RESISTANCE AND CONDUCTIVITY
_Resistance is that property of a substance that opposes the flow of an
electric current through it._
The practical electrician has to measure electrical resistance,
electromotive forces, and the capacities of condensers. Each of these
several quantities is measured by comparison with ascertained standards,
the particular methods of comparison varying, however, to meet the
circumstances of the case.
Ohm’s law states that the strength of a current due to an electromotive
force falls off in proportion as the resistance in the circuit increases.
It is therefore possible to compare two resistances with one another by
finding out in what proportion each will cause the current of a constant
battery to fall off.[5]
_Silver is taken as the standard, with the percentage of 100, and the
conductivity of all other metals is expressed in hundredths of the
conductivity of silver._
=Conductivity of Metals and Liquids.=--The metals in general, conduct
well, hence their resistance is small, but metal wires must not be too
thin or too long, or they will resist too much, and permit only a feeble
current to pass through them. The liquids in the battery do not conduct
nearly so well as the metals, and different liquids have different
resistances. Pure water will hardly conduct at all, unless the voltage be
very high.
Salt and saltpetre dissolved in water are good conductors, and so are
dilute acids, though strong sulphuric acid is a bad conductor. Gases are
bad conductors.
=Effect of Heat.=--Another very important fact concerning the resistance
of conductors is that the resistance in general increases with the
temperature. While this fact is true regarding metals, it does not apply
to non-metals. The resistance of different metals does not increase in the
same proportion. Iron at 100 degrees C, has lost 39 per cent. of the
conducting power it possessed at zero, while silver loses but 23 per cent.
=Laws of Electrical Resistance.=--Resistances in a circuit may be of two
kinds:
1. Resistance of the conductors;
2. Resistance due to imperfect contact.
The latter kind of resistance is affected by pressure, for when
the surfaces of two conductors are brought into more intimate
contact the current passes more freely from one conductor to the
other.
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 pass 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
[Illustration: 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.
[Illustration: FIG. 84.--Hydraulic analogy 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 quantities 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, assuming 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.
Supposing the current to be divided into 7 parts, 4 of these
parts would flow in the first conductor 2 in the second and 1 in
the third.
The whole current is 42 amperes.
4/7 of 42 = 24.
2/7 of 42 = 12.
1/7 of 42 = 6.
Current in first conductor = 24 amperes. }
“ “ second “ = 12 “ } Ans.
“ “ third “ = 6 “ }