Welcome to Section 1.11: Internal Resistance and Electromotive Force

Have you ever noticed that your phone or a torch battery gets warm when you use it for a long time, or that car headlights dim slightly for a split second when the engine starts up? This happens because real-world power supplies are not "perfect." Every cell, battery, or power pack has its own built-in resistance that affects how much electrical energy it can deliver.

In this chapter of CCEA AS 1 Physics (Forces, Energy and Electricity), you will discover what happens inside an electrical power supply. We will unpack the concepts of Electromotive Force (\(\varepsilon\)), Terminal Potential Difference (\(V\)), Lost Volts, and Internal Resistance (\(r\)). Don't worry if these terms seem a bit overwhelming at first — we will break every single idea down step-by-step with clear equations, real-world examples, and visual graph explanations!

1. Core Concepts: What Happens Inside a Power Supply?

A. Electromotive Force (\(\varepsilon\))

The Electromotive Force (\(\varepsilon\)) is the total chemical energy (or other non-electrical form of energy) converted into electrical energy per unit charge as charge passes through a power source.

• Unit: Volt (\(\text{V}\)) or Joule per Coulomb (\(\text{J}\cdot\text{C}^{-1}\)).
• Alternative Definition: The potential difference across the terminals of a power supply when no current is being drawn (often called the open-circuit potential difference).
• Common Trap: Despite its historical name, EMF is NOT a mechanical force measured in Newtons (\(\text{N}\)). It is an energy transfer per unit charge measured in Volts (\(\text{V}\)).

B. Internal Resistance (\(r\))

A battery is made of real physical materials: metal plates, connecting wires, and chemical electrolytes. When charge flows through the cell, these internal parts resist the flow of current. This is known as internal resistance (\(r\)).

• Definition: The opposition to electric current flowing within the power source itself, arising from the chemicals, electrolyte, and internal connections of the cell/battery.
• Unit: Ohm (\(\Omega\)).
• We usually model a real battery as an "ideal" EMF source (\(\varepsilon\)) placed in series with a small internal resistor (\(r\)).

C. Terminal Potential Difference (\(V\))

Terminal Potential Difference (\(V\)) is the potential difference measured across the external terminals of the supply. It represents the electrical energy transferred per unit charge to the components in the external circuit (the load resistance \(R\)).

D. Lost Volts (\(v\) or \(Ir\))

When current (\(I\)) flows through the circuit, some energy per unit charge is inevitably transformed into heat inside the battery because of the internal resistance (\(r\)). This drop in potential inside the supply is called the lost volts.

• Formula: \(\text{Lost Volts} = Ir = \varepsilon - V\)
• Why "lost"? It is not destroyed (energy is always conserved!), but it is "lost" to the external circuit because it is dissipated as thermal energy inside the power source.

Key Takeaway: When no current flows (\(I = 0\)), there are no lost volts (\(Ir = 0\)), so the terminal p.d. equals the full EMF (\(V = \varepsilon\)). Once current flows, the terminal p.d. drops below the EMF.

2. The Fundamental Equations

By the principle of conservation of energy, the total energy supplied per coulomb of charge (\(\varepsilon\)) must equal the energy delivered to the external circuit per coulomb (\(V\)) plus the energy dissipated inside the battery per coulomb (\(Ir\)).

The Core Mathematical Relationships:

\(\varepsilon = V + Ir\)

Since the terminal potential difference across an external load of resistance \(R\) is given by Ohm's Law (\(V = IR\)), we can substitute this in:

\(\varepsilon = IR + Ir\)

\(\varepsilon = I(R + r)\)

Rearranging to Find Circuit Current (\(I\)):

\(I = \frac{\varepsilon}{R + r}\)

Where:
• \(\varepsilon\) = Electromotive force of the power supply (\(\text{V}\))
• \(V\) = Terminal potential difference across the external circuit (\(\text{V}\))
• \(I\) = Current flowing through the circuit (\(\text{A}\))
• \(R\) = Total resistance of the external load (\(\Omega\))
• \(r\) = Internal resistance of the power supply (\(\Omega\))

Step-by-Step Worked Example:

A battery of EMF \(\varepsilon = 12.0\,\text{V}\) and internal resistance \(r = 0.50\,\Omega\) is connected to an external resistor of \(R = 5.50\,\Omega\). Calculate: (a) the total circuit current, (b) the lost volts, and (c) the terminal potential difference.

