Introduction to Electrodynamics
4th Edition
ISBN: 9781108420419
Author: David J. Griffiths
Publisher: Cambridge University Press
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Chapter 1.3, Problem 1.35P
To determine
To verify: Corollary 1 for the function.
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1.16. Establish thermodynamically the formulae
v (7)=
= S and
v (R),
V
= N.
Express the pressure P of an ideal classical gas in terms of the variables and 7, and verify the
μl
above formulae.
Figure 1.30
Problem 1.15 Calculate the divergence of the following vector functions:
(a) va = x² + 3xz²ŷ - 2xz2.
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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- 1.32 In the Eigen vector equation AX = X, the operator A is given 32 A = 41 Find: (a) The Eigen values (b) The Eigen vector X (c) The modal matrix C and it's inverse C-1 (d) The product C-1 ACarrow_forward1.16. Establish thermodynamically the formulae (F). V = S and v (3) ₁² V T = N. Express the pressure P of an ideal classical gas in terms of the variables μ and T, and verify the above formulae.arrow_forwardFigure 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.arrow_forward
- 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_forwardUse Component and Polygon methodarrow_forwardFigure 1.52 shows a spherical shell of charge, of radiusa and surface density σ, from which a small circular piece of radius b << ahas been removed. What is the direction and magnitude of the fieldat the midpoint of the aperture? Solve this exercise using the relationship for a force on a small patch.arrow_forward
- Construct the 3D rotation ma- trices for rotations about the x-axis, y-axis, and z-axis by an angle o as described in I section 1.1.5. You should end up with three 3x3 matrices, one for each rotation axis. Demonstrate that your matrices do what they are supposed to do by considering >= 90 deg and applying them to an arbitrary vector A. Draw pictures to illustrate your reasoning.arrow_forward1.3. Determine an orthogonal basis for the subspace of C (-1, 1] spanned by functions: {f(x) = x, f(x) = x³, f(x) = x³] using Gram-Schmidt process.arrow_forwardProblem 2.24 Delta functions live under integral signs, and two expressions (D₁ (x) and D₂(x)) involving delta functions are said to be equal if [ + f(x)D₁(x) dx = [+0° f(x)D₂(x) dx. -8 -00 for every (ordinary) function f(x). (a) Show that 8(cx) = -178(x). [2.142] where c is a real constant. (Be sure to check the case where c is negative.) (b) Let 0(x) be the step function: { 0(x)= 1. [2.143] (In the rare case where it actually matters, we define (0) to be 1/2.) Show that de/dx = 8(x). if x > 0. if x < 0. 0.arrow_forward
- A hinged rigid bar of length l is connected by two springs of stiffnesses K1 and K2 and is subjected to a force F as shown in Fig. 1.33(a). Assuming that the angular displacement of the bar (θ) is small, find the equivalent spring constant of the system that relates the applied force F to the resulting displacement x.arrow_forwardConslder the following. TIs the counterdockwise rotation of 45° in R, v = (2, 2). %3D (a) Find the standard matrix A for the linear transformation 7. (b) Use A to find the image of the vector v. 7(v) (c) Sketch the graph of v and its image. y. T(V)arrow_forward1.38 You lay a rectangular board on the horizontal floor and then tilt the board about one edge until it slopes at angle 0 with the horizontal. Choose your origin at one of the two corners that touch the floor, the x axis pointing along the bottom edge of the board, the y axis pointing up the slope, and the z axis normal to the board. You now kick a frictionless puck that is resting at O so that it slides across the board with initial velocity (vox, voy, 0). Write down Newton's second law using the given coordinates and then find how long the puck takes to return to the floor level and how far it is from O when it does so.arrow_forward
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