RESISTORS AND OHM'S LAW

1. Electric current

An electric current is a flow of electric charge through a conductor. For practical purposes, we imagine an electric current to be a flow of positive charges (we call this current the CONVENTIONAL CURRENT, but the reality is that current is carried by the negative charges on electrons within the conductor. Conventional current flows from positive to negative, whereas the electrons flow in the other direction. The unit of current is the AMPERE, A, named in honour of the French scientist, Ampère.

"One ampere is the current in a conductor when a charge of one coulomb passes through a cross section of the conductor each second."

Note that this is consistent with the formal SI definition of the ampere, namely,

"The ampere is the constant current which, when flowing in each of two parallel conductors of infinite length and 1m apart in a vacuum, produces between them a force of 2 x 10-7 N per metre of length".

In order for a current to flow through a conductor, two prerequisites must be present:

Current is measured using an instrument called an AMMETER.

Ammeters are connected in series with the part of the circuit through which one wishes to measure the current, I, and they HAVE NEGLIGIBLE RESISTANCE.

2. Potential difference

Current flows in a circuit as a result of a DIFFERENCE IN POTENTIAL between two points in the circuit.

The potential difference between two points in a conductor is the work per unit charge done by the charge in moving from a point of higher potential to a point of lower potential.

The unit of potential difference is called the VOLT, V, after the Italian scientist Alessandro Volta. One volt of potential difference exists between two points if one joule of work is done by each coulomb of charge in moving between them. Potential difference is measured by an instrument called a VOLTMETER.

Voltmeters are connected in parallel to the component across which one wishes to measure the potential difference. They have a resistance which is several orders of magnitude higher than the resistance of the component. THE CURRENT WHICH FLOWS THROUGH A VOLTMETER IS NEGLIGIBLE.

3. Resistance

It can be shown that for any conductor, the ratio of the potential difference across the conductor and the current flowing through it, is constant. This constant is called the RESISTANCE OF THE CONDUCTOR, R. We can write:

The unit of resistance is the OHM, Ω named after Georg Ohm, who discovered this relationship, which is called Ohm's law (See below).

"A conductor has a resistance of one ohm if on application of a potential difference of one volt, a current of one ampere flows through it."

Resistance is a property of a particular conductor and depends on:

  1. The material of which the conductor is made.
  2. The length, L, of the conductor (R ∝ L).
  3. The cross-sectional area, A, of the conductor (R ∝ A-1)
  4. The temperature of the conductor. (Resistance increases non-linearly with temperature).

4. Ohm's law

Ohm's Law states that

Ohm's Law
"For any particular conductor at a constant temperature, the current that flows through it is directly proportional to the potential difference applied across it."

A conductor that has a significant resistance is called a RESISTOR. Resistors are normally depicted in circuit diagrams in one of two ways, as shown on the right. The symbol (a) is currently used, but in older textbooks the symbol (b) is used.

Conductors made of metals or carbon (graphite) obey Ohms' Law.

5. Resistors in series

Consider a circuit in which three resistors R1, R2 and R3 are connected series, thus

The total, or equivalent, resistance of a set of resistors connected in series (in this case three), is the sum of the resistances of the individual resistors:

(See the derivation)

6. Resistors in parallel

Consider a circuit in which three resistors R1, R2 and R3 are connected PARALLEL, thus

The total or equivalent, resistance of a set of resistors connected in parallel (in this case three) is given by the equation

(See the derivation)

7. Network of resistors

Resistors can be connected in networks in which some are arranged in series and others in parallel, as shown in the example on the left.

In analyzing such a network, it is important to simplify the circuit by recognizing which resistors are in series or parallel, and replacing them by a single EQUIVALENT RESISTANCE, as shown in the worked example.

Also remember that for resistors in series,

  1. The current flowing through each resistor is the same.
  2. The potential difference, Vi, across each resistor Ri is given by Vi = RiV/R, where V is the potential drop across the whole circuit, and R is the resistance of the circuit.
  3. The total potential difference V is the sum of the potential differences across each of the resistors.





See the conventions used in drawing circuit diagrams.

What is a "voltage divider"?

(Click here for a discussion)

8. Problems involving light bulbs

In many problems that are encountered, the resistors are light bulbs, which are represented as shown here on the right.

Light bulbs are considered to have a constant resistance, and therefore obey Ohm's law. This is not strictly true as the resistance will in fact increase as the temperature of the filament rises - however it is a useful simplification.

The brightness of a light bulb increases as more current flows through it, and decreases as the current flowing though it decreases.

9. The Wheatstone bridge

The Wheatstone bridge is a circuit specially designed to give an accurate value of unknown resistances. Referring to the diagram on the right, X is an unknown resistance, while P, Q and R are resistances of known value. G is a sensitive galvanometer, able to detect very small currents.

The resistances P, Q and R are varied until the reading on G is zero, a condition that obtains when the potential difference between the points B and D is zero.

This means that the potential differences across X and Q are the same, and the current through X (let's call it I1) must equal the current through P.

Using the same argument, the current through Q must equal the current through R (let's call it I2).

We can then write

I1X = I1P, and
I2Q = I2R      

I1X/I2Q = I1P/I2R
X/Q = P/R
X = PQ/R

10. Additional questions


Derivation of series resistors formula

The current, I, which flows through the circuit, must flow through each of the resistors

The total resistance between the points A and B, call it R, will produce a potential drop between the test points, where V = IR. (1)


The potential difference between each of the individual resistors will be

The total potential drop between A and B will be the sum of the individual potential drops across each resistor. Thus,


Derivation of parallel resistors formula

At the point A, the current, I, divides into three smaller currents I1, I2, and I3, in such a way that the total current is given by I = I1 + I2 + I3 (1)

If this were not so, some current would disappear or accumulate somewhere!

The potential difference across the points A and B will be V, and will be equal to the current I multiplied by the total resistance of the parallel resistors, R. Thus, V = IR.

Since each resistor is connected across the points A and B, the potential difference across each is the same, namely, V.

The currents through each resistor in turn will be I1 = V/R1, I2 = V/R2 and I3 = V/R3.


Some conventions used in drawing circuit diagrams

Circuit diagrams can be extraordinarily complex. In particular, lines that designate conducting wires may cross other lines, and be or may not be connected at the crossing point. The following are the conventions that are used: