Introduction to Emf and Internal Resistance
Welcome to a crucial part of the Electricity section! So far in your Physics journey, you might have treated batteries and cells as "perfect" sources of energy. However, in the real world, things are a bit more complicated—and a lot more interesting. Have you ever noticed that your phone or laptop battery gets warm when you use it heavily? Or that the headlights of a car might dim slightly when the engine starts? These effects happen because of Internal Resistance.
In this chapter, we will learn why batteries aren't 100% efficient and how we can mathematically model the energy they "lose" before it even reaches the circuit.
Electromotive Force (Emf)
The term Electromotive Force, or emf (\(\varepsilon\)), is a bit of a historical accident—it isn't actually a "force" in the mechanical sense (measured in Newtons). It is measured in Volts (\(V\)).
Definition: The emf of a source is the total amount of chemical energy converted into electrical energy per unit charge that passes through it.
Think of the emf as the "maximum potential" of the battery. It is the voltage across the terminals of the cell when no current is flowing. As soon as you connect a wire and current starts to move, things change.
Internal Resistance: The Energy Thief
Every power source (like a chemical cell or a generator) is made of materials. These materials—whether they are chemicals inside a battery or wires in a generator—have their own resistance. We call this Internal Resistance (\(r\)).
Because the current has to flow through the battery to get to the rest of the circuit, some of the energy is "wasted" as heat inside the battery itself. This leads to two very important concepts:
- Terminal Potential Difference (Terminal pd): This is the actual voltage (\(V\)) delivered to the external circuit. It is what you would measure if you put a voltmeter across the battery terminals while a current is flowing.
- Lost Volts: This is the energy lost per unit charge due to the internal resistance. We represent this as \(v\) or \(I \times r\).
Analogy: Imagine a delivery van carrying 100 packages (the emf). To leave the warehouse, the van has to navigate a bumpy, difficult driveway (internal resistance). By the time it reaches the main road, 5 packages have fallen off and broken (lost volts). The van arrives at the customers' houses with only 95 packages (terminal pd).
The Key Equation
We can combine these ideas into one simple energy conservation equation. The total energy provided by the source (\(\varepsilon\)) must equal the energy used in the external circuit (\(V\)) plus the energy wasted inside the source (\(Ir\)).
\(\varepsilon = V + Ir\)
Since \(V = IR\) (where \(R\) is the total resistance of the external circuit, often called the load resistance), we can also write:
\(\varepsilon = I(R + r)\)
Key Takeaway: If the current (\(I\)) increases, the "lost volts" (\(Ir\)) also increase. This means the terminal pd (\(V\)) must decrease.
Measuring Emf and Internal Resistance (Required Practical 6)
In your lab work, you will be required to find the emf and internal resistance of a cell. You can't just look inside a battery with a microscope to see the resistance, so we use a graphical method.
The Setup
You connect a cell in series with an ammeter and a variable resistor (rheostat). You then connect a voltmeter in parallel across the cell.
The Method
- Vary the resistance of the variable resistor to change the current (\(I\)) in the circuit.
- For each change, record the current (\(I\)) and the terminal pd (\(V\)).
- Plot a graph of \(V\) on the y-axis and \(I\) on the x-axis.
Analyzing the Graph
We can rearrange our main equation \(\varepsilon = V + Ir\) into the format of a straight-line graph (\(y = mx + c\)):
\(V = -rI + \varepsilon\)
When you look at this equation:
- The y-intercept is the emf (\(\varepsilon\)). This makes sense: when current is zero, the terminal pd equals the emf.
- The gradient (slope) is \(-r\) (the negative of the internal resistance).
Common Mistake: Students often forget that the gradient is negative internal resistance. Resistance itself is always a positive value, but the line on your graph will slope downwards.
Cells in Series and Parallel
Sometimes we use more than one cell. Here is how emf and internal resistance behave:
Cells in Series
If you connect \(n\) identical cells in series:
- Total emf = \(n \times \varepsilon\)
- Total internal resistance = \(n \times r\)
Identical Cells in Parallel
If you connect \(n\) identical cells in parallel:
- Total emf = \(\varepsilon\) (the same as a single cell)
- Total internal resistance = \(\frac{r}{n}\)
Why use parallel? It doesn't increase the voltage, but it reduces the total internal resistance, meaning less energy is wasted as heat, and the battery pack can last longer while providing higher currents.
Quick Review & Tips
Don't worry if this seems tricky at first! Just remember that \(V\) is what you "get" and \(\varepsilon\) is what the battery "starts with." The difference is always caused by the internal resistance \(r\).
Summary Table
Emf (\(\varepsilon\)): Total energy per charge; measured when \(I = 0\).
Terminal pd (\(V\)): Energy delivered to the circuit; measured when \(I > 0\).
Internal Resistance (\(r\)): The "built-in" resistance of the power source.
Lost Volts (\(Ir\)): The potential difference "dropped" across the internal resistance.
Common Exam Pitfall
When a switch is open, the ammeter reads \(0\) and the voltmeter across the battery reads the emf. When the switch is closed, the voltmeter reading will drop. This drop is the "lost volts." If an exam question asks "Why does the voltmeter reading decrease when the switch is closed?", the answer is always: "Current flows through the circuit, so potential is dropped across the internal resistance of the cell."