Power in Electric Circuits
A circuit consisting of a battery B that is connected by wires, which we assume have negligible resistance, to an unspecified conducting device. The device might be a resistor, a storage battery (a rechargeable battery), a motor, or some other electrical device. The battery maintains a potential difference of magnitude V across its own terminals. and thus (because of the wires) across the terminals of the unspecified device, with a greater potential at terminal a of the device than at terminal h. Since there is an external conducting path between the two terminals of the battery, and since the potential differences setup by the battery are maintained, a steady current i is produced in the circuit, directed from terminal a to terminal h. The amount of charge do that moves between those terminals in time interval it is equal to i dr. This charge dq moves through a decrease in potential of magnitude V, and thus its electric potential energy decreases in magnitude by the amount.
dU = dq V = i dr V.
The principle of conservation of energy tells us that the decrease in electric potential energy from a to b is accompanied by a transfer of energy to some other.
form. The power P associated with that transfer is the rate of transfer which is
F = iF (rate of electric energy transfer).
Moreover, ‘his power P is also the rate that energy is transferred from the battery to the unspecified device. If that device is a motor connected to a mechanical load, the energy is transferred as work done on the load. If the device is a storage battery that is being charged. the energy is transferred to stored chemical energy in the storage battery. If the device is a resistor, the energy is transferred to internal thermal energy, tending to increase the resistor’s temperature. The unit of power that follows from is the volt-ampere (Y . A). We can write it as
The course of an electron moving through a resistor at constant drift speed is much like that of a stone falling through water at constant terminal speed. The average kinetic energy of the electron remains constant. and its lost electric potential energy appears as thermal energy in the resistor and the surroundings. On a microscopicscale this energy transfer is due to collisions between the electron and the molecules of the resistor. which leads to ‘In increase in the temperature of the resistor lattice. The mechanical <energy thus than-feared to thermal energy is dissipated (lost). because the transfer cannot be reversed. For a r sister or some other device with resistance R. we can combine Eqs.27-8 (R = l’li\ and ~7-21 .to obtain. for the rate of electric energy dissipation due to a resistance. either
You are given a length of uniform heating wire made of a nickel chromium- iron alloy called chlorine: it has a resistance R of T!. n. At what rate is energy dissipated in each of the following situations? (I) A potential difference of 120 V is applied across the full length of the wire. (2) The wire is cut in half. and a potential difference of 120 V is applied across the length of each half.
The Key Idea is that a current in a resistive material produces a transfer of mechanical energy to thermal energy; the rate of transfer (dissipation) is given by Eqs. 27-21 to 27-23. Because we know the potential V and resistance R. we use Eq. 27-23, which yields, for situation I