Step 1: Calculate the current (\(I\))
\(I = \frac{\varepsilon}{R + r} = \frac{12.0}{5.50 + 0.50} = \frac{12.0}{6.00} = 2.0\,\text{A}\)

Step 2: Calculate the lost volts
\(\text{Lost Volts} = Ir = 2.0\,\text{A} \times 0.50\,\Omega = 1.0\,\text{V}\)

Step 3: Calculate the terminal p.d. (\(V\))
Method 1: \(V = \varepsilon - Ir = 12.0\,\text{V} - 1.0\,\text{V} = 11.0\,\text{V}\)
Method 2: \(V = IR = 2.0\,\text{A} \times 5.50\,\Omega = 11.0\,\text{V}\)

3. Graphical Analysis: Determining \(\varepsilon\) and \(r\)

In your CCEA AS 1 theory exam and AS 3 practical exam, you are frequently asked to analyze a graph of Terminal Potential Difference (\(V\)) on the y-axis against Current (\(I\)) on the x-axis.

Linking the Physics to \(y = mx + c\):

Start with the fundamental equation: \(\varepsilon = V + Ir\)
Rearrange to make \(V\) the subject:
\(V = -rI + \varepsilon\)

Comparing this directly to the standard equation of a straight line (\(y = mx + c\)):
• \(y\)-variable: Terminal p.d. \(V\)
• \(x\)-variable: Current \(I\)
• Gradient (\(m\)): \(-r\) (the negative of the internal resistance)
• \(y\)-intercept (\(c\)): \(\varepsilon\) (the electromotive force)
• \(x\)-intercept (\(V = 0\)): The short-circuit current \(I_{\text{sc}} = \frac{\varepsilon}{r}\)

Exam Skill Alert — Handling the Gradient:

Because the graph slopes downwards from left to right, the calculated gradient will be a negative value.
\(\text{Gradient} = \frac{\Delta V}{\Delta I} = -r\)
Therefore, the internal resistance is: \(r = -\text{Gradient}\).
Always remember: resistance \(r\) is a positive physical quantity. If your gradient is \(-1.8\,\text{V}\cdot\text{A}^{-1}\), then \(r = 1.8\,\Omega\).

4. Combinations of Identical Cells

When multiple identical cells are combined, both their EMFs and their internal resistances combine according to standard series and parallel rules.

A. Identical Cells in Series (\(n\) cells)

When \(n\) identical cells (each with EMF \(\varepsilon\) and internal resistance \(r\)) are connected in series facing the same direction:
• Total EMF: \(\varepsilon_{\text{total}} = n\varepsilon\)
• Total Internal Resistance: \(r_{\text{total}} = nr\)
• Total Circuit Current: \(I = \frac{n\varepsilon}{R + nr}\)

Example: Three \(1.5\,\text{V}\) cells with internal resistance \(0.4\,\Omega\) each in series produce \(\varepsilon_{\text{total}} = 4.5\,\text{V}\) and \(r_{\text{total}} = 1.2\,\Omega\).

B. Identical Cells in Parallel (\(m\) branches)

When \(m\) identical cells are connected in parallel:
• Total EMF: \(\varepsilon_{\text{total}} = \varepsilon\) (the EMF remains the same as a single cell)
• Total Internal Resistance: \(r_{\text{total}} = \frac{r}{m}\) (internal resistance is significantly reduced)
• Total Circuit Current: \(I = \frac{\varepsilon}{R + \frac{r}{m}}\)

Benefit of parallel cells: Connecting cells in parallel lowers the internal resistance, allowing the battery pack to supply higher currents with smaller lost volts.

5. Required Practical Method (AS 1 & AS 3)

To determine \(\varepsilon\) and \(r\) experimentally in the laboratory, use the following standard setup and procedure:

Circuit Setup:

1. Connect a cell in series with a switch, an ammeter, and a variable resistor (rheostat or resistance box).
2. Connect a high-resistance voltmeter directly across the terminals of the cell.

