Problem 72 - Physics

Question

A resistor has a resistance \(R\), and a battery has an internal resistance \(r\). When the resistor is connected across the battery, ten percent less power is dissipated in \(R\) than would be dissipated if the battery had no internal resistance. Find the ratio \(r / R\).

Step-by-Step Solution

Answer

The ratio \( r/R \) is \( \frac{1 - \sqrt{0.9}}{\sqrt{0.9}} \).

Step 1: Determine Power with No Internal Resistance

When there is no internal resistance, the voltage across the resistor is equal to the battery's electromotive force (emf), let's denote it as \( V \). The power dissipated in the resistor \( R \) can be calculated by the formula \( P_0 = \frac{V^2}{R} \).

Step 2: Determine Power with Internal Resistance

With internal resistance \( r \), the effective voltage across the resistor is reduced. The total resistance in the circuit is \( R + r \). The voltage across \( R \) is \( V' = \frac{V imes R}{R + r} \). The power dissipated in \( R \) is \( P = \frac{(V')^2}{R} = \frac{V^2 imes R}{(R + r)^2} \).

Step 3: Set Up the Equation for Ten Percent Less Power

We are given that \( P \) is ten percent less than \( P_0 \). This gives us the equation: \( \frac{V^2 imes R}{(R + r)^2} = 0.9 \times \frac{V^2}{R} \).

Step 4: Simplify the Equation

Cancel out \( V^2 \) from both sides and rearrange to get: \( \frac{R^2}{(R + r)^2} = 0.9 \).

Step 5: Solve for the Ratio r/R

Take the square root of both sides to obtain \( \frac{R}{R + r} = \sqrt{0.9} \). Solve for \( r \) in terms of \( R \): \( r = R \left( \frac{1}{\sqrt{0.9}} - 1 \right) \). Simplify this to find \( \frac{r}{R} = \frac{1 - \sqrt{0.9}}{\sqrt{0.9}} \).

Key Concepts

Internal Resistance

Internal resistance is a key factor in understanding the behavior of real-life batteries. It refers to the resistance within the battery itself, which can impede the flow of current. This internal resistance causes the terminal voltage of the battery to be lower than its electromotive force (emf) when current flows through the circuit.
When you connect a resistor to a battery, two types of resistances are at play:

• The resistance of the resistor itself, denoted as \( R \).
• The internal resistance of the battery, denoted as \( r \).

The presence of internal resistance leads to a decrease in the voltage available across the external resistor, reducing the power it can dissipate. Understanding this concept is essential for accurate calculations in circuits involving power sources.

Electromotive Force (emf)

Electromotive force (emf) is a fundamental concept in electrical circuits. Contrary to what its name suggests, it is not actually a "force". Instead, it is the voltage generated by a battery or other power source when no current is flowing. Think of it as the maximum potential difference the source can provide.
The emf is often denoted by \( V \), representing the ideal voltage of the power source without any load. However, once the current starts to flow and the internal resistance comes into play, the effective voltage (or terminal voltage) seen by the circuit is reduced. This reduction is crucial to consider when analyzing circuits as it affects the performance and efficiency of the power source, especially in practical applications.

Circuit Voltage

Circuit voltage, particularly in the context of this problem, refers to the actual voltage across the external resistor when connected to a battery with both resistance and internal resistance. It's not the same as the emf. Instead, it's the voltage after accounting for the voltage drop due to the internal resistance.
The voltage drop across internal resistance can be calculated using Ohm's Law. In our problem, this drop reduces the voltage from its emf value to a lower effective voltage \( V' \) which is given by:

• \( V' = \frac{V \times R}{R + r} \),

where \( V \) is the emf, \( R \) is the resistor's resistance, and \( r \) is the internal resistance. This effective voltage is crucial to determining the actual power being dissipated in the real-world scenario.

Resistance Ratio

The resistance ratio \( \frac{r}{R} \) is an expression that compares the internal resistance of the battery to the resistance of the external resistor. This ratio is significant as it directly influences the circuit's efficiency and performance.
In the provided exercise, we determined this ratio using the fact that the power dissipated across the resistor with internal resistance is ten percent less than if there were no internal resistance at all. By setting up and solving the equation for the power difference, we deduced that the ratio can be expressed as:

• \( \frac{r}{R} = \frac{1 - \sqrt{0.9}}{\sqrt{0.9}} \)

This expression illustrates the impact of internal resistance on the circuit's overall power dissipation. Understanding and calculating resistance ratios are essential for designing efficient electrical systems where power loss needs to be minimized.