We may give an analogy from hydrostatics to make the relation between E. M. F. and A V somewhat clearer.

Let us imagine two reservoirs, the levels of water in the two being L, and L2 respectively, connected by a pipe filled with coarse sand.

A current of water will be urged through the pipe, and its magnitude will depend, cæteris paribus, on the pressure due to the difference of level. The pressure can be measured by the difference of level (L-L); we might, indeed, if our system of units is suitably fixed, say that the pressure 'is' the difference of level; this would be scientifically inaccurate, but-with our fixed system of units would lead to no error in calculations.

So we can measure the E.M.F. of a cell by the AV that appears at the poles when the circuit is broken; or by the greatest V that we discover between two points in the circuit when the current is flowing. And we may sometimes, somewhat inaccurately perhaps, use AV as synonymous with E.M.F.

This E.M.F. of a cell depends solely upon the nature of the cell. As we include the coatings of gases on the plates, &c., in the term 'nature of the cell,' the reader will see that we have taken into account the phenomenon of 'Polarisation' (see Chapter XI. § 10), which diminishes the E.M.F. of the cell from its initial value.

However large a current be flowing, the E.M.F. remains unaltered save by polarisation.

II. Resistance. Somewhat as, in the hydrostatic analogy given above, the pipe will offer more or less resistance to the current of water according to its length, its section, and the closeness with which it is packed with gravel or sand, so a conductor in the electric circuit will offer more or less resistance according to its length, section, material, and temperature.

Resistance is that which hinders the passage of a current and limits its magnitude without producing any tendency to a backcurrent; it is purely passive.

III. Current. This term has been explained earlier. It simply means quantity of electricity passing across any section of the circuit in one second of time. We may measure this quantity as in Chapter V. § 1; but in dealing with electric currents another system of units (explained in Chapters XIV. § 1, and XVIII. $$ 3 and 4) will be employed.

§ 2. Exact Statement of Ohm's Law.-Let AB be a conductor including no source of difference of potential. And let there

be means of maintaining the points

A and B at various ▲ V.s, and of measuring these AV.s by electrometer or other methods; and also of measuring by electrolysis or galvanometer methods the current that then flows in the conductor A B.

Then the experimental law known as 'Ohm's Law' is that

RA

Battery

'So long as the conductor A B remains unaltered in temperature and in all other physical respects, the current flowing in it is directly proportional to the difference of potential maintained between its extremities A and B.

We may in symbols express this by saying that

C∞ (VA-VB).

On trying how the current varies when the conductor is varied, the AV between the extremities being maintained constant, our observations lead us to follow the hydrostatic analogy and to say that the conductor 'offers a greater or less resistance' to the current according to the dimensions, material, temperature, &c. Symbolising this resistance by R, we define RA by the relation

B

Α

It occurs to us to inquire next whether 'Ohm's Law' applies to the entire circuit, including both connecting wires and battery. The difficulty now is to know what replaces the 'VA-VB' of the experimental law given above. Let E represent the electrostatic AV between the terminals of the battery when the circuit is broken. Let B and represent respectively the resistances of battery cell and of connecting wires (see §§ 3 and 4, and Chapter XIV.) Then a great mass of evidence has convinced investigators that when the current is not dense enough to cause polarisation or other alteration in the battery, we may state that

And if there are many cells in the circuit we have

where the E.M.F.s are to be reckoned as negative if they oppose the resulting current.

Note. It is in fact believed that Ohm's law always holds, but that if the current is too dense then E and B have some new values not known from our previous measurements.

§ 3. Resistance further Discussed.-We will now discuss further the nature of resistance, and the physical meaning of Ohm's law.

Ohm's law connects the three quantities C, E, and R ; of which, up to the present point, only the two quantities C and E have received an exact meaning.

We might therefore be inclined to think that the law simply defines R as such that 'C

all, but a mere truism.

=

F

R

;' and therefore is no discovery at

If this were the case the law would be of little use to us, since R might not be a fixed property of the conductor, one to be found once for all, but might depend upon E, so that the law would not then lend itself to the most important problem, viz., that of determining the current from a knowledge of the E.M.F.s and of the nature and dimensions of the conductor.

