiers (our analog of the test charges q) on a ski slope. A skier (of mass m) at the top has gravitational potential energy Ug(y2) = mgy2.
The skier trades potential energy for kinetic energy on the trip down. Neglecting friction and any falls along the way, and assuming the skier starts from rest (K=0 at the top) the skier reaches the bottom with a kinetic energy equal to the decrease in potential energy, K=mg(y2 - y1). Skiers naturally tend toward
locations of lower potential energy because, again, the force points "downhill". Note that the change in kinetic energy depends only on the height of the hill, y2 - y1, not on how high the ski slope is above some arbitrary reference level, such as sea level.
In this analogy, since the potential energy is proportional to the mass of the skier, the quantity analogous to charge must be mass. The strength of the local gravitational field is measured by g, the force per unit mass (or acceleration) and plays the role of E. Note the direction of the gravitational field is down for this situation. Only the vertical displacement, y2 - y1 is relevant because that is the only component parallel to the force of gravity when calculating work done.
[If one found a piece of matter with negative gravitational mass (but still a positive inertial mass), it would, like negative electric charges, run uphill, opposite to the direction of the gravitational field.]
A gravitational analog to electric potential can also be invented by the same scheme as before: dividing the gravitational potential energy by the "gravitational charge" -- the mass of the object, m: Vg = Ug / m = gy. Again, only differences in the gravitational potential are significant; absolute values of Vg depend on where you choose to take y=0, i.e. "true ground".
Continue to the next topic, electric current.