Physics 9702: Study Notes - Potential Dividers (Topic 10.3)

Welcome to the world of Potential Dividers! Don't worry if complex circuits sometimes feel like tangled spaghetti—this chapter is about simplifying and controlling voltage, which is essential for almost every electronic device you use. Think of the potential divider as the electronic equivalent of a volume knob or a dimmer switch!

In this section, we will learn how to precisely control the voltage output of a circuit using resistors, and how to apply this principle in real-world measuring devices and sensing circuits. You’ve got this!

1. Understanding the Potential Divider Principle

A potential divider is a series circuit designed to divide the input potential difference (voltage) into smaller, usable fractions. It’s fundamentally a simple series circuit, but we use it specifically to tap off a certain voltage.

Analogy: The Mountain Waterfall

Imagine a waterfall (this is your supply voltage, \(V_{in}\)). If you want to take water out at different heights, you place different pipes (resistors) at varying levels. The difference in height (potential difference) between the start and the point where you tap the water depends on how far down the "pipe" (resistor) you are.

How the Potential Divider Circuit Works

When two resistors, \(R_1\) and \(R_2\), are connected in series to a supply voltage, \(V_{in}\):

  • The total potential difference \(V_{in}\) is split between \(R_1\) and \(R_2\).
  • The ratio of the p.d. across a resistor is equal to the ratio of its resistance.
  • We measure the output voltage (\(V_{out}\)) across the resistor we are interested in (let's call it \(R_2\)).

The total resistance in the circuit is \(R_{Total} = R_1 + R_2\).
Using Ohm's Law, the current (\(I\)) flowing through the series circuit is constant:

$$I = \frac{V_{in}}{R_1 + R_2}$$

The output voltage, \(V_{out}\) (measured across \(R_2\)), is given by \(V_{out} = I \times R_2\).
Substituting the expression for \(I\), we get the crucial Potential Divider Formula:

$$V_{out} = \left( \frac{R_2}{R_1 + R_2} \right) \times V_{in}$$

Key Point Alert: The resistor in the numerator (\(R_2\)) is always the one across which you are measuring the output potential difference.

Quick Review Box: Calculating \(V_{out}\)
  • If \(R_1 = R_2\), then \(V_{out} = 0.5 \times V_{in}\). The voltage is divided equally.
  • If \(R_2\) is much larger than \(R_1\), then \(V_{out}\) will be close to \(V_{in}\).
  • If \(R_2\) is much smaller than \(R_1\), then \(V_{out}\) will be close to zero.

Did you know? Many modern electronic components require a very specific, stable voltage (like 3.3V or 5V) but are connected to a higher source (like a 12V battery). Potential dividers are used to drop and regulate this voltage for delicate components.

Key Takeaway: A potential divider uses two series resistors to produce an output voltage that is a predictable fraction of the input voltage, calculated by the ratio of the output resistance to the total resistance.

2. The Potentiometer and Null Methods

A potentiometer is a specialized potential divider. While a fixed potential divider gives a fixed output voltage, a potentiometer allows the output voltage to be varied continuously from zero up to the supply voltage.

What is a Potentiometer?

It usually consists of a uniform resistance wire or a track with a sliding contact (wiper). By moving the wiper, you change the resistance ratio, thereby changing the output voltage.

The key principle used here is that for a wire of uniform cross-sectional area and material, the potential difference (voltage) across any length is directly proportional to that length (\(L\)):

$$\frac{V_{out}}{V_{in}} = \frac{L_{out}}{L_{total}}$$
The Potentiometer as a Means of Comparing Potential Differences

Syllabus requirement 10.3(2) asks us to understand its use for comparing potential differences, which is usually done via the Null Method.

In this classic experimental setup, the potentiometer is used to find the ratio of the e.m.f. of two sources (or an e.m.f. and a p.d.).

Step-by-Step: The Null Method

The Null Method involves using a galvanometer to detect when *zero* current is flowing between two points in a circuit.

1. A driver cell (A) powers the main potential divider wire (Potentiometer wire AB).

2. A test cell (B, with unknown e.m.f. \(E_B\)) is connected in series with a galvanometer (G) and its sliding contact (J).

