Potential Dividers and EMF

A real battery has EMF ($\varepsilon$) and internal resistance ($r$).

$\boxed{\varepsilon = I(R + r) = IR + Ir = V + Ir}$

Where:

• $\varepsilon$ = EMF (V) — the total energy per unit charge from the battery
• $V = IR$ = terminal potential difference (what you measure across the battery terminals)
• $Ir$ = "lost volts" — voltage dropped inside the battery

Key Relationships

$V = \varepsilon - Ir$

• When $I = 0$ (open circuit): $V = \varepsilon$ (terminal PD equals EMF)
• As current increases, terminal PD decreases (more lost volts)
• Short circuit ($R = 0$): $I_{\max} = \varepsilon/r$

Measuring EMF and $r$

Plot $V$ against $I$:

• y-intercept = $\varepsilon$ (when $I = 0$)
• Gradient = $-r$
• Equation: $V = \varepsilon - rI$ (straight line, negative gradient)

$\boxed{V_{\text{out}} = V_{\text{in}} \times \frac{R_2}{R_1 + R_2}}$

Two resistors in series share the input voltage proportionally.

With Sensors

LDR in potential divider:

• In dark: LDR resistance high → $V_{\text{out}}$ across LDR is high (or low, depending on position)
• In light: LDR resistance low → $V_{\text{out}}$ changes

Thermistor in potential divider:

• Hot: thermistor resistance low → $V_{\text{out}}$ changes
• Cold: thermistor resistance high → $V_{\text{out}}$ changes

The output voltage can trigger switches, alarms, or be read by a microcontroller.

Loading Effects

When a device is connected across $R_2$ (in parallel), the combined resistance is less than $R_2$ alone, so $V_{\text{out}}$ drops. This is the "loading effect."

• EMF: $\varepsilon = I(R + r)$; terminal PD: $V = \varepsilon - Ir$
• Lost volts = $Ir$; measuring EMF: $V$ vs $I$ graph
• Potential divider: $V_{\text{out}} = V_{\text{in}} \times R_2/(R_1 + R_2)$
• Sensor circuits: thermistor/LDR in divider gives variable output voltage