Introduction to Electrodynamics
4th Edition
ISBN: 9781108420419
Author: David J. Griffiths
Publisher: Cambridge University Press
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Chapter 1.3, Problem 1.30P
To determine
The surface integral of the function for the bottom of the box.
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Check the divergence theorem for the provided function (see attatched image), using as your volume one octant of the sphere of radius R (see attatched image). Make sure you include the entire surface. [Answer: π R4/4]
Problem 4.16 It is desired to find the equation for the shortest distance be-
tween two points on a sphere. Determine the functional for this problem. (Use
spherical coordinates.)
Figure 1.52 shows a spherical shell of charge, of radiusa and surface density σ, from which a small circular piece of radius b << a has been removed. What is the direction and magnitude of the field at the midpoint of the aperture? Solve this exercise using direct integration.
Chapter 1 Solutions
Introduction to Electrodynamics
Ch. 1.1 - Using the definitions in Eqs. 1.1 and 1.4, and...Ch. 1.1 - Prob. 1.2PCh. 1.1 - Prob. 1.3PCh. 1.1 - Prob. 1.4PCh. 1.1 - Prob. 1.5PCh. 1.1 - Prob. 1.6PCh. 1.1 - Prob. 1.7PCh. 1.1 - Prob. 1.8PCh. 1.1 - Prob. 1.9PCh. 1.1 - Prob. 1.10P
Ch. 1.2 - Prob. 1.11PCh. 1.2 - The height of a certain hill (in feet) is given by...Ch. 1.2 - Prob. 1.13PCh. 1.2 - Prob. 1.14PCh. 1.2 - Prob. 1.15PCh. 1.2 - Prob. 1.16PCh. 1.2 - Prob. 1.17PCh. 1.2 - Prob. 1.18PCh. 1.2 - Prob. 1.19PCh. 1.2 - Prob. 1.20PCh. 1.2 - Prob. 1.21PCh. 1.2 - Prob. 1.22PCh. 1.2 - Prob. 1.23PCh. 1.2 - Prob. 1.24PCh. 1.2 - Prob. 1.25PCh. 1.2 - Prob. 1.26PCh. 1.2 - Prob. 1.27PCh. 1.2 - Prob. 1.28PCh. 1.3 - Prob. 1.29PCh. 1.3 - Prob. 1.30PCh. 1.3 - Prob. 1.31PCh. 1.3 - Prob. 1.32PCh. 1.3 - Prob. 1.33PCh. 1.3 - Prob. 1.34PCh. 1.3 - Prob. 1.35PCh. 1.3 - Prob. 1.36PCh. 1.4 - Prob. 1.37PCh. 1.4 - Express the unit vectors in terms of (that is,...Ch. 1.4 - Prob. 1.39PCh. 1.4 - Prob. 1.40PCh. 1.4 - Prob. 1.41PCh. 1.4 - Prob. 1.42PCh. 1.4 - Prob. 1.43PCh. 1.5 - Evaluate the following integrals:
(a)
(b)
(c)...Ch. 1.5 - Prob. 1.45PCh. 1.5 - (a) Show that .
[Hint: Use integration by...Ch. 1.5 - Prob. 1.47PCh. 1.5 - Prob. 1.48PCh. 1.5 - Prob. 1.49PCh. 1.6 - (a) Let and . Calculate the divergence and curl...Ch. 1.6 - Prob. 1.51PCh. 1.6 - Prob. 1.52PCh. 1.6 - Prob. 1.53PCh. 1.6 - Prob. 1.54PCh. 1.6 - Prob. 1.55PCh. 1.6 - Prob. 1.56PCh. 1.6 - Prob. 1.57PCh. 1.6 - Prob. 1.58PCh. 1.6 - Prob. 1.59PCh. 1.6 - Prob. 1.60PCh. 1.6 - Prob. 1.61PCh. 1.6 - Prob. 1.62PCh. 1.6 - Prob. 1.63PCh. 1.6 - Prob. 1.64P
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- Figure 1.52 shows a spherical shell of charge, of radius a and surface density σ, from which a small circular piece of radius b << a has been removed. What is the direction and magnitude of the field at the midpoint of the aperture? Solve this exercise using superposition.arrow_forward1.13 Two infinite grounded parallel conducting planes are separated by a distance d. A point charge q is placed between the planes. Use the reciprocation theorem of Green to prove that the total induced charge on one of the planes is equal to (-q) times the fractional perpendicular distance of the point charge from the other plane. (Hint: As your comparison electrostatic problem with the same surfaces choose one whose charge densities and potential are known and simple.)arrow_forwardProblem 3.35 A solid sphere, radius R, is centered at the origin. The "northern” hemisphere carries a uniform charge density po, and the "southern" hemisphere a uniform charge density -po. Find the approximate field E(r, 0) for points far from the sphere (r » R).arrow_forward
- Figure 1.30arrow_forward(Problem 4.10) A sphere of radius R carries a static radial polarization density P(r) = kr, r < R where k is a constant and r is the radial vector from the center of the sphere. (a) What are the dimensional units of the constant k? (b) Calculate the surface areal bound charge density o(R, 0, ø) and the volume bound charge density p(r). 2 (c) Find the electric field inside and outside the sphere.arrow_forwardI have the problem attached. I also have the formula that will help from the book. I also added some key ideas to help you and you will find that in the attachment of a page from a book, but on the top left. I wrote what you might possibly need. I just think i have my integrals wrong but I do know that the first integral goes to 2 pi.arrow_forward
- Problems 5.1 Determine the material derivative of the flux of any vector property Qj through the spatial area S. Specifically, show that in agreement with Eq 5.2-5. 1999 by CRC Press LLC 5.2 Let the property P in Eq 5.2-1 be the scalar 1 so that the integral in that equation represents the instantaneous volume V. Show that in this case dV = 5.3 Verify the identity !! and, by using this identity as well as the result of Problem 5.1, prove that the material derivative of the vorticity flux equals one half the flux of the curl of the acceleration; that is, show that 5.4 Making use of the divergence theorem of Gauss together with the identity aw at show that w,v, - dtarrow_forwardProblem 4.32 A point charge q is imbedded at the center of a sphere of linear dielectric material (with susceptibility Xe and radius R). Find the electric field, the polarization, and the bound charge densities, ph and op. What is the total bound charge on the surface? Where is the compensating negative bound charge located?arrow_forwardEvaluate the integralsarrow_forward
- Locate the centroid of the plane area/line. Express answer in terms of a and b.arrow_forwardCompleting the square Evaluate the following integrals.arrow_forwardCompute 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_forward
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