# E.m.f and Internal Resistance ## What is Electromotive Force (e.m.f)? - When charge flows through a power source like a cell or battery, it **gains** electrical energy - **Electromotive force (e.m.f)** is the amount of chemical energy converted to electrical energy per coulomb of charge when charge passes through a power supply - Represented by the Greek letter epsilon: $\varepsilon$

E.m.f is the energy transferred from other forms to electrical energy per unit charge: $\varepsilon = \frac{\text{Energy transferred to electrical}}{\text{Charge}}$

- Despite the name, e.m.f is **not a force** – it is measured in **volts (V)** - The e.m.f comes from sources such as batteries, cells, or power supplies - E.m.f equals the potential difference across the cell **when no current is flowing** (open circuit) ### Measuring E.m.f - E.m.f can be measured by connecting a **high-resistance voltmeter** across the terminals of the cell in an **open circuit** - The voltmeter must be in **parallel** with the cell - With the circuit open (switch off), the voltmeter reading gives the e.m.f directly ## Internal Resistance - All real power supplies have some resistance between their terminals - This resistance is called **internal resistance** (symbol: $r$) - Internal resistance causes charge carriers to dissipate some electrical energy as they pass through the power supply itself - This is why batteries and cells become warm after being used for some time - Energy is being transferred to thermal energy inside the cell - Over time, internal resistance causes **loss of voltage** or **energy loss** in the power supply ### Circuit Model - A real cell can be modelled as an ideal e.m.f source with an internal resistance connected in **series** - This is shown in circuit diagrams as: - $\varepsilon$ = the e.m.f of the cell - $r$ = the internal resistance (inside the dotted box representing the cell) - $R$ = the load resistor (external resistance) | Component | Symbol | Meaning | |-----------|--------|---------| | $\varepsilon$ | V | E.m.f of the power supply | | $r$ | Ω | Internal resistance | | $R$ | Ω | Load resistor (external) | | $V$ | V | Terminal p.d (across load) | | $v$ | V | Lost volts (across internal resistance) | ## Terminal Potential Difference (p.d) - **Terminal potential difference** is the potential difference measured **across the terminals** of a cell - If there was no internal resistance, the terminal p.d would equal the e.m.f - Defined by: $V = IR$ Where: - $V$ = terminal p.d (V) - $I$ = current (A) - $R$ = load resistance (Ω)

If a cell has internal resistance, the terminal p.d is always **lower** than the e.m.f

- When you connect a load resistor $R$ across the cell's terminals, the terminal p.d $V$ is **also** the p.d across the load resistor ## Lost Volts - In a closed circuit, current flows through the cell and a potential difference develops across the **internal resistance** - Since resistance opposes current, this reduces the energy per unit charge (voltage) available to the rest of the external circuit - This reduction is called the **"lost volts"** (symbol: $v$)

Lost volts is the voltage lost in the cell due to internal resistance: $v = Ir$

- From **conservation of energy**: $$v = \varepsilon - V$$ - Using Ohm's law across the internal resistance: $$v = Ir$$ Where: - $v$ = lost volts (V) - $I$ = current (A) - $r$ = internal resistance (Ω) - $\varepsilon$ = e.m.f (V) - $V$ = terminal p.d (V) ## The E.m.f Equation - Combining the relationships above gives the fundamental e.m.f equation: $$\varepsilon = V + v$$ - Substituting $V = IR$ and $v = Ir$: $$\varepsilon = IR + Ir = I(R + r)$$ - This can also be rearranged as: $$\varepsilon = V + Ir$$ **Summary of key equations:** | Equation | Description | |----------|-------------| | $\varepsilon = I(R + r)$ | E.m.f in terms of total resistance | | $\varepsilon = V + Ir$ | E.m.f as terminal p.d plus lost volts | | $v = \varepsilon - V$ | Lost volts from energy conservation | | $V = IR$ | Terminal p.d across load resistor | | $v = Ir$ | Lost volts across internal resistance |

If a battery has "negligible internal resistance", then $\varepsilon = V$ and internal resistance calculations are not needed

## Energy Distribution in the Circuit - The e.m.f represents the **total energy per unit charge** supplied by the cell - This energy is distributed between: - **Useful energy**: delivered to the external load ($V = IR$) - **Wasted energy**: dissipated as heat in the internal resistance ($v = Ir$) - As current increases: - Lost volts ($Ir$) increases - Terminal p.d ($V$) decreases - More energy is wasted internally - This explains why: - A battery's voltage "drops" under heavy load (high current) - Old batteries with higher internal resistance perform poorly - Short-circuiting a battery (very high current) can cause dangerous heating ### Effect of Current on Voltage Distribution The chart below illustrates how terminal p.d. and lost volts vary with current for a cell with e.m.f of 12V and internal resistance of 2Ω:

- Notice that the terminal p.d. ($V$) decreases linearly as current increases - Simultaneously, the lost volts ($v$) increase linearly with current - The sum $V + v$ always equals the e.m.f ($\varepsilon = 12V$) - At zero current (open circuit), terminal p.d equals e.m.f and there are no lost volts - At high currents, most of the e.m.f is "lost" across the internal resistance, leaving little useful voltage for the external circuit