doPhysics · Electromagnetism 1
Cells, Bridges and Measuring Instruments
The emf and internal resistance of a cell, terminal voltage, power and maximum power transfer; cells in series, in parallel and in rows; the Wheatstone bridge, the meter bridge and the post office box; the potentiometer; and how a galvanometer becomes an ammeter or a voltmeter.
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Why this chapter
Why this chapter exists
Every circuit so far had a perfect cell and perfect meters. Real cells lose part of their work inside themselves, and real meters change the circuit they measure. This chapter looks at both. It explains why headlights dim when an engine starts, and how to group cells for the largest current. It shows how a sliding contact on a bare wire measures a resistance or an emf without disturbing it. The meter bridge and the potentiometer are in the practical syllabus of Class 12, where every student meets them at the bench. The JEE asks about them too. The post office box of Lesson 4 is beyond both, and is marked so.
You will learn
By the end of this chapter you will
- Explain the emf and internal resistance of a cell. Use \(V = \varepsilon - Ir\) while it discharges and \(V = \varepsilon + Ir\) while it is charged.
- Use the power balance \(\varepsilon I = VI + I^2 r\), and find the load that takes the largest power and the efficiency there.
- Find the equivalent cell of cells in series, in parallel and in rows, and choose the grouping that gives the largest current.
- Use the balance P/Q = R/S of a Wheatstone bridge, measure a resistance with a meter bridge, and correct for end errors.
- Use the potential gradient of a potentiometer to compare emfs and find an internal resistance, and say what moves the null point.
- Turn a galvanometer into an ammeter with a shunt and into a voltmeter with a series resistance. Judge how much a meter disturbs a circuit.
Before you begin
Before you begin
You should know Ohm's law, resistivity and the power \(I^2 R\) from the chapter Electric Current and Resistance. From the chapter Circuits: Kirchhoff's Laws, Networks and RC Transients, you should be able to combine resistances and walk round a loop. The balance of a Wheatstone bridge was met there too; here it becomes an instrument.
Lessons
1
EMF, Internal Resistance and Terminal Voltage
A cell does chemical work on every coulomb it moves, its emf, but part of that work heats its own internal resistance. So the terminal voltage is \(\varepsilon - Ir\) while the cell drives a current and \(\varepsilon + Ir\) while it is being charged. The power balance \(\varepsilon I = VI + I^2 r\) explains dimming headlights, and a load takes the most power when it matches \(r\).
Emf and terminal voltageMeasuring internal resistancePower balance of a cellMaximum power transfer
2
Combinations of Cells
Cells in series add their emfs with sign and their internal resistances always. Equal cells in parallel keep the emf and divide r; unequal ones give a weighted average. Rows of cells in parallel give the largest current when their internal resistance matches the load, and the load decides whether series or parallel wins.
Cells in seriesCells in parallelMixed grouping of cellsWhich grouping wins
3
The Wheatstone Bridge and the Meter Bridge
A Wheatstone bridge balances when P/Q = R/S, because the two middle points are then at one potential and the galvanometer carries nothing. The meter bridge makes two of the arms from one uniform wire, so X/R = l/(100 − l). End corrections, the choice of a null near the middle and the effects of warming decide how good the answer is.
Balance of the wheatstone bridgeThe meter bridgeEnd corrections and errors
4
The Post Office Box
A post office box is a Wheatstone bridge in a box of plug keys. Ratio arms of 10, 100 and 1000 Ω and a resistance box that moves in whole ohms bracket an unknown resistance ever more closely, one decimal place per stage. The battery key is always pressed before the galvanometer key.
Ratio arms and the procedureWhy 10 100 and 1000Battery key before galvanometer key
5
The Potentiometer
A steady current along a uniform wire makes the potential fall by the same amount per metre, the potential gradient. A cell balanced against a length of that wire draws no current, so the length measures its emf itself. The same wire compares two emfs, finds an internal resistance, and tells what moves the null.
Potential gradientComparing emfs and why it beats a voltmeterInternal resistance by potentiometerHow the null point shifts
6
Galvanometer to Ammeter and Voltmeter
A galvanometer reads full scale at about a milliampere and a fraction of a volt. A small shunt in parallel turns it into an ammeter, and a large resistance in series into a voltmeter. Every real meter changes the circuit a little: an ammeter should have almost no resistance and a voltmeter almost infinite resistance.
