Page:Elementary Text-book of Physics (Anthony, 1897).djvu/336

322 be steady, this number is the same whatever be the radius of the circle; it therefore expresses also the number of unit tubes which enter the wire in unit of time.

We have already seen that the energy carried into the conductor by $$Q$$ unit tubes, when the difference of potential is maintained constant, is equal to $$Q (V_{1} - V_{2})$$ that is, is twice the energy associated with these tubes when at rest. It has also been shown that the energy in unit length of a unit tube at rest is $$\frac{F}{2}$$ and $$F$$ therefore measures the energy carried into the conductor by unit length of each unit tube. In the case before us, the energy transferred in one second through a cylindrical surface of unit height and of radius $$r,$$ concentric with the wire, is $$2\pi rv NF.$$ Now, on the view of the current here taken, the number of unit tubes which disappear in one second is equal to the current strength, so that $$2\pi rv N = I.$$ The energy introduced through the cylindrical surface is therefore $$FI.$$ Since in this case the difference of potential equals the electrical force multiplied by the length of the wire, the energy introduced into the whole wire is $$I (V_{1} - V_{2}).$$ The energy which passes through unit area of the cylindrical surface is $$\frac{FI}{2\pi r}\cdot$$ It may be shown that the magnetic force due to the current at the distance $$r$$ is $$P = \frac{2I}{r},$$ and hence the energy which passes through unit area may also be represented by $$\frac{FP}{4\pi}\cdot$$

The example here given is a special case of a general theorem due to Poynting. This theorem asserts that the energy expended in the current enters the conductor from the dielectric, passing at right angles to the lines of electrostatic force and the lines of magnetic force, and that the amount of energy which passes perpendicularly through unit area is proportional to the electrostatic force and to the magnetic force.