Thus it might have been the case that when once a current was started with a certain E.M.F. E, the resistance of the conductor was once for all broken down; so that an E. M. F. of 2E with the same circuit might give more than twice the current.

But direct experiment gives us two important results.

(i.) That R is a definite quantity for each conductor, and capable of measurement as such. Hence we can predict the current which any given E.M.F. will drive through any given conductor.

(ii.) That R depends in a very simple way on the length, cross section, temperature, and material of the conductor.

§ 4. The Exact Conditions on which Resistance Depends.— If the resistance of such bodies as wires be examined experimentally, it is found that R is directly proportional to the length of the

wire, inversely proportional to its sectional area A, and depends also on its 'specific resistance' c, and on its temperature.

We will define later the exact meaning of 'specific resistance,' and we will for the present omit the factor expressing the dependence of the resistance on temperature. Thus we may state that

Here k is a constant depending on the unit of resistance employed. The nature of the experimental methods of proving the above is indicated in what follows.

Experiments.—(i.) External resistance.- If we wish to examine most simply the law of resistance of such external conductors as wires, we must get rid of the internal resistance B of the battery (see § 2, equation (ii.)).

We can practically make B = by employing, instead of an ordinary battery, a thermo-cell. This can be constructed of metallic bars of such thickness as to make their resistance relatively negligible; and a constant E. M. F. can be insured by keeping the two sets of junctions at two fixed temperatures (see Chapter XVI.).

With such a battery the formula of Ohm's law becomes C

=

E

We can measure C by means of a galvanometer, as will be explained fully in Chapter XVII. Then, by varying the length, material, and cross section of c. l the wire, we are led to the result that r = k A

It is to be noticed that this result proves that the current flows through the whole body of a wire equally, and not along its surface. Provided that the temperature of the wire is constant, it makes no difference whether a cross section of (e.g.) 1 sq. cm. be circular, square, or of the form of a flat rectangle. Here is a notable difference between conductors as used for electrostatic purposes, and as used for conveying a current, respectively.

(ii.) Internal resistance of a battery.—The internal resistance of a battery is a far more difficult matter to investigate. Indeed the conditions are so complex, that it is invariably the rule to find experimentally the resistance of a cell or battery, instead of to calculate it- -as we do in the case of a wirefrom a knowledge of the materials and their dimensions.

But still it is possible to show that in all probability the same general laws hold; or that B = k.

c.l
A

To investigate internal resistance we get rid of the external resistance r by employing very thick copper conductors, of no appreciable resistance, outside

the cell. The cell itself is constructed with plates of various sizes (so as to vary A), whose distance apart can be altered (so as to vary /). The same general results are found to hold; though, since we cannot confine the current to that portion of the liquid which lies directly between the opposed plates, we cannot arrive at very exact results.

It is important to note that—

(i.) n cells in series offer to the current n times the resistance of one cell, since, in the former, the length / of liquid traversed is n times that in the latter.

(ii.) If n cells be coupled zincs to zincs and coppers to coppers, or 'in parallel,' we practically make one large cell of plates n times the area of those of one cell. Hence the area A of the column of liquid traversed is increased n-fold; and the resistance is th of

I

that of one cell, and of that offered by n cells in series.

I

R❜

I

R

n

§ 5. Conductivity. Since C = E., we may call by the name 'conductivity,' and say that the current is directly proportional to the conductivity, instead of inversely proportional to the resistance, of the circuit.

...

In § 9, fig. iii., we represent two points A and B in a circuit joined by several wires of resistances 1, 2, 7‍3, 74 respectively. We may call their conductivities C, C2, C3, and c1 . . . respectively; where ‹1 = -, &c.

I

Now it is possible to replace these wires by one wire of resistance R', such that the current is unaltered. When this is the

case we call R' the equivalent resistance; and if c' =

will be the equivalent conductivity.

Now it hardly seems to require proof that

I

R'

then c'

which enables us to determine that resistance which is equivalent

to, and might replace, the many branches.