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# Potential divider circuits

## What is a potential divider?

A potential (or voltage) divider is a type of circuit. It produces an output voltage that is lower than the input voltage.

Potential dividers can be used to power things that need a smaller voltage than the voltage of the source.

A potential divider circuit is made up of two resistors, $R_{1}$ and $R_{2}$. They are connected in series to a power source supplying a voltage $V_{s}$.

The voltage provided by the battery or power source is divided up between the two resistors.

A potential divider circuit can be compared to a large water pipe that splits to supply two showers. If one shower is switched on, the pressure in the other shower decreases.

Depending on the use of the potential divider, the resistors can be fixed resistors, variable resistors, LDRs or thermistors.

Potential divider circuits without a moveable connector (left) and with a moveable connector (right)

## Potential divider voltage calculations

One type of potential divider circuit is made up of two fixed resistors, $\Tred{R_{1}}$ and $\Tblue{R_{2}}$. They are connected in series to a power source supplying a voltage $V_{s}$.

The potential is divided between the two resistors depending on their resistances.

If $\Tred{R_{1}}$ is larger than $\Tblue{R_{2}}$ then the potential difference across $\Tred{R_{1}}$ will be larger than across $\Tblue{R_{2}}.$

If the resistances of the resistors are equal then the potential differences across them will be equal as well.

The potential difference across $\Tred{R_{1}}$ and $\Tblue{R_{2}}$, denoted by $\Tred{V_{1}}$ and $\Tblue{V_{2}}$, is given by: $$\Tred{V_{1}}=\frac{\Tred{R_{1}}}{\Tred{R_{1}}+\Tblue{R_{2}}}\times V_{s}\quad\quad \Tblue{V_{2}}=\frac{\Tblue{R_{2}}}{\Tred{R_{1}}+\Tblue{R_{2}}}\times V_{s}$$

A potential divider circuit with two fixed resistors.

## Movable connectors and rheostats in potential divider circuits

Some potential divider circuits use a moveable connector, which enables the output voltage to be varied (unlike a standard potential divider which has a fixed voltage ratio).

The pointer on the slider shows how much of the resistance of a resistor is added between the output terminals.

A potential divider circuit with a moveable connector.

If $R_{1} = 100 \, \Omega$ then the connector can be moved to vary the resistance between $0$ and $100 \, \Omega.$

When the connector is positioned between the two resistors $V_{1} = 0$. As the connector is moved upwards, the voltage $V_{1}$ increases up to its maximum value.

A similar effect can be accomplished using a variable resistor in the place of one of the fixed resistors.

## Light-dependent resistors in potential dividers

Light-dependent resistors (LDRs) are often set up using a potential divider circuit.

An LDR is used instead of a fixed resistor so that the output voltage changes depending on how dark or bright it is.

LDR potential divider circuits are used in street lights. The lights turn on when the light level outside decreases beyond a certain value.

The resistance of an LDR decreases as it gets brighter. This means that the voltage across the LDR also decreases as it gets brighter.

The voltage across the fixed resistor therefore increases with increasing light levels.

The output voltage is taken across either the fixed resistor or the LDR depending on the use of the circuit.

Left: Output voltage increases with increasing brightness. Right: Output voltage decreases with increasing brightness.

## Thermistors in potential dividers

Thermistors are often used in potential divider circuits in electronic thermometers.

A thermistor is used instead of a fixed resistor so that the output voltage changes depending on temperature.

The resistance of a thermistor decreases as it gets hotter. This means that the voltage across the thermistor also decreases as it gets hotter.

The voltage across the fixed resistor therefore increases with increasing temperature.

The output voltage is taken across either the fixed resistor or the thermistor depending on the use of the circuit.

Left: Output voltage increases with increasing temperature. Right: Output voltage decreases with increasing temperature.

## Potentiometers

A potentiometer is a type of potential divider circuit that uses a resistance wire (often constantan or nichrome) and a moveable connector to change the output voltage.

These circuits are used in light dimmers and volume controls.

The resistance wire used is of high enough resistivity that it cannot be ignored in calculations. By moving the connector along the wire, the resistance and voltage between the output terminals can be changed.

The resistance of a wire is given by the equation $R=\rho \ell/A$. This implies that resistance is proportional to length: $R\propto\ell$. The potential difference across the output terminals of a potentiometer is given by: $$\frac{V_{1}}{V_{s}}=\frac{\ell_{1}}{\ell}$$ $V_{1}$ is the voltage across the output terminals, $V_{s}$ is the voltage of the source, $\ell_{1}$ is the distance between the moveable connector and the fixed terminal and $\ell$ is the total length of the wire.

Potentiometer diagram

## Potentiometers and e.m.f. measurement

Potentiometers can be used to measure the e.m.f. of a cell. It is often assumed that the e.m.f. of a cell without internal resistance is equivalent to the terminal potential difference.

However, in reality, some amount of current flows through the voltmeter used to measure the terminal potential difference, leading to an inaccurate value.

The e.m.f. of a cell is only equivalent to the terminal potential difference if no current flows through the cell (i.e. $I=0$). This arises from the equation $E=V+Ir$.

To measure the e.m.f. of a cell, one must first connect the cell in parallel to the resistance wire in a potentiometer with its terminals in the opposite direction of the current through the resistance wire.

E.m.f. measurement potentiometer

At a point Y along the wire, the potential difference across XY becomes equal to the e.m.f. of the cell. When this occurs, no current flows through the cell. The potential difference across XY is equal to the e.m.f. of the cell (recall $V_{1}/V_{s}=\ell_{1}/\ell$).