Introduction: The "Push" Behind the Sensor

In the previous chapters of Sensing and Materials, we looked at how current and potential difference (p.d.) work in circuits. But where does that energy come from? Whether it’s a battery in a portable light sensor or a power supply in a lab, no energy source is "perfect."

In this chapter, we explore why batteries get warm when you use them, why the voltage of a battery "drops" when you turn a device on, and how we calculate the power used by our components. Understanding these concepts is vital for designing reliable sensors and electronic devices.

1. Electromotive Force (e.m.f.)

We often use the words "voltage" or "potential difference" to describe a battery, but in Physics B, we use a more precise term for the total energy a source provides: Electromotive Force or e.m.f.

Definition: The e.m.f. (\(\epsilon\)) is the energy transferred from chemical (or other) forms to electrical energy per unit charge.

Don't let the name fool you—it isn't actually a "force" measured in Newtons. It is measured in Volts (V), where \(1\text{ V} = 1\text{ J C}^{-1}\).

Key Formula:

\(\epsilon = \frac{W}{Q}\)

Where:

  • \(\epsilon\) = Electromotive force (V)
  • \(W\) = Work done or energy transferred (J)
  • \(Q\) = Charge (C)

Note: The syllabus uses the Greek letter epsilon (\(\epsilon\)) for e.m.f. to avoid getting it confused with \(E\) for energy.

2. Internal Resistance (\(r\))

Have you ever noticed that a phone or a laptop battery gets warm after heavy use? This happens because of internal resistance. No battery is a perfect conductor; the chemicals and components inside the battery itself resist the flow of charge.

We model a real battery as an "ideal" source of e.m.f. in series with a small resistor, which we call the internal resistance (\(r\)).

  • Terminal Potential Difference (\(V\)): This is the voltage measured across the terminals of the battery when it is supplying current. This is the "useful" voltage available to the rest of the circuit.
  • "Lost Volts" (\(Ir\)): This is the potential difference dropped across the internal resistance. This energy is usually "lost" as heat inside the battery.
The Core Equation:

The total energy supplied by the battery (\(\epsilon\)) must equal the energy used in the external circuit (\(V\)) plus the energy lost inside the battery (\(Ir\)).

\(\epsilon = V + Ir\)

Or, more commonly used for calculations:

\(V = \epsilon - Ir\)

Analogy: Think of e.m.f. as your "Gross Salary" (the total money you earn). Internal resistance is like "Tax" (money taken away before you get it). The Terminal p.d. is your "Net Pay" (the money you actually get to spend in the shops).

3. Power Dissipation

Power is the rate at which energy is transferred. In a circuit, we want to know how much energy our sensor or component is using every second.

Calculating Power:

Using the relationships from the "Sensing" module, we have three main ways to calculate power (\(P\)):

  1. \(P = IV\) (The standard formula: Current \(\times\) Potential Difference)
  2. \(P = I^2R\) (Useful when you know the resistance and the current)
  3. \(P = \frac{V^2}{R}\) (Useful when you know the voltage and the resistance)

Why \(I^2R\)? This formula is particularly important for understanding internal resistance. The power wasted as heat inside a battery is \(P = I^2r\). If you double the current, you quadruple the power wasted! This is why high-current devices drain batteries so quickly and make them so hot.

Energy Transfer:

To find the total energy (\(W\)) transferred over a period of time (\(t\)), use:

\(W = VIt\)

4. Practical Application: Measuring \(\epsilon\) and \(r\)

In your practical work (PAG4), you may be asked to determine the e.m.f. and internal resistance of a cell. We do this by changing the external resistance and measuring how the terminal p.d. changes with current.

The Graph Method:

If we rearrange our formula \(V = \epsilon - Ir\) into the form \(y = mx + c\):

\(V = (-r)I + \epsilon\)

When you plot a graph of Terminal p.d. (\(V\)) on the y-axis against Current (\(I\)) on the x-axis:

  • The y-intercept is the e.m.f. (\(\epsilon\)). (This is the voltage when no current flows).
  • The gradient of the line is \(-r\) (the negative of the internal resistance).

Common Mistake: Students often forget that the gradient is negative. Because \(V\) drops as \(I\) increases, the line will slope downwards. If your gradient is \(-1.5\), then the internal resistance is \(1.5\ \Omega\).

5. Summary and Key Takeaways

Quick Review Box:

  • e.m.f. (\(\epsilon\)): Total energy per unit charge supplied by the source.
  • Internal Resistance (\(r\)): The resistance inside the power source itself.
  • Terminal p.d. (\(V\)): The actual voltage delivered to the circuit.
  • Lost Volts (\(Ir\)): The voltage used up overcoming internal resistance.
  • Power (\(P\)): Measured in Watts (W), calculated as \(IV\) or \(I^2R\).

Don't forget: In a perfect world, \(V\) would always equal \(\epsilon\). In the real world of Physics B, the more current you draw, the more voltage you "lose" to the internal resistance!

For more information on how these components fit into wider circuits, see the chapter on "Resistivity, conductance and potential dividers".