Problem 73 - Physics for Scientists and Engineers with Modern Physics

Question

Terminal voltage of mercury cell \((3.99 \mathrm{~V})\) is closer to \(4.0 \mathrm{~V}\) than terminal voltage of dry cell \((3.84 \mathrm{~V})\)

Step-by-Step Solution

Answer

The terminal voltage of the mercury cell (3.99 V) is closer to 4.0 V than that of the dry cell (3.84 V).

Step 1: Understand the problem statement

We need to determine if the terminal voltage of a mercury cell, which is given as \( 3.99 \text{ V} \), is closer to \( 4.0 \text{ V} \) compared to the terminal voltage of a dry cell, which is given as \( 3.84 \text{ V} \).

Step 2: Calculate the difference for the mercury cell

Calculate how far away the mercury cell's terminal voltage is from \( 4.0 \text{ V} \): \( |4.0 - 3.99| = 0.01 \text{ V} \).

Step 3: Calculate the difference for the dry cell

Calculate how far away the dry cell's terminal voltage is from \( 4.0 \text{ V} \): \( |4.0 - 3.84| = 0.16 \text{ V} \).

Step 4: Compare the differences

Compare the calculated differences: \( 0.01 \text{ V} \) for the mercury cell and \( 0.16 \text{ V} \) for the dry cell. The smaller the difference, the closer the terminal voltage is to \( 4.0 \text{ V} \).

Step 5: Conclusion

Since the difference for the mercury cell is smaller than the difference for the dry cell, the terminal voltage of the mercury cell is closer to \( 4.0 \text{ V} \).

Key Concepts

Understanding Electric Cells

Electric cells are crucial components in numerous electronic devices, designed to convert chemical energy into electrical energy. When we talk about electric cells, it is essential to know that they come in different types. For instance:

• Primary cells – These are not rechargeable and are used once. Examples include alkaline and zinc-carbon cells.
• Secondary cells – These are rechargeable cells like lithium-ion and nickel-cadmium batteries.

The **mercury cell** in our exercise is a primary cell, well-known for its stable voltage output, making it a preferred choice for small devices such as watches and hearing aids. Understanding the type of electric cell is essential for comprehending why it behaves the way it does in various applications.

Exploring Voltage Difference

Voltage difference is a fundamental parameter in understanding how electric cells perform. It is essentially the difference in electric potential between two points, usually the terminals of the cell. This voltage difference is what enables electric current to flow when the cell is connected in a circuit. In the given exercise, we examined the terminal voltage difference between two cell types: a mercury cell and a dry cell. The mercury cell has a terminal voltage of 3.99 V, which is extremely close to its nominal voltage of 4.0 V, suggesting very minimal voltage drop due to internal resistance. In contrast, the dry cell showed a larger voltage difference, having a terminal voltage of 3.84 V. This indicates a greater voltage drop which can be due to factors like higher internal resistance or degradation over time. Understanding voltage difference helps in assessing the efficiency and suitability of cells for specific devices.

Performing a Comparative Analysis

When we look at a problem involving multiple variables, such as terminal voltages of different electric cells, a comparative analysis helps clarify which option is more suitable for a specific application. In our example, the exercise involves comparing the difference between the actual terminal voltages of both the mercury cell and the dry cell from the nominal voltage of 4.0 V. A smaller numerical difference indicates higher precision in retaining its voltage closer to the rated value. Here:

• The mercury cell with a difference of 0.01 V shows it retains its intended voltage much better than the dry cell.
• The dry cell with a larger difference of 0.16 V suggests a potential inefficiency or older technology.

This comparison allows users to decide which cell is preferable based on specific precision and stability requirements. Such analyses are invaluable in technology selection for both manufacturers and consumers.