Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 25: Problem 64

Show that a battery delivers the most power when the load resistance across its terminals is equal to its internal resistance. (This is not the way to treat a battery, but it's the basis for load matching in amplifiers; see Problem \(65 .\) )

Short Answer

Expert verified
It has been proven that the maximum power is delivered by a battery when the load resistance equals the internal resistance. This was obtained by deriving the power expression in respect of Load resistance and setting it to zero.

Step by step solution

01

Represent given parameters

Let's denote the internal resistance of the battery as \(r\) and the load resistance as \(R\). The battery produces an electromotive force, \(E\).
02

Set up the equations

The current in the circuit, according to Ohm's law, can be expressed as \(I=E/(r+R)\), and power (P) delivered to the load is given by \(P=I^2.R\). We can substitute the expression for I from the current equation into the power equation, and we get: \(P=E^2.R/(r + R)^2\).
03

Find the maximum power

To find the maximum power, we should derive the power expression with respect to the load resistance R and equate it to zero, as the maximum or minimum of a function occurs where its derivative is zero. Doing the necessary differentiation, we get: \(dP/dR=(E^2)/(R+r)^3 - 2*E^2*R/(R+r)^3\). Set this to 0 and solve for R. You will find that \(R=r\).
04

Check the validity of the result

To ensure that the value obtained is for maximum power and not minimum, we double differentiate the power equation and substitute R=r. If the second derivative is negative, it means we have a maximum point, which is what we should expect. The double differentiation of P in respect of R gives \(d^2P/dR^2=-2*E^2/(R+r)^3 + 6*E^2*R/(R+r)^4\). If we substitute R=r in the above equation we get it to be negative confirming that it is a maximum point.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Show that a capacitor is charged to approximately \(99 \%\) of the applied voltage in five time constants \((5 R C)\)

Sketch a circuit diagram for a circuit that includes a resistor \(R_{\square}\) connected to the positive terminal of a battery, a pair of paralle resistors \(R_{2}\) and \(R_{3}\) connected to the lower-voltage end of \(R_{1}\) and then returned to the battery's negative terminal, and a capacitor across \(R_{2}\)

Sketch a diagram for a circuit consisting of two batteries, a resistor, and a capacitor, all in series. Does the circuit description allow you any flexibility?

Can the voltage across a battery's terminals be higher than the battery's rated voltage? Explain.

When a large electric load such as a washing machine or oven comes on, lights throughout a house often dim. Why?
See all solutions

Recommended explanations on Physics Textbooks

Astrophysics

Read Explanation

Medical Physics

Read Explanation

Energy Physics

Read Explanation

Electric Charge Field and Potential

Read Explanation

Dynamics

Read Explanation

Modern Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free