ON ELECTRICAL WORK AND ENERGY

Excerpts

ON ELECTRICAL WORK AND ENERGY
Since an electrified system is subject to the law of Conservation of Energy, the work expended in charging it is entirely stored up in the system in the form of electrical energy.
ON ELECTRICAL WORK AND ENERGY
If the electrified body is moved in such a way as to remain on the same equipotential surface, no work is done by the electric forces or against them.
ON ELECTRICAL WORK AND ENERGY
There are many such reciprocal relations. They occur in every branch of science, and they often enable us to deduce the solution of new electrical problems from those of simpler problems with which we are already familiar.
ON ELECTRICAL WORK AND ENERGY
Thus when a fish has swallowed the angler’s hook and swims off, the angler following him for fear his line should break, the fish is doing work against the angler, but when the fish becomes tired and the angler draws him to shore, the angler is doing work against the fish.
ON ELECTRICAL WORK AND ENERGY
Work is always measured by the product of the change of configuration into the force which resists that change. Thus, when a man lifts a heavy body, the change of configuration is measured by the increase of distance between the body and the earth, and the force which resists it is the weight of the body. The product of these measures the work done by the man. If the man, instead of lifting the heavy body vertically upwards, rolls it up an inclined plane to the same height above the ground, the work done against gravity is precisely the same; for though the heavy body is moved a greater distance, it is only the vertical component of that distance which coincides in direction with the force of gravity acting on the body.
ON ELECTRICAL WORK AND ENERGY
In fact, this doctrine is the one generalised statement which is found to be consistent with fact, not in one physical science only, but in all. When once apprehended it furnishes to the physical enquirer a principle on which he may hang every known law relating to physical actions, and by which he may be put in the way to discover the relations of such actions in new branches of science.
ON ELECTRICAL WORK AND ENERGY
*The electric potential at a given point of the field is measured by the amount of work which must be done by an external agent in carrying one unit of positive electricity from a place where the potential is zero to the given point.*
ON ELECTRICAL WORK AND ENERGY
Hence the magnitude of the electric force may be found by dividing the difference of the potentials of two neighbouring equipotential surfaces by the distance between them, the distance being, of course, very small, and measured perpendicularly to either surface.
ON ELECTRICAL WORK AND ENERGY
By increasing without limit the number of equal parts into which the charge is divided, the breadth of the parallelograms will be diminished without limit. In the limit, therefore, the difference of the two values of the work vanishes, and either value becomes ultimately equal to the area CAFGHBD, bounded by the curve, the extreme ordinates, and the base line.
ON ELECTRICAL WORK AND ENERGY
This is the first instance we have met with of the *reciprocal* relation of two bodies. There are many such reciprocal relations. They occur in every branch of science, and they often enable us to deduce the solution of new electrical problems from those of simpler problems with which we are already familiar.
ON ELECTRICAL WORK AND ENERGY
Hence, when the potential of each conductor is maintained constant during a displacement in which a quantity of work, W, is done, the voltaic batteries which are employed to keep the potentials constant must do an amount of work equal to 2W. Of this energy supplied to the system, half is spent in increasing the energy of the system, and the other half appears as mechanical work.

