Question

The capacity of batteries such as the typical AA alkaline battery is expressed in units of milliamp-hours (mAh). An "AA" alkaline battery yields a nominal capacity of 2850 mAh. (a) What quantity of interest to the consumer is being expressed by the units of \(\mathrm{mAh}\) ? (b) The starting voltage of a fresh alkaline battery is \(1.55 \mathrm{~V}\). The voltage decreases during discharge and is \(0.80 \mathrm{~V}\) when the battery has delivered its rated capacity. If we assume that the voltage declines linearly as current is withdrawn, estimate the total maximum electrical work the battery could perform during discharge.

Short answer

(a) The unit milliamp-hours (mAh) represents the battery capacity, indicating how long the battery will last while providing a specific current before it is discharged. (b) First, we calculate the average voltage: \(V_{avg} = (1.55 + 0.80) / 2 = 1.175 V\). Next, we convert the capacity (2850 mAh) to Coulombs: \(Q = 2850 \,\mathrm{mAh} \times \frac{1 \,\mathrm{A}}{1000 \,\mathrm{mA}} \times 3600 \,\mathrm{s} = 10.26\ C\). Finally, we calculate the total maximum electrical work: \(W_{max} = Q \times V_{avg} = 10.26 \,C \times 1.175 \,V = 12.05 \,\mathrm{J}\).

Step-by-step solution

Part (a): Understanding Milliamp-hours (mAh)

The unit mAh is used to express the capacity of a battery, which basically tells us how much electrical charge the battery can store. It is the product of the current (in milliamps) and the time (in hours) that the battery can deliver before it is discharged. For the consumer, it is an indicator of how long the battery will last while providing a specific amount of current.

Part (b) Step 1: Calculate the average voltage

As the problem states that the voltage declines linearly, we can calculate the average voltage during the discharge process by taking the mean of the starting voltage and the ending voltage: \[V_{avg} = \frac{V_{start} + V_{end}}{2}\] where \(V_{start} = 1.55 V\) and \(V_{end} = 0.80 V\).

Part (b) Step 2: Calculate the total charge delivered

First, we need to calculate the total charge delivered by the battery. Using the given capacity (2850 mAh), we can convert it to Coulombs using the formula: \[Q = I \times t\] where \(Q\) is the total charge, \(I\) is the current, and \(t\) is the time. We have the battery capacity in mAh, which is equal to the current (in milliamps) multiplied by the time (in hours). Convert the given 2850 mAh to Coulombs: \[Q = 2850 \,\mathrm{mAh} \times \frac{1 \,\mathrm{A}}{1000 \,\mathrm{mA}} \times 3600 \,\mathrm{s}\]

Part (b) Step 3: Calculate the total maximum electrical work

Now, we will use the average voltage calculated in Step 1 and the total charge calculated in Step 2 to estimate the total maximum electrical work the battery can perform during discharge. The formula for the electrical work is: \[W_{max} = Q \times V_{avg}\] Substitute the values of \(Q\) and \(V_{avg}\) into the formula to get the total maximum electrical work.

Key concepts

Milliamp-hours (mAh)

Understanding the milliamp-hour (mAh) rating of battery capacity begins with recognizing it as a reflection of energy storage capability. In essence, an AA battery rated at 2850 mAh can theoretically provide a current of 2850 milliamps for one hour, or a proportional amount of current for a different period, say 285 milliamps for 10 hours, before it is completely discharged.

For consumers, this rating is crucial as it directly correlates to the usage time of their electronic devices; higher mAh values suggest longer battery life between charges. However, actual performance may vary based on device power requirements and battery discharge characteristics.

Electrical Work Calculation

Electrical work is determined by the amount of electric charge that moves and the voltage it moves across. The unit used is the joule, which corresponds to one coulomb of charge moving across an electric potential difference of one volt. When calculating the total work done by a battery, we consider the average voltage across which the charge is moved and the total amount of charge the battery can deliver.

The simple formula to remember here is:
\[W = Q \times V_{avg}\]
where W is the work in joules, Q is the charge in coulombs, and V_{avg} is the average voltage in volts. This calculation provides an estimate of the maximum energy a battery can supply during its life cycle.

Voltage and Charge Relationship

The relationship between voltage and charge is fundamental to understanding electrical circuits and battery operation. Voltage, measured in volts (V), can be considered as the 'pressure' that pushes electrical charges through a conductor. The charge, measured in coulombs (C), represents the quantity of electricity transported.

As charge flows through a circuit, the voltage may not remain constant in a real-world battery; it typically decreases as the battery discharges. A linear decline assumption simplifies calculations and offers a reasonable approximation for the average voltage during discharge, which is essential for estimating the work a battery can perform as its stored energy is used up.

Battery Discharge Characteristics

Battery discharge characteristics explain how the output of a battery changes over the course of its discharge cycle. These characteristics are influenced by various factors including the battery chemistry, the discharge rate, and temperature. It is common for batteries, like the 'AA' alkaline mentioned, to exhibit a gradual decline in voltage as they discharge.

Understanding this behavior is important for consumers and engineers alike, as it affects the performance of the battery in practical use. Devices typically require a minimum voltage to operate correctly; hence, even though a battery has residual charge, it might not be sufficient to power a device once below a certain voltage threshold.