Many students are comfortable working with individual resistors, but run into confusion when those resistors are combined into networks. A common point of misunderstanding is the voltage divider: specifically, why the current through both resistors is the same even when their resistances differ.

At first glance, it can seem contradictory. Using Ohm’s law, students often reason that different resistances should imply different currents, or that the voltages across the resistors must somehow be equal. The key is to recognise how series circuits behave.

 Vin
  o
  |
 [R1]
  |
  o------ Vout
  |
 [R2]
  |
  o
 GND

Step 1: Think of the total resistance

The two resistors are in series, so their total resistance is:

$$ R_{\text{total}} = R_1 + R_2 $$

Step 2: Find the current

The same current flows through all elements in a series path (provided no current is drawn from the midpoint node $V_{out}$). Therefore, the current through both resistors is:

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

This is the critical point: series elements share current, not voltage.

Step 3: Find the output voltage

The output voltage $V_{out}$ is simply the voltage across $R_2$. Using Ohm’s law:

$$ V_{out} = I \cdot R_2 $$

Substituting the expression for $I$:

$$ V_{out} = \frac{V_{in}}{R_1 + R_2} \cdot R_2 $$

$$ V_{out} = V_{in} \cdot \frac{R_2}{R_1 + R_2} $$

Key insight

• The current is the same through both resistors because there is only one path for charge to flow.
• The voltage divides between the resistors in proportion to their resistances.

A larger resistor drops more voltage, but it does not carry more current in a series circuit.

Once this distinction clicks—current continuity in series versus voltage division—the behaviour of resistor ladders becomes much more intuitive.