Welcome to Internal Resistance & Electromotive Force

Have you ever noticed that a phone battery gets warm when running demanding apps, or that a car's dashboard lights dim slightly for a split second when the engine starts? In introductory physics, we often pretend batteries and power supplies are "perfect" sources of voltage. However, in the real world, power supplies are made of physical materials that have their own internal resistance.

In this chapter of AS 1: Forces, Energy and Electricity, you will learn why real power supplies behave the way they do, how energy is conserved across a circuit, and how to determine the key electrical characteristics of a cell experimentally.

1. Electromotive Force (\(\mathcal{E}\)) vs. Potential Difference (\(V\))

Don't worry if these two terms seem almost identical at first—they both share the same unit, the Volt (\(\text{V}\)), which is equivalent to joules per coulomb (\(\text{J}\cdot\text{C}^{-1}\)). However, they describe energy transfers going in opposite directions.

Electromotive Force (\(\mathcal{E}\))

• Definition: The total energy transferred from chemical (or other non-electrical forms) into electrical potential energy per unit charge (\(\text{C}\)) passing through the source.
• Key Idea: \(\mathcal{E}\) is the energy "pushed into" the electrical circuit by the battery or cell.

Terminal Potential Difference (\(V\))

• Definition: The energy transferred from electrical energy into other forms (such as heat or light) per unit charge across the external load resistance (\(R\)).
• Key Idea: \(V\) is the actual voltage available to the external components connected across the terminals of the power supply.

Examiner Warning: What's in a Name?

Despite having the word "force" in its name, Electromotive Force is NOT a force measured in Newtons (\(\text{N}\)). It is an energy per unit charge quantity measured in Volts (\(\text{V}\) or \(\text{J}\cdot\text{C}^{-1}\)). Always state the direction of energy conversion when defining these terms in an exam.

Section Key Takeaway: \(\mathcal{E}\) converts non-electrical energy \(\to\) electrical energy. Terminal p.d. \(V\) converts electrical energy \(\to\) non-electrical forms in the external circuit.

2. Internal Resistance (\(r\)) and "Lost Volts"

What is Internal Resistance?

A battery is not an ideal magical device; it is made of real physical components such as chemical electrolytes, metal plates, and connecting terminals. As charge carriers (electrons and ions) flow through the inside of the cell itself, they collide with particles and encounter resistance.

• Internal Resistance (\(r\)): The inherent resistance to the flow of charge inside the source of \(\text{EMF}\) itself, leading to energy dissipation within the supply (measured in \(\Omega\)).

Understanding "Lost Volts" (\(Ir\))

Because the cell has internal resistance \(r\), when a current \(I\) flows around the circuit, some energy must be used just to push the charges through the battery itself. This internal potential drop is called lost volts:

\(V_{\text{lost}} = Ir\)

These volts are not literally "lost" from the universe—they represent energy converted into heat inside the battery due to work done against its internal resistance.

The Governing Conservation of Energy Equation

From the law of conservation of energy, the total energy supplied per coulomb of charge by the cell (\(\mathcal{E}\)) must equal the energy delivered to the external circuit per coulomb (\(V\)) plus the energy dissipated internally per coulomb (\(Ir\)):

\(\mathcal{E} = V + Ir\)

Since the external terminal potential difference is given by Ohm's law as \(V = IR\) (where \(R\) is the external load resistance), we can write this relationship in several equivalent forms:

\(\mathcal{E} = IR + Ir\)

\(\mathcal{E} = I(R + r)\)

Two Important Circuit Conditions

• Open-Circuit Condition (\(I = 0\)): If the circuit is broken or disconnected, no current flows (\(I = 0\)). Therefore, the lost volts term is zero (\(Ir = 0\)), which means:
\(V = \mathcal{E}\)
A high-resistance (ideal) voltmeter connected across an isolated cell directly measures the \(\text{EMF}\) because it draws negligible current.

• Short-Circuit Condition (\(R = 0\)): If the terminals of the cell are connected directly with a wire of zero resistance, the maximum possible current flows through the cell:
\(I_{\text{max}} = \frac{\mathcal{E}}{r}\)

Section Key Takeaway: As external current \(I\) increases, the lost volts (\(Ir\)) increase, which causes the terminal potential difference \(V\) to drop.

3. Experimental Determination of \(\mathcal{E}\) and \(r\)

Circuit Setup

To determine the \(\text{EMF}\) and internal resistance of a cell experimentally in the laboratory, connect the following in series:
• The cell under test (with \(\text{EMF}\) \(\mathcal{E}\) and internal resistance \(r\))
• A variable resistor / rheostat (\(R\))
• An ammeter (to measure circuit current \(I\))
• A switch
Connect a high-resistance digital voltmeter directly across the terminals of the cell to measure the terminal potential difference \(V\).

