exact field due to the rod

College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
icon
Related questions
Question

A thin plastic rod of length L has a positive charge Q uniformly distributed along its length. We will
calculate the exact field due to the rod in the next homework set. In this set, we will approximate
the rod as several point sources and develop the Riemann sum as an intermediate step on the way
to writing an integral.
For those aiming at a P rating, you may use L = 3.0m , Q = 17 mC, and y = 0.11m to calculate the
answer numerically first and substitute variables for them only as required in the problem statement.
For those aiming at an E rating, leave L, Q and y as variables. Substitute numbers only where
required in the problem statement, and only as a last step 

L
(a) Divide the rod into 7 equally sized pieces, each with charge Q/7. Treat each piece as if it were
a point source centered in the middle of the piece, so we have 7 point sources, located along
the length of the original rod at equal intervals (so if x=0 at the center, we have charges at
x = −L/2, –L/3,–L/6, 0, +L/6, +L/3, and +L/2.)
*P
Draw a diagram of the electric field vectors at the test point.
Draw a separate diagram for just the vector of the field due to the source charge at L/3, showing
its x and y components as well as the vector r from the one source and its x and y components.
Notice that the two triangles formed by each vector and its components (E, E₂, E, and r, x, y)
are similar triangles.
(b) Find the magnitude and x and y components of the fields due to each piece and from those, the
magnitude and direction of the net electric field at the test point, due to these 7 point sources.
When finding components, do not calculate each angle! Instead, use the similar triangles and
ratios of lengths of the r, z, y to find the value of the appropriate trigonometric functions more
directly.
For E level work, first work the problem in terms of variables (i.e. expressed as a function of Q.
L, y and k.) Only then calculate a numerical value for the net |E| from your resulting expression
for the case L = 3.0m, Q = 17 mC, and y = 0.11m.
(c) Generalize the calculation you just did for 7 slices to an arbitrary number N of slices at a
distance y above the center of a rod length L using the following: The ith source is a slice length
Ar = L/N of the rod, has charge q, which we will treat as a point source, and is centered at a
distance z from the center. Express the charge of each slice, q, in terms of the total charge Q
on the rod, the length of the rod, L, and the width of each slice, Ar. Use this and the analysis
you just did for N=7 to guide you in writing a sum (use the Σ notation) representing a general
expression for the electric field at the test point for any number (N) of pieces, any amount of
total charge Q, any length L, and any distance y. This means your sum is over the counter i
which goes from 0 to N, and the only variables in the terms in the sum should be k, Q, L. y.
Ti, and Az. This is the Riemann sum that, in an upcoming homework, we will turn into an
integral when Az → 0.
Transcribed Image Text:L (a) Divide the rod into 7 equally sized pieces, each with charge Q/7. Treat each piece as if it were a point source centered in the middle of the piece, so we have 7 point sources, located along the length of the original rod at equal intervals (so if x=0 at the center, we have charges at x = −L/2, –L/3,–L/6, 0, +L/6, +L/3, and +L/2.) *P Draw a diagram of the electric field vectors at the test point. Draw a separate diagram for just the vector of the field due to the source charge at L/3, showing its x and y components as well as the vector r from the one source and its x and y components. Notice that the two triangles formed by each vector and its components (E, E₂, E, and r, x, y) are similar triangles. (b) Find the magnitude and x and y components of the fields due to each piece and from those, the magnitude and direction of the net electric field at the test point, due to these 7 point sources. When finding components, do not calculate each angle! Instead, use the similar triangles and ratios of lengths of the r, z, y to find the value of the appropriate trigonometric functions more directly. For E level work, first work the problem in terms of variables (i.e. expressed as a function of Q. L, y and k.) Only then calculate a numerical value for the net |E| from your resulting expression for the case L = 3.0m, Q = 17 mC, and y = 0.11m. (c) Generalize the calculation you just did for 7 slices to an arbitrary number N of slices at a distance y above the center of a rod length L using the following: The ith source is a slice length Ar = L/N of the rod, has charge q, which we will treat as a point source, and is centered at a distance z from the center. Express the charge of each slice, q, in terms of the total charge Q on the rod, the length of the rod, L, and the width of each slice, Ar. Use this and the analysis you just did for N=7 to guide you in writing a sum (use the Σ notation) representing a general expression for the electric field at the test point for any number (N) of pieces, any amount of total charge Q, any length L, and any distance y. This means your sum is over the counter i which goes from 0 to N, and the only variables in the terms in the sum should be k, Q, L. y. Ti, and Az. This is the Riemann sum that, in an upcoming homework, we will turn into an integral when Az → 0.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 5 steps with 1 images

Blurred answer
Knowledge Booster
Properties of electric charge
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
College Physics
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
Physics for Scientists and Engineers
Physics for Scientists and Engineers
Physics
ISBN:
9781337553278
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:
9780321820464
Author:
Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:
Addison-Wesley
College Physics: A Strategic Approach (4th Editio…
College Physics: A Strategic Approach (4th Editio…
Physics
ISBN:
9780134609034
Author:
Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:
PEARSON