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Potential Divider

Suppose I have a battery of 10 V.  Suppose that I only need 5 V.  I cannot cut the battery into half.  That would destroy the battery.  There is another way.  I can connect the battery to two equal resistances in series.  A voltage will appear across each resistor.  Because the resistances are equal, the 10 V from the battery will be divided into 5 V across each of the two resistors.  Then we can actually use each resistor as if it is a battery of 5 V.  This setup is called a potential divider - it literally divides up the voltage from the battery.
What if we need different voltages?  Then instead of using equal resistances, we can use different resistances R₁ and R₂.  These resistances will divide the battery voltage in the ratio R₁ : R₂.  As an example, suppose the battery is 10 V and the resistances are 100 Ohms and 400 Ohms.  The ratio is 1:4.  Then the voltages will be 2 V and 8 V, because these keep the same ratio of 1:4 and they also add to give the original 10 V.

The reason why the voltages across the resistances must add to give the battery's voltage has been explained in the page on Potential Differences in a Series Circuit.  To understand the reason for the ratio, we use Ohm's law.  According to Ohm's law, the relations between current I and voltage across each resistor in the above picture are:


If we divide left side of first equation by left side of second equation, and do the same for the right side, we get


Therefore, the ratio of voltages is equal to the ratio of resistances.

It would be even better if we can somehow get any voltage we want (up to the battery voltage) without having change the resistances each time.  There is a way:  connect a single resistor that is in the form of a long wire to the battery.  We know that different lengths of the wire has different resistances.  So we can select two points along the wire and use these as two poles of a battery.  If we want to change the voltage, we just slide the poles along the resistor wire to a different length between them.  This form of the potential divider is called a potentiometer.
Potential dividers and potentiometers are widely used.  In practice, the resistors or wire are often rolled up into a circle with the slider is attached to a knob.  When you rotate the knob, the slider contact rotates and touches different parts of the wire.  This has the same effect as the picture above, changing the voltage between the slider and the other pole.  You may have seen this used in switches for ceiling lights, volume control for music players and heating control for electric cookers.  (These are increasingly being replaced by digital controls in newer models.)