Transforming the Equation to a Straight Line

We start with our energy conservation equation and rearrange it into the standard straight-line form \(y = mx + c\):

\(\mathcal{E} = V + Ir\)

\(V = -rI + \mathcal{E}\)

Comparing this directly with \(y = mx + c\):
• \(y\)-axis variable: Terminal potential difference, \(V\) (in \(\text{V}\))
• \(x\)-axis variable: Current, \(I\) (in \(\text{A}\))
• Gradient (\(m\)): \(-r\) (the negative of internal resistance)
• \(y\)-intercept (\(c\)): \(\mathcal{E}\) (the Electromotive Force)

Interpreting the Graph

• Finding \(\text{EMF}\) (\(\mathcal{E}\)): Read the value where the line of best fit crosses the vertical \(V\)-axis (\(y\)-intercept). At this point, \(I = 0\), so \(V = \mathcal{E}\).
• Finding Internal Resistance (\(r\)): Calculate the gradient of the straight line. Since the line slopes downward, the gradient is negative:
\(r = -\text{gradient}\)
Remember that internal resistance is a physical quantity and must always be stated as a positive value in \(\Omega\).
• Finding Short-Circuit Current (\(I_{\text{sc}}\)): The intercept on the horizontal \(I\)-axis (where \(V = 0\)) gives the theoretical short-circuit current \(I_{\text{sc}} = \frac{\mathcal{E}}{r}\).

Practical Tips & Common Experimental Errors

• Switch Off Between Readings: Always open the switch between taking measurements. If current flows continuously, the cell and circuit components heat up, altering the resistance and prematurely discharging the cell.
• Voltmeter Resistance: An ordinary, low-resistance voltmeter draws current from the cell, creating internal lost volts and underestimating \(\mathcal{E}\). A high-resistance or digital voltmeter ensures current drawn by the meter is negligible.

Section Key Takeaway: Plotting \(V\) against \(I\) yields a downward-sloping straight line where the \(y\)-intercept is \(\mathcal{E}\) and the internal resistance is \(r = -\text{gradient}\).

4. Combinations of Identical Cells

When working with circuits containing multiple identical power sources, use these standard rules:

Identical Cells in Series

When \(n\) identical cells, each with \(\text{EMF}\) \(\mathcal{E}\) and internal resistance \(r\), are connected in series (positive terminal to negative terminal):

\(\mathcal{E}_{\text{total}} = n\mathcal{E}\)

\(r_{\text{total}} = nr\)

Example: Three \(1.5\text{ V}\) cells each with \(0.4\ \Omega\) internal resistance connected in series provide a total \(\text{EMF}\) of \(3 \times 1.5 = 4.5\text{ V}\) and a total internal resistance of \(3 \times 0.4 = 1.2\ \Omega\).

Identical Cells in Parallel

When \(n\) identical cells, each with \(\text{EMF}\) \(\mathcal{E}\) and internal resistance \(r\), are connected in parallel across identical branches:

\(\mathcal{E}_{\text{total}} = \mathcal{E}\)

\(r_{\text{total}} = \frac{r}{n}\)

Example: Two \(6.0\text{ V}\) cells each with \(0.8\ \Omega\) internal resistance connected in parallel provide a total \(\text{EMF}\) of \(6.0\text{ V}\), but the total internal resistance is halved to \(\frac{0.8}{2} = 0.4\ \Omega\). This allows the circuit to deliver higher currents without excessive internal voltage drop.

Section Key Takeaway: Series combinations boost total \(\text{EMF}\) but also increase total internal resistance. Parallel combinations keep \(\text{EMF}\) the same while reducing total internal resistance.

5. Quick Summary & Examiner Pitfall Checklist

Before sitting your AS 1 exam, double-check that you can avoid these classic traps:

• Trap 1: Stating that terminal p.d. \(V\) is constant. Correction: As the load draws more current \(I\), lost volts (\(Ir\)) increase, so terminal p.d. \(V\) decreases.
• Trap 2: Writing that \(r = \text{gradient}\) on a \(V\) vs. \(I\) graph. Correction: The line has a negative slope; \(r = -\text{gradient}\).
• Trap 3: Confusing energy directions in definitions. Correction: \(\text{EMF}\) converts non-electrical \(\to\) electrical; p.d. converts electrical \(\to\) other forms.
• Trap 4: Forgetting units in final answers. Correction: \(\text{EMF}\) and p.d. in \(\text{V}\); current in \(\text{A}\); resistance in \(\Omega\).