A 45 ampere-hour battery will supply 45 A for \(1 \mathrm{~h}, 2 \mathrm{~A}\) for \(22.5 \mathrm{~h}\), etc. If such a battery has an electromotance of \(12 \mathrm{~V}\), find (a) the total charge which can be supplied, (b) the total energy available, (c) how long the sidelights of a car could be operated if they consumed \(24 \mathrm{~W}\), (d) how long the battery could operate the starter motor taking \(135 \mathrm{~A}\). Assume that the potential difference at the terminals is \(12 \mathrm{~V}\) and remains at this value until complete discharge.

Understanding electrical charge is essential for using and calculating the capacity of batteries. In the context of batteries, the charge is an indicator of how much electricity the battery can store and deliver. It's measured in coulombs (C), with an ampere being one coulomb per second passing a given point. A battery's ampere-hour (Ah) rating is a measure of its electrical charge capacity, essentially telling us how many amperes a battery can provide for one hour.

Our example is a 45 Ah battery. To find out how many coulombs of charge this battery holds, we multiply the ampere-hour rating by the number of seconds in an hour. Here's how it looks: \[ \text{Charge (Q)} = 45\text{ Ah} \times 3600\text{ s/h} = 162000\text{ C} \]
By making these calculations, we outline a fundamental aspect of a battery's capability — its ability to supply a specific amount of electric current over a period.

After understanding charge, the next step is energy calculation. Energy is the potential a battery has to do work, and it's determined by both the charge and the electromotive force (EMF) or voltage (V) of the battery. The energy of a battery tells us how much work can be done before the battery is depleted.

To find the total energy a battery can provide, multiply the charge by the electromotive force, using the formula: \[ \text{Energy (E)} = \text{Charge (Q)} \times \text{EMF (V)} \]
In our exercise, with a charge of 162000 C and a voltage of 12 V, the energy calculation is straightforward: \[ \text{E} = 162000\text{ C} \times 12\text{ V} = 1944000\text{ J} \]
This step essentially gives us a quantifiable figure for the potential energy available in the battery, which can then be related to its ability to perform work, such as powering electronic devices.

Electric power, measured in watts (W), refers to the rate at which energy is used or transferred. It can be calculated by multiplying voltage and current together, with the formula: \[ \text{Power (P)} = \text{Voltage (V)} \times \text{Current (I)} \]
For our car's sidelights consuming 24 W and a battery providing a consistent 12 V, we need to find out how much current is running through them. The calculation is quite simple: \[ \text{I} = \text{P} / \text{V} = 24\text{ W} / 12\text{ V} = 2\text{ A} \]
This step is vital for determining how long a device can be run using the battery. The power consumption of devices, like the sidelights of a car, must be known to calculate the duration of their operation on a single battery charge.

Battery capacity is the measure of the charge a battery can store and is denoted in ampere-hours (Ah). It helps us understand the duration of use for a given device. To determine how long a battery can operate a particular device, we divide the battery's capacity by the device's current draw.

In our exercise, we have two scenarios. For sidelights drawing 2 A, we can calculate the duration of operation using:\[ \text{Duration for sidelights (t)} = \frac{45\text{ Ah}}{2\text{ A}} = 22.5\text{ h} \]
Conversely, for a starter motor drawing a hefty 135 A, the duration of operation would be:\[ \text{Duration for starter motor (t)} = \frac{45\text{ Ah}}{135\text{ A}} = 0.33\text{ h} \]
Thus, the battery capacity gives us a direct relationship between the device's current requirement and how long the battery can sustain its operation. Understanding the battery capacity allows users to manage their expectations regarding battery life and plan accordingly.