Internal resistance — OCR A-Level Physics
Test yourself on Internal resistance with OCR A-Level practice questions.
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Internal resistance explained
A source of e.m.f. is a device that converts other forms of energy into electrical energy, maintaining a potential difference across its terminals.
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Real sources have internal resistance, which is the resistance to current flow within the source itself. The e.m.f. (ε) is the energy transferred per unit charge by the source, while the terminal p.d. is the energy transferred per unit charge by the source to the external circuit. When current flows, some energy is dissipated inside the source, so the terminal p.d. is less than the e.m.f. The relationship is ε = I(R + r), where R is external resistance and r is internal resistance. For example, a battery with ε = 9.0 V and r = 1.0 Ω connected to a 4.0 Ω resistor gives I = 9.0 / (4.0 + 1.0) = 1.8 A.
(b) terminal p.d.; 'lost volts'
Terminal p.d. is the potential difference across the external circuit, measured between the terminals of a source when current flows. 'Lost volts' is the potential difference across the internal resistance of the source, equal to Ir, where I is the current and r is the internal resistance. The relationship is terminal p.d. = e.m.f. - lost volts, or V = ε - Ir. For example, a cell with ε = 1.5 V and r = 0.50 Ω delivering a current of 2.0 A has lost volts = 2.0 × 0.50 = 1.0 V, so terminal p.d. = 1.5 - 1.0 = 0.50 V. When no current flows, lost volts is zero and terminal p.d. equals e.m.f.
(c)
This guided reading covers the techniques for investigating internal resistance and e.m.f. experimentally. You should understand how to set up a circuit with a variable resistor, a cell, an ammeter and a voltmeter to measure terminal p.d. and current. By varying the resistance, you obtain pairs of values for current I and terminal p.d. V. Plotting V against I gives a straight line with equation V = ε - Ir, so the y-intercept is the e.m.f. and the gradient is -r. Alternatively, plot 1/I against R to find ε and r. You should also be able to identify and correct common experimental errors, such as voltmeter loading and heating effects. This reading supports practical work and data analysis for internal resistance.
(i) the equations E = I ( R + r ) and E = V + I r
A real source of e.m.f. has internal resistance r, so some energy per unit charge is dissipated inside the source. The e.m.f. E equals the total potential difference per unit charge around the circuit: E = I(R + r), where R is the external resistance and I the current. The terminal potential difference V is the p.d. across the external resistance only, so V = IR. Substituting gives E = V + Ir, showing that V is less than E whenever current flows, and that V equals E only when I = 0, as on open circuit. The lost volts Ir increase with current. Rearranging, r = (E − V)/I, which is the basis of the experimental determination of internal resistance.
(ii) techniques and procedures used to determine the internal resistance of a chemical cell or other source of e.m.f.
To determine internal resistance, connect the cell in series with a variable resistor, an ammeter and a closed switch, and connect a voltmeter in parallel with the cell to measure terminal p.d. V. For several settings of the variable resistor, record I and V. Since E = V + Ir, a graph of V against I is a straight line of gradient −r and intercept E on the V-axis. Alternatively, plot 1/I against R; the intercept gives r/E and the gradient gives 1/E. Taking repeat readings and using a range of currents reduces random uncertainty. The cell must not be short-circuited, and the switch should be closed only briefly to limit heating and polarisation.
Your focus
- Define e.m.f. and internal resistance.
- Apply the equation ε = I(R + r) to solve problems involving internal resistance.
- Distinguish between e.m.f. and terminal p.d. in practical circuits.
Show all 15 objectives
- Define terminal p.d. and 'lost volts'.
- Apply the equation V = ε - Ir to calculate terminal p.d. or lost volts.
- Explain how terminal p.d. varies with current in a real source.
- Describe an experiment to determine the e.m.f. and internal resistance of a cell.
- Interpret a graph of terminal p.d. against current to find e.m.f. and internal resistance.
- Identify and correct common sources of error in such experiments.
- Recall and apply E = I(R + r) and E = V + Ir to circuit problems.
- Explain why terminal p.d. is less than e.m.f. when current flows.
- Rearrange the equations to determine internal resistance from measured values.
- Describe a valid circuit and procedure for measuring internal resistance.
- Interpret a V–I graph to obtain e.m.f. and internal resistance.
- Justify experimental precautions that improve the reliability of results.
Internal resistance exam tips
Marking Points
- Define e.m.f. as the energy transferred per unit charge by the source.
- Define internal resistance as the resistance within the source that causes a potential drop when current flows.
- State the relationship ε = I(R + r) for a source with internal resistance r and external resistance R.
- Explain that terminal p.d. equals ε minus the p.d. across the internal resistance (Ir).
- Calculate current, terminal p.d. or internal resistance using ε = I(R + r) and V = ε - Ir.
- Define terminal p.d. as the potential difference across the external circuit when current flows.
- Define 'lost volts' as the potential difference across the internal resistance, equal to Ir.
