Problem 14 - Physics for Scientists and Engineers with Modern Physics

Question

(II) The performance of the starter circuit in an automobile can be significantly degraded by a small amount of corrosion on a battery terminal. Figure 38 depicts a properly functioning circuit with a battery \((12.5-\mathrm{V}\) emf, \(0.02-\Omega\) internal resistance \()\) attached via corrosion-free cables to a starter motor of resistance \(R_{\mathrm{S}}=0.15 \Omega\) Suppose that later, corrosion between a battery terminal and a starter cable introduces an extra series resistance of just \(R_{C}=0.10 \Omega\) into the circuit as suggested in Fig. 38 \(\mathrm{b}\) . Let \(P_{0}\) be the power delivered to the starter in the circuit free of corrosion, and let \(P\) be the power delivered to the circuit with corrosion. Determine the ratio \(P / P_{0}\) .

Step-by-Step Solution

Answer

The power ratio \( P / P_0 \) is approximately 0.396.

Step 1: Initial Setup and Definitions

Let's first define the variables: the battery voltage is \( V = 12.5 \text{ V} \), the internal resistance of the battery is \( r = 0.02 \Omega \), the resistance of the starter motor is \( R_s = 0.15 \Omega \), and the resistance due to corrosion is \( R_c = 0.10 \Omega \). We'll calculate the power delivered to the starter motor both before and after the corrosion is introduced.

Step 2: Calculate Initial Power (Power without Corrosion)

For the circuit without corrosion, the total resistance \(R_0\) is the sum of the battery's internal resistance and the starter motor's resistance: \( R_0 = r + R_s = 0.02 \Omega + 0.15 \Omega = 0.17 \Omega \).The current through the circuit is \( I_0 = \frac{V}{R_0} = \frac{12.5}{0.17} \approx 73.53 \text{ A} \).The power delivered to the starter motor is \( P_0 = I_0^2 R_s = (73.53)^2 \times 0.15 \approx 810.40 \text{ W} \).

Step 3: Calculate Power with Corrosion

With corrosion, the total resistance \(R\) becomes \( R = r + R_s + R_c = 0.02 \Omega + 0.15 \Omega + 0.10 \Omega = 0.27 \Omega \).The new current is \( I = \frac{V}{R} = \frac{12.5}{0.27} \approx 46.30 \text{ A} \).The new power delivered to the starter motor is \( P = I^2 R_s = (46.30)^2 \times 0.15 \approx 321.42 \text{ W} \).

Step 4: Calculate the Power Ratio \( P / P_0 \)

The ratio of the power with corrosion to the power without corrosion is given by \( \frac{P}{P_0} = \frac{321.42}{810.40} \approx 0.396 \). This means the power delivered to the starter motor decreases significantly when there is corrosion.

Key Concepts

Electrical Resistance

Electrical resistance is a fundamental concept in physics that measures how strongly a material opposes the flow of electric current. It's like a narrow passage in a water pipe that makes it harder for water to flow through. Resistance is measured in ohms (Ω) and is an essential parameter in designing and analyzing electrical circuits.

In our exercise, we see two types of resistance: internal and external.

• **Internal resistance**: This is the resistance within the battery itself, denoted as **r**. It slightly reduces the voltage provided by the battery to the rest of the circuit.
• **External resistance**: This includes the resistance of components connected to the battery, like the starter motor **Rs**, and any additional resistance from corrosion **Rc**.

When you combine internal and external resistance, you get the **total resistance** in the circuit. An increase in total resistance, as caused by corrosion, reduces the current flowing through the circuit.

Circuit Analysis

Circuit analysis helps us understand how different elements of a circuit interact. The goal is to find out how the voltages and currents distribute across the various components.

In our example, the problem starts by considering a circuit consisting of:

• **A battery** with a voltage (EMF) of 12.5 volts
• **Internal resistance** of the battery, **r**, which adds to the total resistance in the circuit
• **A starter motor** with resistance **Rs**
• **Corrosion** adds extra resistance **Rc**

The circuit is initially analyzed without corrosion, where the total resistance **R0** is the sum of the internal resistance and the starter motor's resistance. This allows us to calculate the initial current flow using Ohm's Law: \[ I_0 = \frac{V}{R_0} \]When corrosion is added, the total resistance increases to **R = R0 + Rc**. The new current **I** through the circuit changes accordingly. Optimizing these values allows us to determine how much power is ultimately delivered or lost due to added resistances.

Power Calculation

Power in electrical circuits is a measure of how much electrical energy is being used per unit of time. It's vital for understanding the efficiency of circuits, especially in practical applications like the starting system of a car.

Power is calculated with the formula: \[ P = I^2 \times R \]where **I** is the current flowing through the component, and **R** is the resistance of the component where the power is used, in this case, the starter motor's resistance, **Rs**.
Initially, without corrosion, the power delivered to the starter motor **P0** is calculated using the initial current **I0** and the known resistance of the starter motor:\[ P_0 = I_0^2 \times R_s \]With corrosion, the new power **P** is reduced due to the increase in total resistance, resulting in lesser current **I**:\[ P = I^2 \times R_s \]The power ratio \( \frac{P}{P_0} \) gives insight into how efficiency drops with added resistance, showing that even a small corrosion can significantly lower the performance of an electrical system. Understanding these calculations helps in diagnosing and preventing issues in real-world circuits.