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# Resistance

## Definition of resistance

Resistance can be thought of as a quantity that prevents or slows down the flow of charge.

The higher the resistance of a wire, the more energy is lost as a charge flows through it.

Objects lodged in water pipes prevent the flow of water. Similarly, metal atoms in wires prevent the flow of electrons.

The resistance ($R$) of a conductor is the ratio of the potential difference ($\Tblue{V}$) across the conductor to the current ($\Tred{I}$) flowing through it.

$$\text{resistance} = R =\frac{\Tblue{\text{potential difference}}}{\Tred{\text{current}}} = \frac{\Tblue{V}}{\Tred{I}}$$

The SI unit for resistance is the ohm $(\Omega)$. One ohm is equivalent to one volt per ampere (i.e. $1\text{ }\Omega=1\text{ V } / \text{ A}$).

Metals like copper or gold have low resistance while non-metals like plastic and rubber have high resistance.

A common resistor. The coloured bands on the resistor indicate its resistance (given by a colour code chart).

## Resistance measurement

Resistance measurement circuit

The resistance of a resistor can be determined experimentally by setting up a simple circuit.

An ammeter (which measures current) is connected in series to the fixed resistor and a voltmeter (which measures voltage) is connected in parallel to the resistor.

A rheostat (variable resistor) and a battery are connected in series with the resistor.

The rheostat can then be adjusted to vary the overall resistance in the circuit, changing the current $I$ and voltage $V$ across the resistor.

The readings must be taken quickly because if the resistor heats up, its resistance will increase.

After recording a range of values of $I$ and $V$, a graph of $V$ vs $I$ can be plotted. The resistance $R$ can then be obtained by taking the gradient (slope) of the graph.

If you just take one measurement there is a high risk of it being incorrect! Taking several values is more accurate than simply taking one measurement of $V$ and one measurement of $I$ and then calculating the resistance.

## Resistance of a wire

The resistance of a wire is proportional to its length $\ell.$

Longer wires have a higher resistance than shorter ones.

It takes more energy to pump water through a longer pipe than it does to pump it through a shorter one.

The resistance of a wire is also inversely proportional to its cross sectional area $A$.

Narrower wires have a higher resistance than wider ones.

It takes more energy to pump water through a narrow pipe than it does a wide one.

This can be summarised by the relation: $$\text{resistance of a wire} \propto \frac{\Tblue{\text{length}}}{\Tred{\text{area}}}$$

## Resistivity and the resistance of a wire

Resistivity ($\rho$) measures a material's ability to resist the flow of charges.

Copper is a good electrical conductor and has a low resistivity.

Plastic is a poor electrical conductor and has a high resistivity.

Resistivity is a property of a material and does not depend on size or shape.

The resistance of a wire $R$ is a property of the wire and does depend on its size and shape:

$$R= \frac {\text{resistivity}\times\text{length}}{\text{cross-sectional area}} = \frac{\rho\ell}{A}$$

The SI unit of resistivity is the ohm metre ($\Omega\text{ m}$).

## Ohm's law

Ohm's law states that the electric current flowing through a conductor is directly proportional to the potential difference across it.

This law only applies to a certain class of conductors called ohmic conductors. It also only applies when all external physical quantities such as temperature are constant.

Ohm's law generally applies to metallic conductors but not to semiconductors or insulators.

Ohm's law is expressed mathematically as $$I\propto V \quad \quad I=\frac{V}{R} \quad \quad R=\text{constant}$$ $I=$current; $R=$resistance; $V=$voltage.

The term $1/R$ in Ohm's Law is the constant of proportionality.

Graphical representation of Ohm's law for two different ohmic conductors. The graph of $I$ vs $V$ has a constant gradient of $1/R$

## Effect of temperature on resistance

The resistance of a conductor increases as its temperature increases.

This is the net result of two different phenomena involving increased energy:

• An increase in temperature increases the kinetic energy of the metal atoms in a conductor, resulting in more free electrons and slightly decreasing the resistance.

• The increased kinetic energy of the metal atoms causes them to vibrate more strongly. This hinders the flow of charges and significantly increases the resistance.

The increase in resistance caused by the reduction in charge flow is greater than the decrease in resistance from the larger number free electrons. This results in an overall increase in resistance.

The resistance of a conductor increases as the temperature increases.

## Internal resistance and e.m.f.

All power sources have an internal resistance, denoted by the symbol $r$. Unless otherwise mentioned, this resistance must be taken into account when performing calculations.

This resistance can be thought of as an additional resistor connected in series to the power producing section of the source.

This is the reason why the potential difference across the terminals of a battery connected in a circuit is always lower than the e.m.f.

The e.m.f. $\mathcal{E}$ of a source with an internal resistance is given by: $$\mathcal{E}=V+Ir$$ $V$ is the potential difference across the terminals of the battery and $r$ is the internal resistance of the battery.