or W may be found by multiplying the square of the current by the resistance of the cell, so that 123. A Current Generator may Abstract Energy from a Circuit even when its E.M.F. Helps the Current. When a current is passing through a generator in the direction of its E. M. F. the excess of the potential at the terminal by which the current leaves the generator over the potential at the terminal by which it enters it represents that portion of the E.M.F. of the generator which, not having been used in sending the current through the generator itself, is available for sending it through the external circuit. If, however, the potential at the terminal of the generator by which the current leaves it is lower than the potential at the terminal by which the current enters it, electric energy is absorbed by the generator, whether the generator be joined up so as to help or to oppose the current. For example, the first, second, third, and fourth cells. in Fig. 174 (page 378) are all joined up the same way, and in each of them there is a conversion of chemical energy into electric energy; but in the case of the first cell the potential at the leaving terminal c is cq, a negative quantity, and is, therefore, lower than the potential at the entering terminal K or B, which has been arbitrarily taken as our zero of potential. Hence the drop of potential in this first cell is actually greater than BL the E.M.F. of the cell, for it is equal to cQ + B L. So that, if V is the P.D. in volts measured by a voltmeter attached to the terminals of this first cell, = for this first cell, and not V E - A c, which is the usual relationship when a current is passing through a cell in the direction of its E.M.F. Let W be the rate in watts at which electric energy is taken from the outside circuit by this cell, then W = AV. Also, if W1 is the rate in watts at which chemical energy is converted into electric energy in this cell, W1 = AE; and if W is the rate in watts at which electric energy is converted into heat in the cell, 2 Therefore, combining these last three equations with the equation given above for V, we have Hence, the electric energy which is converted into heat in this cell is greater than that produced by the whole chemical action taking place in it, and, although the current is passing through this first cell in the direction of its E.M.F., the current is greater than would be produced if the cell were short-circuited; hence, the cell abstracts electric energy from the circuit instead of giving electric energy to it, which is what a cell usually does when the current is passing through it in the direction of its E. M. F. If we now substitute in the preceding equations the values of c and E, which are 5 ohms and 1 volt for this first cell, and of A, which is 0.273 ampere for the whole circuit, we have as given above. = 0.0996 0.273 = 0.3726 watt, Next, let us consider the fifth cell, which has been inserted between the points I and J so as to oppose the current. In this case, not merely is the potential J z at the terminal J by which the current leaves the cell lower than the potential IX at the terminal I by which the current enters the cell, but the E.M.F. of the cell opposes the current. Hence, in the cell electric energy will be withdrawn from the circuit and converted into heat, and electric energy will also be withdrawn from the circuit and converted into chemical energy. If V is the P.D. measured by a voltmeter attached to the terminals of the cell, which we have already seen (§ 119, page 367) is the formula for a cell of resistance c ohms when a current of A amperes passes it in opposition to its E. M.F. of E volts. For this cell, E is by hypothesis 1·5 volt and c 1·5 ohm, therefore, V 15+0.273 × 1.5 = 1.909 volt. Hence, if W is the rate in watts at which electric energy is given to this cell, W = 0.273 × 1.909 = 0.5211 watt; also if W1 represents the rate in watts at which electric energy is converted into chemical energy in the cell, W1 = 0.273 × 1.5 = 0 4995 watt, = 2 and if W, represents the rate in watts at which electric energy is converted into heat in the cell, or W2 may be found by multiplying the square of the current by the resistance W, 0.2733× 1.5, = of = 0.1116 watt, as before. When a current is passing through a cell there are four possible distributions of potentials represented by Figs. 175, 176, 177, and 178, the current in each case flowing from left to right. In Fig. 175 the rate of transformation of chemical energy into electric energy exceeds the rate of conversion of electric energy into heat in the cell, and the cell gives electric energy to the external circuit. In Fig. 176 the rate of transformation of chemical energy into electric energy is exactly equal to the rate of conversion of electric energy into heat in the cell, and the cell neither gives energy to the outside circuit nor receives energy from it. This is what happens when a cell is short-circuited with an extremely short thick copper wire, or when the current passing through a cell in a circuit containing other current generators is such that the potential at the terminal of Figs. 175, 176, 177, 178. the cell by which the current leaves the cell is exactly equal to the potential at the terminal by which it enters. In Fig. 177 the rate of transformation of chemical energy into electric energy is less than the rate of conversion of electric energy into heat in the cell, and the cell abstracts energy from the outside circuit. In Fig. 178 the rate at which the cell receives energy from the outside circuit equals the sum of the rates at which electric energy is converted into chemical energy in the cell, and at which electric energy is converted into heat in the cell. Fig. 175 may be likened to a pump which raises water, and also wastes some energy in friction; Fig. 176 to a pump which raises no water, but wastes all the energy it receives in friction in its own mechanism; Fig. 177 to a pump which raises water, wasting some energy in friction, and which is partly driven by allowing the water to subsequently fall to a greater distance than that to which the pump has raised it; Fig. 178 corresponds with a turbine which is driven by falling water, some of the energy of the falling water being wasted in friction of the mechanism. Example 96.-What are the maximum currents that can be passed through the following cells, if they are not to abstract energy from the circuit when they are joined up so that their E.M.F. tends to assist the current? Cell (a) has E.M.F. 1·1 volt and resistance 75 ohm. Cell (b) has E.M.F. 1.5 volt and resistance ·3 ohm. Cell (c) has E.M.F. 2.1 volt and resistance 1 ohm. Answer. (a) 1·47 ampere. 124. External Circuit that Receives Maximum Power from a Given Current Generator.-Let E be the E.M.F. of the current generator in volts, and b its resistance in ohms, then, if A is the current in amperes produced when the terminals of the generator are connected N |