Hawkins Electrical Guide, Number One Part 11

The following are the laws of the resistance of conductors:

1. _The resistance of a conducting wire is proportional to its length._

If the resistance of a mile of telegraph wire be 13 ohms, that of fifty miles will be 50 13 = 650 ohms.

2. _The resistance of a conducting wire is inversely proportional to the area of its cross section_, and therefore in the usual round wires _is inversely proportional to the square of its diameter._

Ordinary telegraph wire is about 1/6th of an inch thick; a wire twice as thick would conduct four times as well, having four times the area of cross section; hence an equal length of it would have only 1/4th the resistance.

3. _The resistance of a conducting wire of given length and thickness depends upon_ the material of which it is made--that is, upon the specific resistance _of the material_.

=Conductivity.=--This is _the inverse of resistance_. The term expresses the capability of a substance to conduct the electric current.

If the symbol Y represent the conductivity of a substance, and I the current then:

I/Y = its resistance;

and if R represent the resistance of a substance, then

I/R = its conductivity.

Good conductors of heat are also good conductors of electricity.

=Specific Conductivity.=--The figure which indicates the relation between one substance and another as to their capacity to conduct electricity is called _specific_ or _relative conductivity_. Taking the specific conductivity of silver as 100, that of pure copper is 96.

The specific resistance of a substance is the reverse of its relative conductivity. The specific resistance of a metal is generally expressed in millionths of an ohm as the resistance of a centimeter cube of that metal between opposite sides.

The following table gives the data for a few metals:

Specific Resistance Specific Substance. in Microhms. Conductivity.

Silver 1.609 100.

Copper 1.642 96.

Gold 2.154 74.

Iron (soft) 9.827 16.

Lead 19.847 8.

German Silver 21.470 7.5 Mercury (liquid) 96.146 1.6

The specific resistance of copper is therefore:

1.642 / 1,000,000 ohms, or 1.642 microhms.[6]

=Divided Circuits.=--If a circuit be divided, as in fig. 83, into two branches at A, uniting again at B, the current will also be divided, part flowing through one branch and part through the other.

_The relative strength of current in the two branches will be proportional to their conductivities._

This law will hold good for any number of branch resistances connected between A and B. Conductivity is, as shown before, the reciprocal of resistance.

EXAMPLE--If, in fig 83, the resistance of R = 10 ohms, and R' = 20 ohms, the current through R will be to the current through R'

as 1/10 to 1/20; or, as 2:1, or, in other words, 2/3 of the total current will pa.s.s through R and 1/3 through R'. The joint resistance of the two branches between A and B will be less than the resistance of either branch singly, because the current has increased facilities for travel. In fact, the joint conductivity will be the sum of the two separate conductivities.

Taking again the resistance of R = 10 ohms and R' = 20 ohms, the joint conductivity is

1/10 + 1/20 = 3/20

and the joint resistance is equal to the reciprocal[7] of 3/20

or 6-2/3

[Ill.u.s.tration: FIG. 83.--Divided circuit with two conductors _in parallel_.]

In most cases the resistance of the different branches will be alike. This simplifies the calculations considerably. Take, for instance, two branches of 100 ohms resistance each and find the joint resistance.

SOLUTION: 1/100 + 1/100 = 2/100; the reciprocal is 100/2 = 50 ohms, or, in other words, the joint resistance is one-half of the resistance of a single branch, and each branch, of course, will carry one-half of the total current in amperes.

With three branches of equal resistance, the joint resistance will be 1/3; with four branches 1/4; with 100 branches 1/100 of the resistance of a single branch.

[Ill.u.s.tration: FIG. 84.--Hydraulic a.n.a.logy for divided circuits. In the system of pipes shown, water flows from A B to C D through the six vertical pipes 1 to 6, the greatest amount going through the one which offers the least resistance. If pipes 1 to 6 all have the same dimensions, equal quant.i.ties of water will flow through them. It follows that the resistance which the water encounters diminishes with the increase in the number of pipes between A B and C D. The electrical circuit presents the same conditions: the greater the number of parallel connections (corresponding to the pipes 1 to 6) the less is the resistance encountered by the current.]

If, for instance, the resistance of an incandescent lamp hot be 180 ohms, the joint resistance of 100 such lamps connected in multiple is

180/100 = 1.8 ohms.

If the electromotive force of the system is to be, say 110 volts, then, according to Ohm's law, the current for 100 lamps is:

110/1.8 = 61.11 amperes.

giving for each lamp a current of

110/180 = .61 ampere.

In the case of two branches only, the following rule may be applied also:

_Multiply the two resistances and divide the product by their sum._

Written as a formula:

Joint resistance = (R R')/(R + R')

Again, a.s.suming that R = 10 ohms and R' = 20 ohms:

Joint resistance (10 20)/(10 + 20) = 200/30 = 6-2/3 ohms.

This rule _cannot_ be employed for more than two branches at a time.

EXAMPLE--A current of 42 amperes flows through three conductors in _parallel_ of 5, 10 and 20 ohms resistance respectively. Find the current in each conductor.

SOLUTION: Joint Conductance = 1/5 + 1/10 + 1/20 = 7/20.