the energy transferred from other forms (e.g. chemical) to electrical energy per unit charge in driving charge round a complete circuit.
Potential difference
the energy transferred from electrical to other forms per unit charge.
Internal resistance (r)
the resistance of the source itself, which causes energy to be dissipated inside the source when it delivers current.
Terminal p.d.
the p.d. across the terminals of a source when it delivers a current: V = E - Ir.
Key points
e.m.f.: energy into electrical form; p.d.: energy out of electrical form. Both in V (J C-1).
Ir is the lost volts: energy per coulomb wasted as heat inside the source.
As current increases, lost volts rise and the terminal p.d. falls.
On open circuit (I = 0), terminal p.d. = e.m.f.; a high-resistance voltmeter alone reads close to E.
Experiment: vary the load resistor, record V and I; plot V against I: y-intercept = E, gradient = -r.
Low r matters for large currents (car starter battery); high r is a safety feature for EHT supplies.
A real cell = ideal e.m.f. E in series with internal resistance r (dashed box). The voltmeter across the load reads the terminal p.d. V (ideal ammeter).
Terminal p.d. against current: the y-intercept is the e.m.f. E and the gradient is -r. The gap below E is the lost volts Ir.
Watch out!
A voltmeter across a cell in use reads the terminal p.d., not the e.m.f.
Gradient of V against I is -r: quote r as a positive value.
10.2aKirchhoff's laws
Definitions
Kirchhoff's first law
the sum of the currents into a junction equals the sum of the currents out of the junction.
Kirchhoff's second law
around any closed loop in a circuit, the sum of the e.m.f.s equals the sum of the p.d.s.
Key points
First law is a consequence of conservation of charge.
Second law is a consequence of conservation of energy.
Choose a direction round the loop; an e.m.f. counts positive if the loop passes through it from - to +.
An IR term is positive if the loop goes with the assumed current through that resistor.
A current that comes out negative just flows the opposite way to the one assumed.
Parallel branches have the same p.d.; series components carry the same current.
First law: the current I arriving at the junction splits so that I = I1 + I2.
Second law round a single loop: the e.m.f. equals the sum of the p.d.s, E = IR1 + IR2.
Watch out!
Going round a loop against the assumed current, the IR term is subtracted.
Count each e.m.f. once per loop, with the correct sign for its direction.
10.2bResistors in series and in parallel
Definitions
Combined (equivalent) resistance
the single resistance that would draw the same current from the same p.d. as the combination.
Key points
Series: same current I; p.d.s add (second law), V = IR1 + IR2, so R = R1 + R2.
Parallel: same p.d. V; currents add (first law), I = V/R1 + V/R2, giving the reciprocal rule.
Parallel combination is always less than the smallest branch resistance.
n equal resistors R in parallel: R/n. Two equal in parallel: half of one.
Adding a branch in parallel lowers total resistance and increases the supply current.
Solve networks from the inside out: combine parallel groups first, then add series parts.
Series: resistances add. Parallel: reciprocals add, so the total is below the smallest branch.
Watch out!
After adding reciprocals, remember to invert: 1/R = 0.5 Ω-1 means R = 2 Ω.
Two 10 Ω resistors in parallel make 5 Ω, not 20 Ω.
10.3aPotential dividers and sensor circuits
Definitions
Potential divider
two or more resistors in series across a supply, so that the supply p.d. is shared between them in the ratio of their resistances.
Key points
Series resistors share the supply p.d. in the ratio of their resistances: V1/V2 = R1/R2.
The larger resistor takes the larger share of the p.d.
Replace one resistor with a thermistor or LDR to make a temperature or light sensor.
Output across the fixed resistor rises when the sensor resistance falls (hotter, or brighter).
A variable resistor or rotary potentiometer gives a continuously variable output from 0 to Vin.
Formula assumes the output draws no current; a low-resistance load lowers Vout.
Potential divider: R2 gets its share of the input, Vout = Vin R2/(R1 + R2).
Temperature sensor: when the thermistor warms, its resistance falls, so a larger share of the p.d. appears across the fixed resistor and Vout rises.
Watch out!
Check which resistor the output is across: swapping the sensor and fixed resistor reverses the response.
Vout depends on the ratio R2/(R1 + R2), not on either resistance alone.
10.3bThe potentiometer and null methods
Definitions
Potentiometer
a uniform resistance wire with a steady current in it, so that the p.d. between two points is proportional to the length of wire between them.
Null method
a method in which the length is adjusted until a galvanometer shows zero current (balance).
Balance length
the length of potentiometer wire at which the galvanometer reads zero.
Key points
Driver cell sets a steady current along the wire; p.d. per unit length is constant, so V ∝ L.
Connect like terminals (+ to +) of driver and test cell to the same end A of the wire.
At balance the galvanometer reads zero: no current in the test cell, so no lost volts and it measures the e.m.f.
Compare two e.m.f.s: find the balance length for each; E1/E2 = L1/L2.
Can also compare two p.d.s, e.g. across two resistors carrying the same current, to compare resistances.
If no balance point exists: driver e.m.f. is too small, or a cell is connected the wrong way round.
Potentiometer: slide the jockey J until G reads zero. The balance length AJ is proportional to the e.m.f. of the test cell.
Watch out!
At balance the test cell delivers no current, so the reading is its e.m.f., not a terminal p.d.
The driver cell must have a larger e.m.f. than the cell being tested, or no balance point is found.