Potential dividers

Sharing in the ratio of the resistances

Put two resistors in series across a supply and one current threads both of them. Each takes a pd of V = IR, so the shares come out proportional to the resistances. The battery fixes only the total; the resistors negotiate the split between themselves. Taking the output across R₂ gives

Sanity-check the formula at its limits. Shrink R₂ to nothing and the output falls to zero; grow it without bound and the output climbs towards the full supply. Any voltage between those two is available, provided the output feeds something that draws almost no current; connect a real load and the output sags, which is the subject of its own section below.

A dial for voltage

Make the split point movable and the divider becomes adjustable. A sliding contact on a resistance track divides it into two parts whose ratio changes as the contact moves, so the output follows the slider, continuously, from zero to the full supply.

Volume knobs, dimmer switches and joystick axes are all built this way, on one track and one wiper.

Make that track a uniform wire, one material at one cross-section for its whole length, and the dial acquires a scale. A single current threads the wire end to end, and R = ρL/A gives every centimetre of it the same resistance, so every centimetre takes the same pd. The potential therefore falls at a steady rate along the wire, from the full supply at the end joined to the positive terminal down to zero at the far end, and the pd across any part of it is proportional to the length of that part. Writing V for the pd across the whole length L, the pd V_x across a length x measured from the far end is

Plot potential against distance and the graph is a straight line, whose gradient is the potential gradient along the wire, in volts per metre. Halfway along is half the pd, a fifth of the way along is a fifth of it, and that proportionality is the reason a wiper on a uniform track reads off the fraction of the way it has travelled. It also makes a wire laid beside a metre rule into a voltage scale, which is what the closing section of this lesson puts to work.

What a load does to the output

Everything so far has assumed the output goes nowhere, and that assumption is what a good question breaks. Connect a real component across the output and it sits in parallel with whichever resistor the output is taken across. The pair has a lower resistance than that resistor alone, so its share of the supply shrinks and the measured output sags below the value the ratio alone predicts.

Numbers show the scale of it. Two equal resistors give half the supply with nothing attached. Hang a load of the same resistance across the lower one and the output falls from half the supply to a third of it. Make the load ten times the divider resistance and the sag is only a few per cent.

So a design has two escapes, and either earns the mark. Build the divider from resistances well below whatever will be connected, or feed the output into something that draws almost no current, a high-resistance voltmeter or an amplifier input. Low divider resistances cost you a current running permanently through the chain, which matters in anything battery powered.

Sensing circuits

Replace one resistor with a component whose resistance responds to the world and the divider becomes a sensor. An ntc thermistor's resistance falls as it warms; a light dependent resistor (LDR) behaves the same way towards light, its resistance falling as the illumination rises. Either way, the changing resistance moves the ratio, and the output voltage tracks the environment.

Design comes down to one decision, which resistor the output is taken across. Put the output across the fixed resistor and warming the thermistor raises it, the right way round for switching on a fan. Move the output to the thermistor and the response inverts, which suits a heater instead. The component senses; its position in the divider sets the direction of the change.

The potentiometer and null methods, a CIE extension

One board takes the divider further. CIE asks for the potentiometer as a means of comparing potential differences, and for the part a galvanometer plays in a null method, so read this section only if CIE is your specification. OCR mentions the potentiometer as well, but only as the three-terminal potential divider you met with the sliding contact above. The uniform wire it is built on belongs in the main body above, and Edexcel asks for that much; what is CIE's alone is the null method laid on top of it.

The arrangement is that same metre of uniform resistance wire, now laid along a scale, with a driving cell across its ends keeping a steady current in it. The potential falls uniformly along it, so the pd between one end and any point is proportional to the length of wire between them. Tapping the wire at 65.0 cm of a 100.0 cm run driven at 2.00 V gives 1.30 V, and the wire has become a divider with a scale printed alongside it. The sliding contact earlier in this lesson supplied a pd you could dial; this one is a pd you are going to balance something against.

Now the measurement. Connect the cell you want to measure between that same end of the wire and a sliding contact, the jockey, with a galvanometer in the branch, and connect it so that its emf opposes the pd along the wire. Slide the jockey until the galvanometer reads exactly zero. That is the balance point, and at balance the two pds are equal and opposite, so the cell's emf equals the pd across the tapped length.

The point of all that sliding is what happens at zero. No current flows in the branch, so the cell being measured drops no lost volts inside itself and you read its emf rather than its terminal pd. Nor does the branch load the wire, so the divider ratio is the plain length ratio. A voltmeter cannot do either of those things perfectly, because a real voltmeter has finite input resistance and so draws some current, tiny for a modern digital meter, larger for a moving-coil needle.

Comparing two potential differences is then a matter of two balance lengths. Balance one source at l₁, swap it for the other, rebalance at l₂, and divide.

Everything about the driving circuit cancels, which is why the method is so hard to fool. It also means both balances must be taken with the driving circuit undisturbed, since a driving cell that runs down between the two readings changes the volts per centimetre and quietly breaks the comparison.

The galvanometer earns its place by being sensitive and centre-zero, and by needing no calibration whatsoever. You never read a value off it, only a zero, so any error in its scale drops out of the result, and its own resistance cannot shift where the balance sits, though it does set how sharply the balance is felt, since it decides how much current a given imbalance drives. That is what a null method means: the instrument locates a balance instead of measuring a quantity. Which side it kicks tells you which way to slide, and a kick the same way at both ends of the wire means the driving pd is smaller than the emf you are trying to balance, or the cell is in the wrong way round.

One practical warning about the drawing. The galvanometer above carries the circle-and-G used everywhere on this site, because one symbol set has to serve everyone reading these pages. The glyph printed on your own paper is the one that counts, so check the symbol list published with your specification and be ready to recognise the detector in whatever form it arrives.