Equations

ON ELECTRICAL WORK AND ENERGY
Q = \tfrac{1}{2}\sum(EP)
The electrical energy of a system of conductors is half the sum of the products of each conductor's charge and potential.
ON ELECTRICAL WORK AND ENERGY
Q' = Q + \tfrac{1}{2}\sum\{(E' - E)(P' + P)\}
After the charges change, the energy equals the original energy plus half the sum of each charge increment times the sum of its initial and final potentials.
ON ELECTRICAL WORK AND ENERGY
\frac{Q' - Q}{E' - E} = \tfrac{1}{2}(P' + P)
When all charges but one are held constant, the rate of increase of energy with that charge equals half the sum of the initial and final potentials.
ON ELECTRICAL WORK AND ENERGY
\frac{dQ_e}{dE} = P
The rate at which the electric energy grows with the charge of one conductor, the others held constant, equals that conductor's potential.
ON ELECTRICAL WORK AND ENERGY
\sum(EP') = \sum(E'P)
In a fixed system of conductors, the sum of original charge times final potential equals the sum of final charge times original potential.
ON ELECTRICAL WORK AND ENERGY
\tfrac{1}{2}\sum\{(E' - E)(P' + P)\} = Q' - Q = \tfrac{1}{2}\sum\{(E' + E)(P' - P)\}
The increment of energy of a fixed system equals half the sum of charge increments times summed potentials, and equally half the sum of potential increments times summed charges.
ON ELECTRICAL WORK AND ENERGY
\frac{Q' - Q}{P' - P} = \tfrac{1}{2}(E' + E)
When all potentials but one are held constant, the rate of increase of energy with that potential equals half the sum of the initial and final charges.
ON ELECTRICAL WORK AND ENERGY
\frac{dQ_p}{dP} = E
The rate at which the electric energy grows with the potential of one conductor, the others held constant, equals that conductor's charge.
ON ELECTRICAL WORK AND ENERGY
E_t{P_t}' = 0\quad\text{and}\quad {E_t}'P_t = 0
If a conductor is insulated and uncharged in both the initial and final states, the terms involving it vanish from the reciprocal relation.
ON ELECTRICAL WORK AND ENERGY
E_r{P_r}' + E_s{P_s}' = {E_r}'P_r + {E_s}'P_s
With all but two conductors insulated and uncharged or earthed, the reciprocal relation holds between the two remaining conductors.
ON ELECTRICAL WORK AND ENERGY
E_r{P_r}' = {E_s}'P_s
Charge on A_r times its final potential equals the final charge on A_s times the potential of A_s, in the two-conductor case.
ON ELECTRICAL WORK AND ENERGY
\frac{P_s}{E_r} = \frac{{P_r}'}{{E_s}'}
The ratio of the potential of A_s to the charge on A_r equals the ratio of the potential of A_r to the charge on A_s.
ON ELECTRICAL WORK AND ENERGY
E_r = {E_s}'
The charge on A_r equals the final charge on A_s, in the reciprocity case of Theorem V.
ON ELECTRICAL WORK AND ENERGY
P_s = {P_r}'
The potential of A_s produced by a charge on A_r equals the potential of A_r produced by an equal charge on A_s.
ON ELECTRICAL WORK AND ENERGY
0 = {E_r}'P_r + {E_s}'P_s
With A_r charged and A_s earthed, the final charges and original potentials of the two conductors sum to zero.
ON ELECTRICAL WORK AND ENERGY
\frac{P_s}{P_r} = -\frac{{E_r}'}{{E_s}'}
The ratio of the two potentials equals minus the ratio of the induced final charges.
ON ELECTRICAL WORK AND ENERGY
P_s = nP_r
The potential of A_s is n times the potential of A_r in the induction case.
ON ELECTRICAL WORK AND ENERGY
{E_r}' = -n{E_s}'
The charge induced on A_r is minus n times the charge induced on A_s, where n is the potential ratio.
ON ELECTRICAL WORK AND ENERGY
Q = W + Q'
The original energy of an insulated system equals the work done by the electric forces plus the final energy, by conservation of energy.
ON ELECTRICAL WORK AND ENERGY
W = \tfrac{1}{2}\sum[E(P - P')]
The work done during a displacement of an insulated system is half the sum of each charge times the potential drop of its conductor.
ON ELECTRICAL WORK AND ENERGY
Q' = \tfrac{1}{2}\sum(EP')
The energy remaining in the system after a displacement is half the sum of charge times final potential.
ON ELECTRICAL WORK AND ENERGY
W = \tfrac{1}{2}\sum[E(P - P_1)]
For a small displacement of an insulated system, the work is half the sum of charge times the potential change.
ON ELECTRICAL WORK AND ENERGY
\sum(EP - E_1P_1) = 0
Restoring the original potentials by changing charges leaves the sum of charge times potential unchanged.
ON ELECTRICAL WORK AND ENERGY
W = \tfrac{1}{2}\sum[(E_1 - E)P_1]
The work of an alternating displacement and charge adjustment is half the sum of the charge change times the potential.
ON ELECTRICAL WORK AND ENERGY
W = \tfrac{1}{2}\sum[(E' - E)P]
For a displacement with each conductor's potential held constant, the work is half the sum of the charge increment times the constant potential.
ON ELECTRICAL WORK AND ENERGY
W = \tfrac{1}{2}\sum(E'P) - \tfrac{1}{2}\sum(EP)
The work done with constant potentials equals the final electric energy minus the initial electric energy.
ON ELECTRICAL WORK AND ENERGY
W = Q' - Q
With each conductor's potential held constant, the work done by the electric forces equals the increase of electric energy.

Problems

No exercises in this chapter.