Sensitivity of a galvanometerThe shunt makes an ammeterThe series resistance makes a voltmeterHow a meter disturbs the circuit
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- ammeter
- A meter joined in series to measure current: a galvanometer with a small shunt in parallel. Its resistance should be as small as possible.
- balanced
- Said of a bridge whose galvanometer carries no current, because the two points it joins are at the same potential.
- battery
- A group of cells joined to act as one, such as the six lead-acid cells in a \(12\,\text{V}\) car or inverter battery.
- current sensitivity
- The deflection of a galvanometer per unit current through it, \(\theta/I\), for example in divisions per milliampere.
- driver cell
- The cell that sends the steady current along a potentiometer wire and so sets its potential gradient.
- efficiency
- The share of a cell's power that reaches the load, \(V/\mathcal E = R/(R + r)\). It is only \(50\,\%\) when the load takes the largest power.
- emf
- The work done by the non-electrostatic forces of a cell on each coulomb carried through it, \(\mathcal E = W/q\), in volts. It equals the terminal voltage when no current flows.
- end corrections
- Small extra lengths at the two ends of a meter bridge wire. They stand for the resistance of the joints and of the unmarked wire under the clamps.
- equivalent cell
- The single cell, with one emf and one internal resistance, that could replace a combination of cells without any change outside.
- figure of merit
- The current that moves a galvanometer's pointer by one division, the reciprocal of its current sensitivity.
- galvanometer
- A sensitive detector of small currents, a coil that turns in a magnetic field and carries a pointer. It shows both the size and the direction of a current.
- ideal meter
- A meter that does not disturb the circuit: an ideal ammeter has zero resistance and an ideal voltmeter infinite resistance.
- internal resistance
- The resistance of the electrolyte and electrodes inside a cell, \(r\). It acts in series with the emf and cannot be reached by any wire.
- jockey
- The sliding contact of a meter bridge or a potentiometer, pressed onto the wire at the point being tested.
- meter bridge
- A Wheatstone bridge whose two lower arms are one uniform wire \(100\,\text{cm}\) long. At the null point \(X/R = l/(100 - l)\).
- mixed grouping
- An arrangement of equal cells in \(m\) parallel rows of \(n\) cells in series, equivalent to one cell of emf \(n\mathcal E\) and internal resistance \(nr/m\).
- multiplier
- The large resistance joined in series with a galvanometer to make a voltmeter, \(R = V/I_g - G\).
- null point
- The point on the wire of a bridge or a potentiometer at which the galvanometer shows no deflection.
- post office box
- A Wheatstone bridge built into one box of plug-in coils. Its ratio arms take \(10\), \(100\) or \(1000\,\Omega\), and its resistance box moves in whole ohms.
- potential gradient
- The fall of potential per unit length along a potentiometer wire, \(k = V_{AB}/L\), in V/m.
- potentiometer
- A long uniform wire carrying a steady current, against a length of which an unknown voltage is balanced without drawing current from it.
- ratio arms
- The two arms \(P\) and \(Q\) of a post office box, set to \(10\), \(100\) or \(1000\,\Omega\). Their ratio sets how finely the box's steps divide the unknown.
- short-circuit current
- The current a cell drives when its terminals are joined by a wire of no resistance, \(\mathcal E/r\). It is the largest current the cell can give, and all its power then heats the cell.
- shunt
- A small resistance joined in parallel with a galvanometer so that it carries all but a small fixed share of the current, \(S = I_gG/(I - I_g)\).
- terminal voltage
- The potential difference across the terminals of a cell in use: \(V = \mathcal E - Ir\) while it drives a current, \(V = \mathcal E + Ir\) while it is being charged.
- voltage sensitivity
- The deflection of a galvanometer per unit voltage across it, \(\theta/V = \theta/(IG)\).
- voltmeter
- A meter joined in parallel to measure a potential difference: a galvanometer with a large resistance in series. Its resistance should be as large as possible.
- Wheatstone bridge
- Four resistors \(P\), \(Q\), \(R\), \(S\) in a diamond with a cell across one diagonal and a galvanometer across the other. It is balanced when \(P/Q = R/S\).