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exact field due to the rod

exact field due to the rod

Principles of Physics: A Calculus-Based Text
Principles of Physics: A Calculus-Based Text
5th Edition
ISBN:9781133104261
Author:Raymond A. Serway, John W. Jewett
Publisher:Raymond A. Serway, John W. Jewett
Chapter19: Electric Forces And Electric Fields
Section: Chapter Questions
Problem 6P
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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.
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