Physics Problem: Battery EMF and Internal Resistance

Physics Problem: Battery EMF, Terminal Voltage, and Internal Resistance

#Physics #Electromagnetism #EMF #InternalResistance #BatteryProblems #TajSirPhysics #PhysicsLab #Resnick

Problem Statement

Course: Physics Lab
Instructor: Taj Sir
Source: Resnick Halliday Walker

A battery has an EMF of 15.0 V. The terminal voltage of the battery is 11.6 V when it is delivering 20.0 W of power to an external load resistor \( R \).

(a) What is the value of \( R \)?
(b) What is the internal resistance of the battery?

Solution

Step 1: Find the Current in the Circuit

The power delivered to the load resistor is given by:

\[ P = V_t \cdot I \]

Where:
- \( P = 20.0 \, \text{W} \)
- \( V_t = 11.6 \, \text{V} \)

Solving for current \( I \):

\[ I = \frac{P}{V_t} = \frac{20.0}{11.6} \approx 1.72414 \, \text{A} \]

Step 2: Calculate the Load Resistance \( R \)

Using Ohm's Law:

\[ V_t = I \cdot R \]
\[ R = \frac{V_t}{I} = \frac{11.6}{1.72414} \approx 6.728 \, \Omega \]
Answer to part (a): \( \boxed{6.73\ \Omega} \)

Step 3: Determine the Internal Resistance \( r \)

The relationship between EMF and terminal voltage is:

\[ \mathcal{E} = V_t + I \cdot r \]

Where:
- \( \mathcal{E} = 15.0 \, \text{V} \)
- \( V_t = 11.6 \, \text{V} \)
- \( I \approx 1.72414 \, \text{A} \)

Solving for internal resistance \( r \):

\[ 15.0 = 11.6 + (1.72414) \cdot r \]
\[ r = \frac{15.0 - 11.6}{1.72414} \approx \frac{3.4}{1.72414} \approx 1.972 \, \Omega \]
Answer to part (b): \( \boxed{1.97\ \Omega} \)

Final Answers

(a) Load resistance \( R \approx 6.73\ \Omega \)
(b) Internal resistance \( r \approx 1.97\ \Omega \)

Key Concepts

  • EMF (\( \mathcal{E} \)): The maximum potential difference when no current flows
  • Terminal Voltage (\( V_t \)): The actual voltage across battery terminals when current flows
  • Internal Resistance (\( r \)): Resistance within the battery that causes voltage drop under load
  • Power Delivery: \( P = V_t \cdot I = I^2 R \)

This problem demonstrates the practical effect of internal resistance on battery performance, a crucial concept in circuit analysis and electrical engineering.

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