Q74 P Refer to figure 6.86. Calculate ... [FREE SOLUTION]

Chapter 6: Q74 P (page 282)

Refer to figure 6.86. Calculate the change in electric energy along the two different paths in moving charge away from charge $Q$ from A to B along a radial path, then to C along a circle centered on , then to D along a radial path. Also calculate the change in energy in going directly from A to D along a circle centered at $Q$ . Specifically. What are $U_{B} - U_{A}, U_{C} - U_{B}, U_{D} - U_{C}$ and their sum? What is $U_{D} - U_{A}$ ? Also, calculate the round-trip difference in the electrical energy when moving charge along the path from A to B to C to D to A

Short Answer

Expert verified

The electrical energy change along the path ABCD is 0, and the change in electrical energy for the direct path AD is also $0$ For the one round trip ABCDA, the difference in electrical energy is also $0$

Step by step solution

Given Data

Two charges q and Q are given.

The radii of two circular paths AD and BC are $r_{1} and r_{2}$ respectively.

Concept

The total work done in carrying a positive charge q from infinity to a point inside an electric field due to another charge positive charge $Q$ is known as the potential energy of the system. The difference in potential energy between two points gives the work done in carrying the charge q between these points.

The potential energy at a point is mathematically given as:

$U = \frac{1}{4 {蟺蔚}_{o}} \frac{qQ}{r}$

Calculation

The potential energy at points A, B, C, and D, for the system of charges $Q a n d q$ are:

role="math" localid="1668506217310" $U_{A} = \frac{1}{4 蟺蔚_{o}} \frac{Q q}{r_{1}} U_{B} = \frac{1}{4 蟺蔚_{o}} \frac{Q q}{r_{2}} U_{C} = \frac{1}{4 蟺蔚_{o}} \frac{Q q}{r_{2}} U_{D} = \frac{1}{4 蟺蔚_{o}} \frac{Q q}{r_{1}}$

Now, change in electrical energy along AB is given as:

$U_{B} - U_{A} = \frac{1}{4 蟺蔚_{o}} \frac{Q q}{r_{2}} - \frac{1}{4 蟺蔚_{o}} \frac{Q q}{r_{1}} = \frac{Q q}{4 蟺蔚_{o}} (\frac{1}{r_{2}} - \frac{1}{r_{1}})$

The change in electrical energy along BC is given as:

$U_{C} - U_{B} = \frac{1}{4 蟺蔚_{o}} \frac{Q q}{r_{2}} - \frac{1}{4 蟺蔚_{o}} \frac{Q q}{r_{2}} = \frac{Q q}{4 蟺蔚_{o}} (\frac{1}{r_{2}} - \frac{1}{r_{2}}) = 0$

The change in electrical energy along CD is given as:

$U_{D} - U_{C} = \frac{1}{4 蟺蔚_{o}} \frac{Q q}{r_{1}} - \frac{1}{4 蟺蔚_{o}} \frac{Q q}{r_{2}} = \frac{Q q}{4 蟺蔚_{o}} (\frac{1}{r_{1}} - \frac{1}{r_{2}})$

The change in electrical energy along AD is given as:

$U_{D} - U_{A} = \frac{1}{4 蟺蔚_{o}} \frac{Q q}{r_{2}} - \frac{1}{4 蟺蔚_{o}} \frac{Q q}{r_{2}} = \frac{Q q}{4 蟺蔚_{o}} (\frac{1}{r_{2}} - \frac{1}{r_{2}}) = 0$

The total change in electrical energy along the path ABCD is given as:

$(U_{B} - U_{A}) + (U_{C} - U_{B}) + (U_{D} - U_{C}) = \frac{Q q}{4 蟺蔚_{o}} (\frac{1}{r_{2}} - \frac{1}{r_{1}}) + 0 + \frac{Q q}{4 蟺蔚_{o}} (\frac{1}{r_{1}} - \frac{1}{r_{2}}) = 0$

The total change in electrical energy along the complete round trip ABCDA will be:

$((U_{B} - U_{A}) + (U_{C} - U_{B}) + (U_{D} - U_{C}) + (U_{A} - U_{D})) = \frac{Q q}{4 蟺蔚_{o}} (\frac{1}{r_{2}} - \frac{1}{r_{1}}) + 0 + \frac{Q q}{4 蟺蔚_{o}} (\frac{1}{r_{1}} - \frac{1}{r_{2}}) - 0 = 0$

Conclusion

The change in electrical energy along the path ABCD is 0, and the change in electrical energy for the path AD is also $0$ . The change in electrical energy for the one round trip ABCDA is $0$

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Short Answer

Step by step solution

Given Data

Concept

Calculation

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Measurements

Energy Physics

Classical Mechanics

Fluids

Translational Dynamics

Rotational Dynamics

Study anywhere. Anytime. Across all devices.