Introduction to Electrodynamics
4th Edition
ISBN: 9781108420419
Author: David J. Griffiths
Publisher: Cambridge University Press
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Chapter 3.4, Problem 3.53P
To determine
The expression of electric potential.
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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.
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.
Problem 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).
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...
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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 in three ways: a) direct integration, b) by superposition, and c) using the relationship for a force on a small patch.arrow_forward2.6.1 Divergence Find hv.Edz, where E is the electric field due to a point charge q at x' and V is a sphere centered on the origin with radius b> |x'). please solve thisarrow_forwardA point charge + Q is placed at the centre of an uncharged spherical conducting shell of inner radius a and outer radius b as shown in Fig. 2.51. Fig. b WODY 1. Find the electric field for rarrow_forward1.2 Consider an infinite straight line of constant, positive charge density o (in units of C/m). 1. Using a symmetry argument, show that the electric field set up by the charge den- sity is in cylindrical coordinates of the form E(r) = E(r) e, (meaning it points away from the line and does not depend on z, nor on the angle ø). 2. Using Gauss's Law, find an expression for E(r). As volume N use a cylinder of height h and radius r, and with axis the z-axis, as shown in the picture. 3. Using the relation V (r2) – V(r1) = - E dr find an expression for the potential difference. For C, use a straight line.arrow_forward• P Figure 2.10arrow_forwardProblem 2.7 Find the electric field a distance z from the center of a spherical surface of radius R (Fig. 2.11), which carries a uniform charge density o. Treat the case z R (outside). Express your answers in terms of the total charge q on the sphere. [Hint: Use the law of cosines to write r in terms of R and 0. Be sure to take the positive square root: VR? + z? - 2Rz = (R – z) if R > z, but it's (z – R) if R < z.]arrow_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_forwardProblems 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 2.20 One of these is an impossible electrostatic field. Which one? (a) E =k[xyÂ+2yzý+3xz2]; (b) E = k[y² + (2xy + z²)ý + 2yz 2). Here k is a constant with the appropriate units. For the possible one, find the potential, using the origin as your reference point. Check your answer by computing VV. [Hint: You must select a specific path to integrate along. It doesn't matter what path you choose, since the answer is path-independent, but you simply cannot integrate unless you have a particular path in mind.]arrow_forwardProblem 2.9Suppose the electric field in some region is found to be E = kr³î, in spherical coordinates (k is some constant). (a) Find the charge density p. (b) Find the total charge contained in a sphere of radius R, centered at the origin. (Do it two different ways.)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 << 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_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_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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