Electricity Tutorial 2 - Ohm’s Law
Ohm's Law
Resistance is the opposition to the flow of an electric current. Resistance in a conductor is thought to arise due to the collisions between the charge carriers and the ions in the lattice. The internal energy rises, so the conductor gets hot. The hotter the conductor, the greater the probability of a collision between an ion and an electron. The resistance in hot conductors rises.
Resistance is also the ratio of the voltage to the current, described in the simple equation R = V/I. In a metallic conductor, we find that if we alter the voltage or the current, the other variable changes in such a way that the ratio remains constant.
This
is Ohm’s Law, which states:
The current in a metallic conductor is directly proportional to the potential difference between its ends provided that the temperature and other physical conditions are the same.
A conductor that obeys Ohm’s
Law is called an ohmic conductor.
However, if the temperature does not stay the same, Ohm’s Law does not apply, and it is easy to end up with anomalous results. When doing experiments on Ohm’s Law, it is a good idea to turn the power supply off because it:
Stops the batteries from running down;
Prevents the resistor getting hot and increasing its resistance.
Ohm's Law applies to metallic conductors and carbon. It does not apply to conducting materials like liquid electrolytes, for example a solution of sodium chloride. Semi-conductors also show different behaviour, as we will see later.
What are the key points to Ohm’s Law? |
Conductance is the reciprocal of resistance. It’s a term often used by electrical engineers in preference to resistance. Conductance is given the physics code G and the units Siemens (S). It is related to resistance by the equation:
So it doesn’t take a genius to see that:
Make sure you read carefully the axes on the graph.
In the exam, they could be either way round. |
A component takes a current of 0.35 A from a 12 V supply. What is the resistance of the component? |
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And what is its conductance? |
Reactance (A-level only)
Certain components that are used with Alternating currents (AC) have a kind of resistance called reactance. We will use the definition of resistance as:
The ratio of the potential difference to the current.
We find that that the reactance is the ratio of the RMS voltage to the RMS current. We will study this in more detail in the AC Theory tutorials.
For the reactance of a capacitor, physics code X_{C}, we can write:
The units for reactance are Ohm (W), just like resistance.
Similarly for the reactance of an inductor, X_{L}, we can write:
Changing Physical Conditions (Extension)
When we defined Ohms Law for a metallic conductor, we had a proviso that the temperature and other physical conditions stayed the same. What happens if they don't remain the same? The simple answer is that the resistance changes. The most obvious ways of changing the physical conditions are:
Heating the wire up;
Stretching the wire.
If the wire gets hot, the internal energy increases so that the chances of a collision between an ion and a proton are greater. Therefore the resistance of the wire increases.
If we stretch the wire, which has a fixed volume, the length will increase. Therefore the width will decrease. Therefore the resistance will increase. We will think about this case more with resistivity.
The way the resistance of a wire changes with temperature depends on:
the change in temperature;
the material of the wire itself.
When we refer to a resistance of a resistor, the convention is that the resistance is assumed to be measured at 20^{ o}C, although some authorities use 0^{ o}C. You don't need to know this for A-level. The change in resistance at other temperatures is given by the formula:
The terms are:
R - resistance (W);
R_{0} - reference resistance (W);
a - temperature coefficient of resistance for the conductor (^{o}C^{-1});
Dq - the temperature change (^{o}C).
The temperature change is the difference between the quoted temperature (q) and the reference temperature (q_{0}). It does not matter if you quote temperatures in Kelvin (K) or Celsius; it's the difference that matters.
Here are some values for the temperature coefficients:
Material |
Element/Alloy |
a /^{o}C^{-1} |
Nickel |
E |
0.005866 |
Iron |
E |
0.005671 |
Molybdenum |
E |
0.004579 |
Tungsten |
E |
0.004403 |
Aluminium |
E |
0.004308 |
Copper |
E |
0.004041 |
Silver |
E |
0.003819 |
Platinum |
E |
0.003729 |
Gold |
E |
0.003715 |
Zinc |
E |
0.003847 |
Steel |
A |
0.003000 |
Nichrome |
A |
0.00017 |
Nichrome V |
A |
0.00013 |
Manganin |
A |
0.000015 |
Constantan |
A |
0.000074 |
Source: https://www.allaboutcircuits.com/textbook/direct-current/chpt-12/temperature-coefficient-resistance/
Constantan makes a particularly good resistance wire as the temperature coefficient is very low. Therefore the resistance remains constant (to 2 s.f.) over a wide range of temperatures.
Worked Example A resistor has a value of 15 W at a temperature of 20^{o}C. The resistor is made of iron wire which has a temperature coefficient (a) of 0.005671 ^{o}C^{-1}. What is the resistance if the temperature is raised to 400 ^{o}C? |
Answer Temperature change: Dq = 400 ^{o}C - 20 ^{o}C = 380 ^{o}C
Use:
Substitute: R = 15 W × [1 + (0.005671 ^{o}C^{-1} × 380 ^{o}C)] = 47.3 W |
You can see that the resistance increases about three times.
The graph below shows the change in resistance with temperature:
The graph is linear, but NOT directly proportional. The gradient gives a value for the temperature coefficient.
The reference resistance is 20 W. The graph shows the extremes on the list, Nickel (a = 0.005866 ^{o}C^{-1}) and Constantan (a = 0.000074 ^{o}C^{-1}). You can see that the change in Nickel is quite substantial, with an increase in resistance of about 3 times. For constantan, the change is insignificant - which is why the alloy is called constantan. Its resistance is constant. The limit for the temperature range is from -273 ^{o}C (0 K) to the melting point of the metal.