Internal Resistance and EMF
Internal Resistance and EMF
Every real source of electrical energy (battery, cell, generator) has some internal resistance (r). This means the voltage available to the external circuit is always less than the electromotive force (EMF) of the source.
Definitions
Electromotive force (EMF), ε: The total energy transferred per unit charge by the source. It equals the potential difference across the terminals when no current flows (open circuit). Measured in volts (V).
Internal resistance, r: The resistance within the source itself, caused by the resistance of the chemicals, wires, and connections inside.
Terminal potential difference, V: The voltage measured across the terminals of the source when current is flowing. This is less than the EMF because some energy is dissipated inside the source.
The EMF Equation
Applying Kirchhoff's voltage law around a circuit:
ε = V + Ir or equivalently ε = I(R + r)
where:
- ε = EMF (V)
- V = terminal p.d. = IR (V)
- I = current (A)
- R = external resistance (Ω)
- r = internal resistance (Ω)
- Ir = "lost volts" (voltage dropped across internal resistance)
Rearranging: V = ε − Ir
This is the equation of a straight line: V = (−r)I + ε
Graphical Determination of ε and r
Plotting V against I gives a straight line with:
- y-intercept = ε (when I = 0, V = ε)
- gradient = −r
- x-intercept = ε/r (this is the short-circuit current, when V = 0)
Worked Example
A cell of EMF 1.50 V and internal resistance 0.80 Ω is connected to a 5.0 Ω resistor. Calculate:
(a) The current:
ε = I(R + r) → I = ε/(R + r) = 1.50/(5.0 + 0.80) = 0.259 A
(b) The terminal p.d.:
V = ε − Ir = 1.50 − (0.259 × 0.80) = 1.50 − 0.207 = 1.29 V
(c) The power dissipated in the internal resistance:
P_r = I²r = 0.259² × 0.80 = 0.054 W
(d) The power delivered to the external resistance:
P_R = I²R = 0.259² × 5.0 = 0.335 W
(e) The efficiency:
η = P_R / P_total = P_R / (P_R + P_r) = 0.335/0.389 = 86.1%
Maximum Power Transfer
The power delivered to the external resistance is:
P = I²R = [ε/(R + r)]² × R = ε²R/(R + r)²
Differentiating and setting dP/dR = 0 shows that maximum power transfer occurs when R = r (external resistance equals internal resistance).
At maximum power transfer, the efficiency is only 50% — half the power is wasted internally. This is why power stations use very low internal resistance (high efficiency is more important than maximum power transfer).
Cells in Series
For n identical cells, each with EMF ε and internal resistance r, connected in series:
- Total EMF = nε
- Total internal resistance = nr
- Current: I = nε/(R + nr)
Cells in Parallel
For n identical cells in parallel:
- Total EMF = ε (unchanged)
- Total internal resistance = r/n
- Current: I = ε/(R + r/n)
Parallel connection is useful when the internal resistance of each cell is significant — it reduces the total internal resistance.
Required Practical: Determining EMF and Internal Resistance
Apparatus: Cell, variable resistor, ammeter (in series), voltmeter (across terminals)
Method:
1. Vary the resistance using the variable resistor
2. Record pairs of V and I
3. Plot V against I
4. Find ε from the y-intercept and r from the negative gradient
Sources of error:
- The voltmeter draws a small current → V reading is slightly low → ε underestimated
- The cell may heat up during the experiment, changing r
- The cell's EMF may decrease if large currents are drawn for too long (cell polarisation)
Improvements:
- Use a high-resistance voltmeter (digital, > 10 MΩ)
- Take readings quickly to avoid heating
- Use a switch to break the circuit between readings
Practical Significance
- Car batteries have very low internal resistance (~0.01 Ω) to deliver the high current needed by the starter motor (hundreds of amps)
- As batteries age, their internal resistance increases, so the terminal voltage drops more under load — this is why old batteries appear "dead" under load but show nearly full EMF on an open-circuit voltmeter
- Solar cells have relatively high internal resistance, which limits the current they can supply