Introduction to Electrodynamics
4th Edition
ISBN: 9781108420419
Author: David J. Griffiths
Publisher: Cambridge University Press
expand_more
expand_more
format_list_bulleted
Concept explainers
Videos
Question
Chapter 3.3, Problem 3.24P
To determine
The solution of the Laplace equation by the separation of the variables in the cylindrical coordinates.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
A cylindrically symmetric charge distribution has charge density p(r) given by
P()=
2na
erla
where Op and a are constants. Calculate the total charge O contained within a cylinder
of length I = 5a and radius 2a, centred on the z-axis, and grve your answer by entering
numbers in both of the boxes in the equation below:
[Note that you must enter a value in each of the boxes, including 1 or 0 if appropriate.
Blank boxes will be marked as incorrect.]
Using the method of integration, what is the electric field of a uniformly charged thin circular plate (with radius R and total charge Q) at xo distance from its center? (Consider that the
surface of the plate lies in the yz plane)
Solution
A perfect approach to this is to first obtain the E-field produced by an infinitesimal charge component of the charge Q.
There will be several approaches to do this, but the most familiar to us is to obtain a very small shape that could easily represent our circular plane. That shape would be a ring.
So for a ring whose charge is q, we recall that the electric field it produces at distance x0 is given by
E = (1/
)(x0q)/(
242)
Since, the actual ring (whose charge is dq) we will be dealing with is an infinitesimal part of the circular plane, then, its infinitesimal electric field contribution is expressed as
= (1/
)(x0
2.
)
We wish to obtain the complete electric field contribution from the above equation, so we integrate it from 0 to R to obtain
E =…
The following charge density function:
(Po (1-r/R) for ≤R
55. The
р
=
0
elsewhere
describes a long cylindrical charge
distribution. Determine
the
electric field
due to this charge distribution, everywhere.
Chapter 3 Solutions
Introduction to Electrodynamics
Ch. 3.1 - Find the average potential over a spherical...Ch. 3.1 - Prob. 3.2PCh. 3.1 - Prob. 3.3PCh. 3.1 - Prob. 3.4PCh. 3.1 - Prob. 3.5PCh. 3.1 - Prob. 3.6PCh. 3.2 - Find the force on the charge +q in Fig. 3.14....Ch. 3.2 - (a) Using the law of cosines, show that Eq. 3.17...Ch. 3.2 - In Ex. 3.2 we assumed that the conducting sphere...Ch. 3.2 - A uniform line charge is placed on an infinite...
Ch. 3.2 - Two semi-infinite grounded conducting planes meet...Ch. 3.2 - Prob. 3.12PCh. 3.3 - Find the potential in the infinite slot of Ex. 3.3...Ch. 3.3 - Prob. 3.14PCh. 3.3 - A rectangular pipe, running parallel to the z-axis...Ch. 3.3 - A cubical box (sides of length a) consists of five...Ch. 3.3 - Prob. 3.17PCh. 3.3 - Prob. 3.18PCh. 3.3 - Prob. 3.19PCh. 3.3 - Suppose the potential V0() at the surface of a...Ch. 3.3 - Prob. 3.21PCh. 3.3 - In Prob. 2.25, you found the potential on the axis...Ch. 3.3 - Prob. 3.23PCh. 3.3 - Prob. 3.24PCh. 3.3 - Find the potential outside an infinitely long...Ch. 3.3 - Prob. 3.26PCh. 3.4 - A sphere of radius R, centered at the origin,...Ch. 3.4 - Prob. 3.28PCh. 3.4 - Four particles (one of charge q, one of charge 3q,...Ch. 3.4 - In Ex. 3.9, we derived the exact potential for a...Ch. 3.4 - Prob. 3.31PCh. 3.4 - Two point charges, 3qand q , arc separated by a...Ch. 3.4 - Prob. 3.33PCh. 3.4 - Three point charges are located as shown in Fig....Ch. 3.4 - A solid sphere, radius R, is centered at the...Ch. 3.4 - Prob. 3.36PCh. 3.4 - Prob. 3.37PCh. 3.4 - Here’s an alternative derivation of Eq. 3.10 (the...Ch. 3.4 - Prob. 3.39PCh. 3.4 - Two long straight wires, carrying opposite uniform...Ch. 3.4 - Prob. 3.41PCh. 3.4 - You can use the superposition principle to combine...Ch. 3.4 - A conducting sphere of radius a, at potential V0 ,...Ch. 3.4 - Prob. 3.44PCh. 3.4 - Prob. 3.45PCh. 3.4 - A thin insulating rod, running from z=a to z=+a ,...Ch. 3.4 - Prob. 3.47PCh. 3.4 - Prob. 3.48PCh. 3.4 - Prob. 3.49PCh. 3.4 - Prob. 3.50PCh. 3.4 - Prob. 3.51PCh. 3.4 - Prob. 3.52PCh. 3.4 - Prob. 3.53PCh. 3.4 - Prob. 3.54PCh. 3.4 - Prob. 3.55PCh. 3.4 - Prob. 3.56PCh. 3.4 - Prob. 3.57PCh. 3.4 - Find the charge density () on the surface of a...
