Problem 48 - College Physics

Question

Electric eels. Electric eels and electric fish generate large potential differences that are used to stun 9 enemies and prey. These potentials are produced by cells that each can generate 0.10 V. We can plausibly model such cells as charged capacitors. (a) How should these cells be connected \((\mathrm{in}\) series or in parallel) to produce a total potential of more than 0.10 \(\mathrm{V} ?\) (b) Using the connection in part (a), how many cells must be connected together to produce the 500 \(\mathrm{V}\) surge of the electric eel?

Step-by-Step Solution

Answer

Cells must be connected in series; 5000 cells are needed to achieve 500 V.

Step 1: Understand the Problem

We need to find out whether the cells should be connected in series or parallel to produce more than 0.10 V. Each cell produces 0.10 V. Also, we need to calculate how many cells are required to achieve a potential difference of 500 V.

Step 2: Review Series and Parallel Connections

In a series connection, the total voltage is the sum of the individual voltages: \( V_{total} = V_1 + V_2 + ... + V_n \). In a parallel connection, the voltage across each component is the same as the voltage of a single component: \( V_{total} = V_1 = V_2 = ... = V_n \).

Step 3: Choose Connection Type for More Than 0.10 V

Since we want a voltage greater than 0.10 V, the cells should be connected in series. This way, the voltages add up and can exceed 0.10 V, meeting the requirement in part (a).

Step 4: Calculate Number of Cells Required for 500 V

Since the cells are connected in series, we use the formula \( V_{total} = n \times V_{cell} \). Here, \( V_{cell} = 0.10\, \text{V} \) and \( V_{total} = 500\, \text{V} \). We solve for \( n \): \[ 500 = n \times 0.10 \]Divide both sides by 0.10:\[ n = \frac{500}{0.10} = 5000 \].Thus, 5000 cells are required to achieve a potential difference of 500 V.

Key Concepts

Series Circuit

When dealing with a series circuit, it's important to recognize how the electrical components are organized. In a series configuration, every component is connected end-to-end. This setup means current flows through each component in the sequence. Such a structure has a direct impact on the overall potential difference (voltage) of the circuit. The key feature of a series circuit is that the total voltage across the circuit is equal to the sum of the voltages across each individual component: \[ V_{\text{total}} = V_1 + V_2 + \ldots + V_n \]. This characteristic is why series connections are ideal for producing higher voltages from multiple sources, like the cells in an electric eel. By arranging cells in series, their individual voltages add together, resulting in a larger total voltage. This is exactly why the electric eel's cells need to be in a series configuration to achieve a potential greater than 0.10 V.

Parallel Circuit

Understanding parallel circuits is crucial when exploring different configurations for electrical components. In a parallel circuit, all components are connected across the same two points, creating multiple paths for the current. The voltage across each component in a parallel circuit is the same as the source voltage. This means that even if many components are added to the circuit, the total voltage remains unchanged: \[ V_{\text{total}} = V_1 = V_2 = \ldots = V_n \]. Parallel circuits are often used when it's important to maintain the same voltage across all components, such as in household wiring systems. However, in the case of electric eel cells, a parallel circuit wouldn't be beneficial for increasing the overall potential above 0.10 V, because the voltage is not additive in parallel circuits.

Voltage Calculations

Voltage calculations are a fundamental part of understanding any electrical system, especially when determining how to achieve a desired potential difference. In the context of electric eel cells, calculating the voltage involves using the series connection concept. For cells connected in series, the calculation of total voltage becomes an exercise in simple arithmetic. If each cell provides 0.10 V, and we need a total voltage of 500 V, we would determine the number of cells needed by solving the equation: \[ V_{\text{total}} = n \times V_{\text{cell}} \]. Substituting the known values for the electric eel, \( V_{\text{cell}} = 0.10 \, \text{V} \) and \( V_{\text{total}} = 500 \, \text{V} \), the formula becomes: \[ 500 = n \times 0.10 \]. Solving for \( n \), the result is \( n = 5000 \) cells. This calculation illustrates how arrangements impact the design of bioelectric systems.

Capacitor Modeling

Capacitor modeling is a vital tool in electrical engineering, particularly when trying to replicate or simulate the behavior of specific systems. A capacitor stores electrical energy and can be used to mimic the function of biological cells that generate voltage, such as those in electric eels. In this exercise, we modeled each cell within an eel's body as a capacitor capable of generating a potential difference of 0.10 V. This model simplifies complex biological processes into manageable electrical terms. When capacitors are connected in series, the total potential difference is the sum of the potential differences of each capacitor, analogous to how the cells in an electric eel align to produce significant voltage. Through capacitor modeling, engineers and scientists can explore, design, and understand not only artificial systems but also derive insights into natural bioelectric phenomena. This practice forms a bridge between theoretical physics, electrical engineering, and biology, helping us grasp how creatures like electric eels can generate substantial electric fields.