## Electromotive Force (emf) The **electromotive force (emf)**, denoted by the symbol $\epsilon$, is the energy provided by a source (such as a battery or generator) per unit of charge that passes through it. Despite its name, emf is **not a force**; it is an energy-related quantity that represents the work done to create a potential difference, which drives an electric current. ### 1. EMF vs. Terminal Potential Difference It is essential to distinguish between the emf of a source and the actual potential difference (voltage) it supplies to an external circuit. - **EMF ($\epsilon$):** This is the ideal or total potential difference a source can provide. It is the voltage across the terminals when **no current is being drawn** ($I = 0$). - **Terminal Potential Difference ($V_T$):** This is the actual voltage measured across the terminals of the source when it is connected to a circuit and current ($I$) is flowing. - **Internal Resistance ($r$):** All real-world sources of emf possess some internal resistance, which opposes the flow of current within the source itself. When current ($I$) flows from a source, some potential is lost or dropped across this internal resistance. The relationship is given by: $$V_T = \epsilon - Ir$$ This equation shows that the terminal voltage is always less than the emf when the source is supplying current. ### 2. Current in a Simple Circuit For a simple circuit with an emf source ($\epsilon$), an internal resistance ($r$), and an external load resistance ($R$), the total resistance of the circuit is $R + r$. Using Ohm's Law, the current ($I$) flowing in the circuit is: $$I = \frac{\epsilon}{R + r}$$ ### 3. Power and Efficiency #### Power Dissipation The total power supplied by the emf source is converted into power dissipated in the external and internal resistances. - **Total Power from Source:** $P_{total} = \epsilon I$ - **Power Delivered to Load:** $P_R = I^2R = V_T I$ - **Power Lost in Source:** $P_r = I^2r$ From the conservation of energy: $$P_{total} = P_R + P_r \implies \epsilon I = I^2R + I^2r$$ #### Efficiency The efficiency of an emf source is the ratio of useful power delivered to the load to the total power supplied by the source: $$\text{Efficiency} = \frac{P_R}{P_{total}} = \frac{V_T I}{\epsilon I} = \frac{V_T}{\epsilon} = \frac{R}{R + r}$$ #### Maximum Power Output The power delivered to the external load ($P_R = I^2R$) is maximized when the **load resistance ($R$) equals the internal resistance ($r$)**: $$R = r \quad \text{(for maximum power transfer)}$$ At this point, the maximum power delivered to the load is: $$P_{max} = \frac{\epsilon^2}{4r}$$ However, at maximum power transfer, the efficiency is only 50%, as half the power is lost in the internal resistance. ## Possible Questions and Answers **Q:** What is the difference between EMF and potential difference? **A:** EMF is the energy source that creates the potential difference and is the cause of current flow. It is the total energy per charge supplied by the source. Potential difference (or voltage) is the energy used per charge as it moves between two points in a circuit; it is the effect. The terminal potential difference of a real battery is its EMF minus the voltage lost to its own internal resistance. **Q:** Can the terminal voltage ever be greater than the EMF? **A:** Yes, but only when the source is being charged. In this case, an external source forces current to flow backwards through the battery (from the positive to the negative terminal). The equation becomes $V_T = \epsilon + Ir$. ## Summary - **Electromotive Force (EMF, $\epsilon$)** is the total energy per unit charge provided by a source to an electric circuit. - **Internal Resistance ($r$)** is the resistance within the emf source, which causes a voltage drop when current flows. - **Terminal Potential Difference ($V_T$)** is the actual voltage supplied by the source to the external circuit and is given by $V_T = \epsilon - Ir$. - The current in a simple circuit is $I = \epsilon / (R + r)$. - **Maximum power** is delivered to an external load when the load resistance equals the internal resistance ($R = r$). | Term | Description | | :--- | :--- | | **Electromotive Force (EMF)** | The energy source that drives charge flow | | **Resistance (R)** | The opposition to the flow of electric charges | | **Current (I)** | The rate of flow of electric charges |

Electromotive Force & Maximum Power Output | Electricity | Physics 11