The terminal voltage of a battery under load is lower than its e.m.f. due to the voltage drop caused by internal resistance. When a current flows through a battery, it encounters resistance within the battery itself. This resistance causes a part of the electrical energy to be dissipated as heat within the battery. This energy loss is manifested as a reduction in the voltage from the e.m.f. to the terminal voltage. The larger the current or the higher the internal resistance, the greater the voltage drop, leading to a more significant difference between the e.m.f. and the terminal voltage.

Internal resistance plays a significant role in the charging of batteries, especially in rapid charging technologies. When charging a battery, internal resistance can lead to heat generation. In rapid charging, the high current involved can cause significant heat generation due to higher internal resistance, potentially damaging the battery if not properly managed. This is why many rapid charging technologies incorporate advanced cooling systems or algorithms to control the charging rate, especially as the battery approaches full charge. Managing the heat generated by internal resistance is crucial to maintain battery health and ensure efficient charging.

A circuit cannot have a potential difference without an e.m.f. source. The e.m.f. source is responsible for creating a potential difference by doing work to move charge through the circuit. Without an e.m.f. source, there would be no energy supplied to the circuit to move charges, and hence no potential difference could be established across any component in the circuit. In essence, the e.m.f. source is the primary driver that initiates and sustains the flow of electric charge, leading to the potential differences observed across various components in the circuit.

Internal resistance is a critical factor in determining a battery's lifespan and efficiency. As a battery ages, chemical reactions within it become less efficient, leading to an increase in internal resistance. This increased resistance results in a greater proportion of the energy being dissipated as heat within the battery itself, rather than being available for external use. Consequently, the battery's ability to deliver a high terminal voltage under load diminishes. This is why older batteries drain faster and take longer to charge. Over time, the internal resistance may increase to a point where the battery can no longer effectively power devices, marking the end of its useful lifespan.

In a circuit with multiple power sources, each source's e.m.f. and internal resistance contribute cumulatively to the circuit's overall performance. The total e.m.f. of the circuit is determined by how the sources are connected. If they are in series, their e.m.f.s add up, while in parallel, the effective e.m.f. is more complex to calculate and depends on the individual e.m.f.s and internal resistances. The total internal resistance is the sum of the internal resistances of the individual sources. This overall e.m.f. and internal resistance combination affects the circuit's total voltage, current distribution, and energy efficiency. Understanding these relationships is crucial for designing circuits with optimal performance and efficiency.