- State the relationship terminal p.d. = e.m.f. - lost volts, or V = ε - Ir.
- Calculate terminal p.d. or lost volts given e.m.f., current and internal resistance.
- Explain that terminal p.d. equals e.m.f. when no current flows (open circuit).
- States that E = I(R + r) applies to the whole circuit, with R the external resistance and r the internal resistance.
- Explains that V = IR is the terminal p.d. across the external resistance only.
- Derives E = V + Ir by substituting V = IR into E = I(R + r).
- States that V equals E only when the current is zero, for example on open circuit.
- Identifies Ir as the lost volts and explains that V falls as I increases.
- Rearranges to r = (E − V)/I for use in experimental determination.
- Describes a circuit with the cell in series with an ammeter, variable resistor and switch, and a voltmeter in parallel with the cell.
- Explains that terminal p.d. V and current I are measured for several values of external resistance.
- Uses E = V + Ir to justify plotting V against I, giving gradient −r and intercept E.
- Describes an alternative graph such as 1/I against R, with intercept r/E and gradient 1/E.
- Explains that the switch is closed only briefly to avoid heating and polarisation of the cell.
- Explains that repeat readings and a range of currents reduce random uncertainty.
Examiner Tips
- 💡Remember that e.m.f. is measured in volts and is the energy per unit charge, not a force.
- 💡When using ε = I(R + r), ensure all resistances are in ohms and current in amperes.
- 💡To find terminal p.d., use V = ε - Ir or V = IR.
- 💡Use V = ε - Ir to find terminal p.d. quickly.
- 💡Remember that lost volts = Ir, so if current is zero, lost volts is zero.
- 💡Check units: e.m.f., terminal p.d. and lost volts are all in volts.
- 💡When plotting V against I, the y-intercept is the e.m.f. and the gradient is -r.
- 💡Use a variable resistor to change current and take multiple readings to improve reliability.
- 💡Keep the current low or intermittent to minimise heating effects on internal resistance.
- 💡Write the two equations side by side and show the substitution step explicitly when asked to derive E = V + Ir.
- 💡Check the rearrangement algebraically before substituting numbers, especially when making r the subject.
- 💡Use the open-circuit condition I = 0 to justify any statement that V equals E.
- 💡Draw a clearly labelled circuit diagram before describing the procedure.
- 💡State the quantity on each axis and explain how the gradient and intercept give r and E.
- 💡Mention a precaution that limits heating or polarisation of the cell.
Common Mistakes
- Confusing e.m.f. with terminal p.d.; correction: e.m.f. is the energy per unit charge supplied by the source, while terminal p.d. is the energy per unit charge available to the external circuit and is less than e.m.f. when current flows.
- Ignoring internal resistance when calculating current; correction: include r in the total resistance, so I = ε / (R + r).
- Assuming internal resistance is always negligible; correction: in many real sources it is significant and must be accounted for.
- Thinking that e.m.f. and terminal p.d. are always equal; correction: they are equal only when no current flows (open circuit).
- Thinking that terminal p.d. is always equal to e.m.f.; correction: terminal p.d. is less than e.m.f. when current flows due to lost volts.
- Confusing 'lost volts' with energy lost; correction: 'lost volts' is a potential difference, not energy, though it represents energy dissipated per unit charge inside the source.
- Forgetting to include internal resistance when calculating terminal p.d.; correction: always subtract Ir from e.m.f. to find terminal p.d. when current flows.
- Assuming lost volts is constant; correction: lost volts depends on current, so it varies with the external circuit.
- Connecting the voltmeter in series with the cell; correction: the voltmeter must be connected in parallel with the cell to measure terminal p.d.
- Assuming the ammeter has zero resistance; correction: in reality, ammeters have some resistance, which can affect measurements, so use a low-resistance ammeter or account for it.
- Ignoring the effect of temperature on internal resistance; correction: as current flows, the cell may heat up, changing r, so take readings quickly or allow cooling.
- Plotting V against I but interpreting the gradient as +r; correction: the gradient is -r, so r is the negative of the gradient.
- Treating V as equal to E at all times: the correction is that V = E − Ir, so V equals E only when I = 0.
- Using E = IR and omitting the internal resistance term: the correction is that the total resistance is R + r, so E = I(R + r).
- Confusing e.m.f. with terminal p.d.: the correction is that e.m.f. is the energy transferred per unit charge by the source, while terminal p.d. is the energy transferred per unit charge in the external circuit.
- Assuming r is always negligible: the correction is that r is a property of the source and becomes significant when the current is large.
- Connecting the voltmeter across the variable resistor instead of across the cell: the correction is that the voltmeter must measure terminal p.d. across the cell.
- Leaving the switch closed continuously: the correction is that this heats the cell and changes its e.m.f., so readings become unreliable.
- Plotting V against I and reading the gradient as +r: the correction is that the gradient is −r because V = E − Ir.
- Using only one pair of readings: the correction is that several readings across a range of currents are needed to identify the trend and reduce uncertainty.