Electrical Potential

The electrical potential at a point is the work which would be done by the electrical forces of the system in moving a unit quantity of positive electricity from the point to the boundary of the field, supposing this could be done without disturbing the electrification of the rest of the bodies in the field.

We may put this in other words, and say that the electrical potential at a point is the E.M.F. between that point and the boundary of the field.

It is clear from this definition that the potential at all points of the boundary is zero.

The work done by the forces of the system, in moving a quantity e of positive electricity from a point at potential V to the boundary, is clearly Ve, and the work done in moving the same quantity from a point at potential V₁ to one at potential V₂ is e(V₁-V₂).

Hence, it is clear that the E.M.F. between two points is the difference of the potentials of the points.

We are thus led to look upon the electric field as divided up by a series of surfaces, over each of which the potential is constant The work done in moving a unit of positive electricity from any point on one of these to any point on another is the same.

When two points are at different potentials there is a tendency for positive electricity to flow from the point at the higher to that at the lower potential. If the two points be connected by a conductor, such a flow will take place, and unless a difference of potential is maintained between the two points by some external means, the potential will become equal over the conductor; for if one part of the conductor be at a higher potential than another, positive electricity immediately flows from that part to the other, decreasing the potential of the one and increasing that of the other until the two become equalised.

Now the earth is a conductor, and all points, not too far apart,⁽¹⁾ which are in metallic connection with the earth are at the same potential.

It is found convenient in practice to consider this, the potential of the earth, as the zero of potential; so that on this assumption we should define the potential at a point as the work done in moving a unit of positive electricity from that point to the earth. If the work done in moving a unit of positive electricity from the earth to the boundary of the field be zero, the two definitions are identical; if this be not the case, the potential at any point measured in accordance with this second definition will be less than its value measured in accordance with the first definition by the work done in moving the unit of positive electricity from the earth to the boundary of the field; but . since electrical phenomena depend on difference of potential, it is of no consequence what point of reference we assume as the zero of potential, provided that we do not change it during the measurements. In either case the E. M. F. between two points will be the difference of their potentials. Potential corresponds very closely to level or pressure in hydrostatics. The measure of the level of the water in a dock will depend on the point from which we measure it, e.g. high watermark, or the level of the dock-sill below high water-mark; but the flow of water from the dock if the gates be opened will depend not on the actual level, but on the difference between the levels within and without the dock, and this will be the same from whatever zero we measure the levels.

Various methods have been discovered for maintaining a difference of potential between two points connected by a conductor, and thus producing between those points a continuous flow of electricity; the most usual are voltaic or galvanic batteries.

For the present, then, let us suppose that two points A and B are connected with the poles of a battery, A and B being points on a conductor, and let us further suppose that the pole of the battery connected with A is at a higher potential than that connected with B. The pole connected with A is said to be the positive pole. A continuous transfer of positive electricity will take place along the conductor from A to B. Such a transfer constitutes an electric current.

Let PQ (fig. 56) be any cross-section of the conductor between the points A and B, dividing it into two parts. Then it is found that during the same interval the quantity of electricity which in a given time (say one second) flows across the section PQ is the same for all positions of PQ, provided only that A and B are on opposite sides of the section. Thus, if in the figure P'Q' be a second section, then at each instant the same quantity of electricity crosses PQ and P'Q' per second.

The laws of the flow of electricity in conductors resemble in this respect those which regulate the flow of an incompressible fluid, such as water, in a tube; thus, if the conductor were a tube with openings at A and B, and if water were being poured in at A and flowing out at B, the tube being kept quite full, then the quantity of water which at any time flows in one second across any section of the tube, such as P Q, is the same for all positions of P Q, and as in the case of the water the quantity which flows depends on the difference of pressure between A and B, so with the electricity, the quantity which flows depends on the E.M.F., or difference of potential between the points.⁽²⁾

(1)	If the points are far apart, electro-magnetic effects are produced by the action of terrestrial magnetism.
(2)	Maxwell's Elementary Electricity, §64.