Knowledge Booster
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
- Repeat the preceding problem, with E=2xi+3x2k.arrow_forwardProblem An insulating sphere with radius R contains a total non-uniform charge (i.e. Hydrogen atom) Q such that its volume charge density is 38 312 where Bis a constant and ris the distance from the center of the sphere. What is electric field at any point inside the sphere? Solution To find the electric field inside the non-uniformly charged sphere, we may apply integration method or the Gauss's Law method. Here, let us use the Gauss's law which is expressed as We will choose a symmetric Gaussian surface, which is the surface of a sphere, then evaluate the dot product to obcain (Equation 1) The issue however is how much charge does the Gaussian surface encoses? Since, our sphere is an insulating material. charges will get distributed non-uniformly within the volume of the object. So, we look into the definition of volume charge density to find the enclosed charge. So, we have dq p= dv Based on the given problem, we can also say that dgenc p= 38 dVarrow_forwardAn electric force is expressed as F = -pz sin o a, + p cos pa,. Transform F into cartesian coordinates and compute its magnitude at point (2,-1).arrow_forward
- Compute the line integral of i = (p cos? 0)p – (p cos e sin 0)ô + 3po, 2 %3D around the path shown in the figure below (the points are labeled by their Cartesian coordinates). Check your answer using the Stokes' the- orem. (0,1,2) (0,1,0) (1,0,0) y Windows'u Etkinarrow_forwardUse the Divergence Theorem to find the outward flux of F = (9x° + 12xy) i+ (5y + 5e'sin z) j+ (9z° + 5e' cos z) k across the boundary of the region D: the solid region between the spheres x + y +z? = 1 and x2 +y? + z? = 2. ..... The outward flux of F = (9x + 12xy²) i+ (5y° + 5e'sin z) j+ (9z° + 5e'cos z) k is (Type an exact anwer, using n as needed.)arrow_forwardCompute for the work done, in millijoules, in moving a 6-nC charge radially away from the center from a distance of 6 m to a distance of 12 m against the electric field inside a non-conducting spherical shell of inner radius 4 m, outer radius 18 m, and total charge 7 mC.answer: -3.1893 mJConsider two concentric conducting spherical shells of negligible thickness separated by vacuum. One of the shells has a radius of 4 m, while the other has radius of 8 m. The total charge of one of the shells is 7 mC, while that of the other is 8 mC. Their respective total charges are distributed evenly on their surfaces. Determine the total potential energy, in nanojoules, stored between the two shells. Use the permittivity of free space as 8.854 × 10-12 F/m.ans: -27.5250 ANSWER BOTHarrow_forward
- (Problems in images below.) Both problems require a volume charge integration of the given charge density functions. Technically, a volume integration is a triple integral. However, since the charge density is spherically symmetric, this reduces to a one-dimensional integration.arrow_forwardDetermine the electrostatic force between a nucleus, consisting of two protons, and one electron separated by a distance of 0.11 nm. Is this force attractive or repulsive? The results of all problems must be expressed in SI units (and some other, if requested). I know Points will be deducted if the result does not indicate its unit of measure.arrow_forwardIf there are a uniform infinite charge sheet with charge of -11nC/m² lie at y = 6 with a uniform infinite line charge of 6nC/m located at y = 11 & z= 3.5. Moreover there is a -12nC) at x 1.5, y = -2.5, & z= 5.5, explore the amout of electrical flux intensity at (1, 1, 1)?arrow_forward
- A sphere of radius R carries a polarization that follows P(7) = ke¯" f, where k is a constant wit the appropriate unit, and 7 is the vector from the center of the sphere. a) Calculate the surface bound charge density o,(7). b) Calculate the volume bound charge density pp(7). (Be careful about the spherica coordinates!) c) Find the potential V(7) inside and outside of the sphere. d) Find the electric field E(7) inside and outside of the sphere.arrow_forwardConsider an electric field in the rectangular coordinates system Ē = Yi – 3y 1) Find the equation of the streamlines corresponding to this field. 2) Deduce the streamline that passes through the point M(2, 8).arrow_forwardThe electric field in a region of space near the origin is given by E(z, y, :) – E, (*) yî+ xî a (a) Evaluate the curl Vx E(x, y, z) (b) Setting V(0, 0, 0) = 0, select a path from (0,0, 0) to (x, y, 0) and compute V (r, y,0). (c) Sketch the four distinct equipotential lines that pass through the four points (a, a), (-a, a), (-a, -a), and (a, -a). Label each line by the value of V.arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Ising model | A Bird's Eye View | Solid State Physics; Author: Pretty Much Physics;https://www.youtube.com/watch?v=1CCZkHPrhzk;License: Standard YouTube License, CC-BY