3. The slider (J) is moved along the wire AB until the galvanometer shows a zero reading. This is the null point.

When the galvanometer reads zero:

The potential difference tapped off the length \(L_B\) of the wire is exactly equal to the e.m.f. \(E_B\) of the test cell.

$$\text{At the Null Point: } V_{L_B} = E_B$$
Why Use the Null Method? (Accuracy!)

This is the most accurate way to measure or compare voltages because, at the null point, no current is drawn from the test cell (B). Why is this critical?

  • Recall that a real source's terminal p.d. is \(V = E - Ir\).
  • If the current \(I = 0\) (which is true at the null point), then \(V = E\).

The null method ensures that the measured potential difference is exactly equal to the source's true electromotive force (e.m.f.), free from any voltage drop caused by internal resistance.

Key Takeaway: A potentiometer is an accurate variable potential divider. Using a galvanometer to find the null point ensures that the measurement of the potential difference is equivalent to the pure e.m.f. of the source, as zero current is drawn.

3. Potential Dividers in Sensing Circuits

Potential dividers are incredibly useful when combined with components whose resistance changes based on external factors (like heat or light). These components, called sensors (or transducers), allow the potential divider to function as a simple control circuit.

The syllabus requires you to understand the use of Thermistors and Light-Dependent Resistors (LDRs) in these circuits.

3.1 Thermistors (Temperature Sensing)

A thermistor is a resistor whose resistance changes significantly with temperature. In the 9702 syllabus, we assume the use of an NTC (Negative Temperature Coefficient) thermistor:

  • Increase in Temperature (\(T\)) $\rightarrow$ Decrease in Resistance (\(R\)).

If we place the thermistor in a potential divider, we can get an output voltage that controls a circuit (like a fan or heater).

Example: Temperature Controlled Fan

We want a fan to turn ON when the temperature gets too HIGH.

1. We set up a potential divider with a fixed resistor \(R_{fixed}\) and the thermistor \(R_{thermistor}\).

2. We measure the output voltage \(V_{out}\) across the fixed resistor \(R_{fixed}\).

When it gets hot:

  • \(T\) increases.
  • \(R_{thermistor}\) decreases (it gets smaller).
  • Since \(R_{thermistor}\) is smaller, it takes a smaller share of \(V_{in}\).
  • Therefore, the fixed resistor \(R_{fixed}\) takes a larger share.
  • \(V_{out}\) across \(R_{fixed}\) increases. This high \(V_{out}\) can then switch on the fan.
3.2 Light-Dependent Resistors (LDRs) (Light Sensing)

A Light-Dependent Resistor (LDR) is a component whose resistance changes based on the intensity of incident light:

  • Increase in Light Intensity $\rightarrow$ Decrease in Resistance (\(R\)).

Example: Automatic Street Lamp

We want a street lamp to turn ON when it gets DARK (low light intensity).

1. We set up a potential divider with a fixed resistor \(R_{fixed}\) and the LDR \(R_{LDR}\).

2. To turn the light ON when dark, we need the output voltage \(V_{out}\) to be HIGH when light is LOW. Therefore, we measure \(V_{out}\) across the LDR itself.

When it gets dark (LOW light):

  • Light intensity decreases.
  • \(R_{LDR}\) increases significantly (it gets bigger).
  • Since \(R_{LDR}\) is larger, it takes a larger share of \(V_{in}\).
  • \(V_{out}\) across the LDR increases. This high \(V_{out}\) can trigger the street light to switch on.
Common Mistake to Avoid!

Students often forget which resistor the output voltage is measured across. Always remember:

If you want the output voltage to increase when the sensor's property (e.g., light) increases, measure \(V_{out}\) across the fixed resistor.

If you want the output voltage to increase when the sensor's property decreases, measure \(V_{out}\) across the sensor itself.

Key Takeaway: By replacing one fixed resistor in a potential divider with a variable sensor (LDR or thermistor), we create a circuit where the output voltage is controlled by an environmental factor, allowing us to automate devices based on temperature or light levels.