Ohm's Law

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Ohm's Law

The Simplest Circuit

We can make current flow in a circle (circuit) by connecting the terminals of a battery together. This will melt the wire, make sparks fly and maybe start a fire, so don't do it. Instead, connect something to control the current. The ability to control current is called resistance, and all materials have it to some degree- in fact we classify materials according to their resistance: those with very low resistance are conductors, those with lots of resistance are insulators. There are devices called resistors that are used in electronic gadgets- they have a resistance that is something between conduction and insulation, and predictable. So here's a safe circuit:

The battery has a certain amount of push, called electromotive force or emf. This is measured in units called volts. We indicate emf (often called voltage) in formulas with the letter E. Voltage has to be measured between two points in the circuit in the same way that height has to be measured between two points on the side of a mountain. There is no such thing as "0 volts" except that the voltage between two points is 0 if they are connected together.

The resistor has a determined amount of resistance measured in units called ohms. We indicate resistance in formulas with the letter R.

When current is flowing, we measure it in units called amperes, and indicate it with a letter I.

The three are related by a simple formula called Ohm's Law:

I=E/R

Also written E = IR or R=E/I.

That tells us the current if we know the voltage and resistance, or the voltage if we know the current and resistance, or the resistance if we know the current and voltage. If this seems a bit circular to you, you're right. We can measure current by the strength of magnetic field it will generate, but there is no yardstick for voltage other than seeing how much current flows through a known resistance. And how is a resistance known? We apply a known voltage and see how much current flows.

The definition of the units is circular too: 1 ampere is the amount of current that flows through a 1 ohm resistor if 1 volt is applied.

Two resistors in series

It gets a bit more complicated if there are two resistors:

Whatever the current is, it's the same at A, B, and C. (There isn't anywhere else for the current to flow.)

The voltage between A and C is equal to that between A and B added to that between B and C.

EAC = EAB + EBC

The voltages add up, just as the height of a house is the sum of the heights of its stories.

The voltage across each resistor is proportional to the resistance of each resistor.

EAB / R1 = EBC/ R2

You see, ohm's law is true for each part of a circuit as well as the circuit as a whole. Whatever current flows, it's the same in each resistor, so the voltages will adjust themselves.

Total resistance is R1 + R2

Two Resistors in Parallel

The current through A equals the current through B plus the current through C. The current splits up and comes together, like water flowing around an island.

The voltage across R1 is the same as the voltage across R2.

EAB = EAC, so IB R1 = ICR2 and IB / R2 = IC/ R1

In other words, the current through each resistor is inversely proportional to the values of the resistors. It's also important to remember a high value resistor passes a small current.

We can solve the above for total current (IB + IC) and get the equivalent resistance for the two resistors:

In the special case where resistors are the same, the equivalent resistance is R1/2. This turns up more often than you might expect.

In another special case, where R2 is more than 100 times the value of R1, R2 accounts for such a small portion of the current that we don't bother to include it in the calculations. Then we say R2 does not load the circuit.

Here's a more complex example

R1 is a special type of resistor that has an adjustable tap in the middle. This really makes R1 behave like two resistors in series. If we say R2 is 100 times R1, we can leave it out of the calculations, and find that the voltage E2 will vary directly with the position of the tap.

If R2 were comparable to R1 in value, we would have to figure it in, first solving R2 and the bottom part of R1 as two resistors in parallel, and using the result of that in a series calculation to find the voltage E2 and the total current. The resulting voltage curve (what you'd get if you plotted E2 for various positions of the tap) is pretty messy, so we really prefer to have an R2 that does not load the circuit.

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