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Chapter 4: Problem 52

A battery having fixed voltage \(V\) and fixed internal resistance \(r\) is connected to a circuit that has variable resistance \(R\) (see figure). By Ohm's law, the current \(I\) in the circuit is \(I=V /(R+r)\). If the power output \(P\) is given by \(P=I^{2} R\), show that the maximum power occurs if \(R=r\).

Short Answer

Expert verified
Maximum power occurs when \( R = r \).

Step by step solution

01

Express Power in Terms of R

The power output \( P \) can be expressed as a function of \( R \) using the equations given in the problem. First, substitute \( I = \frac{V}{R+r} \) into \( P = I^2 R \). This results in: \[ P = \left( \frac{V}{R+r} \right)^2 R = \frac{V^2 R}{(R+r)^2} \].
02

Differentiate P with Respect to R

To find the maximum power, we need to differentiate the power equation with respect to \( R \). Compute \( \frac{dP}{dR} \):\[\frac{dP}{dR} = \frac{d}{dR} \left( \frac{V^2 R}{(R+r)^2} \right)\]Apply the quotient rule:\[ \frac{dP}{dR} = \frac{(R+r)^2 \cdot V^2 - V^2 R \cdot 2(R+r)}{(R+r)^4} \].
03

Simplify the Derivative Expression

Simplify the expression for \( \frac{dP}{dR} \):\[\frac{dP}{dR} = \frac{V^2 (R+r)^2 - 2V^2 R(R+r)}{(R+r)^4} = \frac{V^2 ((R+r)^2 - 2R(R+r))}{(R+r)^4}\]Further simplification yields:\[= \frac{V^2 (R^2 + 2Rr + r^2 - 2R^2 - 2Rr)}{(R+r)^4} = \frac{V^2 (r^2 - R^2)}{(R+r)^4}\].
04

Find Critical Points

Set the derivative \( \frac{dP}{dR} \) to zero to find the critical points:\[\frac{V^2 (r^2 - R^2)}{(R+r)^4} = 0\]Simplifying, this gives:\[ r^2 - R^2 = 0 \]Thus: \( R = r \).
05

Verify Maximum Power Condition

To confirm that \( R = r \) gives a maximum, consider the second derivative test or examine the behavior of \( \frac{dP}{dR}\) around \( R = r \). Since the expression for \( \frac{dP}{dR} \) changes sign, this indicates a maximum at \( R = r \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Output
In any electrical circuit, power output is the rate at which energy is used or produced by components within the circuit. For resistive circuits, this power is calculated using the formula: \( P = I^2R \), where:
  • \( P \) is the power output, measured in watts (W).
  • \( I \) is the current flowing through the circuit in amperes (A).
  • \( R \) is the resistance in the circuit in ohms (\( \Omega \)).
Substituting Ohm's Law, which states that \( I = \frac{V}{R+r} \) (where \( V \) is the voltage across the circuit and \( r \) is the internal resistance), we find that the power output in terms of \( R \) can be expressed as: \[ P = \frac{V^2 R}{(R+r)^2} \] This helps in understanding how various factors in the circuit affect the power output.
Maximum Power Transfer
The maximum power transfer theorem is a crucial concept in electronics that denotes the condition for maximum power output from a source to a load. According to this theorem, maximum power is transferred when the load resistance \( R \) equals the internal resistance \( r \) of the source. This is evident mathematically by setting the derivative of the power function with respect to \( R \) to zero and solving for \( R \). In the provided exercise, this condition is given as \( R = r \). When this equality is met, the power output is at its maximum value, allowing for the most efficient transfer of energy to the load without excessive losses.
Circuit Analysis
Circuit analysis involves the systematic study of electrical circuits to understand and control their behavior. It is about understanding how current flows through various components and how voltages are distributed across these components. In the context of this exercise, circuit analysis helps us explore how the change in a variable resistance \( R \) affects the overall power output of the circuit. By using Ohm's Law and power formulas, one can predict the circuit's response and optimize for desired outcomes, such as maximizing power transfer or minimizing energy waste. Different tools like nodal analysis, mesh analysis, and most importantly, Ohm's Law, form the backbone of understanding circuit behavior.
Variable Resistance
Variable resistance, often referred to as a variable resistor or potentiometer, allows for varying levels of resistance within an electrical circuit. This adjustability is vital for controlling current flow and voltage distribution across a circuit. In our exercise, variable resistance \( R \) is connected to a battery with a fixed internal resistance \( r \). By analyzing how changes in \( R \) affect the power output \( P \), one can determine the conditions under which power transfer is maximized. Adjusting \( R \) involves experimenting by either physically turning a knob on a rheostat or adjusting settings in a simulated circuit environment. By achieving the condition where \( R = r \), the variable resistor is set for optimal energy efficiency, ensuring the best performance of the circuit.

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