Experimental Procedure:

1. Open-Circuit Measurement: With the switch open (\(I = 0\)), record the voltmeter reading to obtain an initial estimate of \(\varepsilon\).
2. Vary Load: Close the switch. Adjust the variable resistor to at least 6 different settings to achieve a wide range of current and voltage values.
3. Simultaneous Readings: For each setting, record the current \(I\) from the ammeter and the terminal potential difference \(V\) from the voltmeter.
4. Plot Graph: Plot a graph of \(V\) (y-axis) against \(I\) (x-axis). Draw a line of best fit.
5. Calculate Results: Determine the y-intercept to find \(\varepsilon\), and calculate the gradient to find \(-r\).

Crucial Experimental Precaution:

Open the switch between readings! Leaving the circuit closed causes current to flow continuously, which heats up the cell and causes it to run down. Heating changes the internal resistance \(r\), which would produce curved or unreliable data instead of a straight line.

6. Real-World Applications: High vs. Low Internal Resistance

A. Why Car Batteries Need VERY LOW Internal Resistance (\(r \approx 0.01\,\Omega\))

A car starter motor requires a massive surge of current (typically \(100\text{--}200\,\text{A}\)) for a brief moment to turn over the engine.
• If the lead-acid battery has \(r = 0.01\,\Omega\) and supplies \(I = 100\,\text{A}\), the lost volts are: \(\text{Lost Volts} = Ir = 100 \times 0.01 = 1.0\,\text{V}\).
• The terminal p.d. remains high (\(12.0\,\text{V} - 1.0\,\text{V} = 11.0\,\text{V}\)), successfully delivering high power to the starter motor.

B. Why Extra High Tension (EHT) Power Supplies Need VERY HIGH Internal Resistance (\(r \approx \text{M}\Omega\))

EHT supplies used in school laboratories generate very dangerous voltages (e.g., \(3000\text{--}5000\,\text{V}\)).
• They are deliberately built with an internal resistance of several megaohms (\(\text{M}\Omega\)).
• Safety Mechanism: If an accidental short-circuit occurs (\(R \approx 0\)), the maximum possible current is limited to: \(I_{\text{max}} = \frac{\varepsilon}{r}\).
• With \(\varepsilon = 5000\,\text{V}\) and \(r = 5\,\text{M}\Omega = 5 \times 10^6\,\Omega\), the maximum short-circuit current is only \(I_{\text{max}} = \frac{5000}{5 \times 10^6} = 1 \times 10^{-3}\,\text{A} = 1\,\text{mA}\), which is well below lethal thresholds!

7. Pitfalls to Avoid & Chief Examiner Tips

• Pitfall 1: Defining EMF poorly. Do not write "EMF is the voltage of the battery." Always write the full formal definition: the energy converted from chemical (or non-electrical) energy to electrical energy per unit charge.
• Pitfall 2: Forgetting to multiply \(r\) in series combinations. If 4 identical cells of \(r = 0.5\,\Omega\) are in series, \(r_{\text{total}} = 4 \times 0.5 = 2.0\,\Omega\). Don't accidentally use just \(0.5\,\Omega\) in your denominator.
• Pitfall 3: Dropping the negative sign on the gradient. Remember that \(\text{gradient} = -r\). Internal resistance \(r\) must always be stated as a positive number in ohms (\(\Omega\)).
• Pitfall 4: Confusing EMF and Terminal p.d. Remember: \(\varepsilon\) is the total electrical energy produced per coulomb inside the source, while \(V\) is the electrical energy delivered to the external load per coulomb.

Quick Summary Checklist

• \(\varepsilon = V + Ir = I(R + r)\)
• Lost Volts \(= Ir = \varepsilon - V\)
• Graph of \(V\) against \(I\): \(\text{y-intercept} = \varepsilon\), \(\text{gradient} = -r\)
• \(n\) identical cells in series: \(\varepsilon_{\text{total}} = n\varepsilon\), \(r_{\text{total}} = nr\)
• \(m\) identical cells in parallel: \(\varepsilon_{\text{total}} = \varepsilon\), \(r_{\text{total}} = \frac{r}{m}\)
• Low \(r\) allows huge currents (car batteries); high \(r\) limits current for safety (EHT supplies).