Introduction to Electrodynamics
4th Edition
ISBN: 9781108420419
Author: David J. Griffiths
Publisher: Cambridge University Press
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Chapter 3.4, Problem 3.36P
To determine
The electric field of a dipole in the form of coordinates free form.
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Problem 2.20 One of these is an impossible electrostatic field. Which one?
(a) Ek[xy x + 2yzý + 3xz2];
(b) E= k[y² + (2xy + z²) ŷ + 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.]
Problem 2.11 Use Gauss's law to find the electric field inside and outside a spherical shell of
radius R, which carries a uniform surface charge density o. Compare your answer to Prob. 2.7.
Problem 2.21 Find the potential inside and outside a uniformly charged solid sphere whose
radius is R and whose total charge is q. Use infinity as your reference point. Compute the
gradient of V in each region, and check that it yields the correct field. Sketch V (r).
An insulating solid sphere of radius 3 m has 15 C of charge uniformly distributed throughout its volume. Calculate the charge contained in a Gaussian surface having a radius 1/2 that of the sphere. Present your answer accurately to 2 decimal numbers i.e 3.20. Do not include units!
1.27| The important dipole field (to be addressed in Chapter 4) is expressed in
spherical coordinates as
E =4 (2 cos 0 a, + sin 0 ag)
where A is a constant, and where r> 0. See Figure 4.9 for a sketch.
(a) Identify the surface on which the field is entirely perpendicular to the xy
plane and express the field on that surface in cylindrical coordinates.
(b) Identify the coordinate axis on which the field is entirely perpendicular
to the xy plane and express the field there in cylindrical coordinates.
(c) Specify the surface on which the field is entirely parallel to the xy plane.
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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- Problem 2.20 One of these is an impossible electrostatic field. Which one? (a) E= k[xy x + 2yz y + 3xz 2]; (b) E= k[y² + (2xy + z²)ŷ + 2yz2]. Here k is a constant with the appropriate units. For the possible one, find the poten- tial, 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 definite path in mind.]arrow_forwardProblem 2.41 Find the electric field at a height z above the center of a square sheet (side a) carrying a uniform surface charge o. Check your result for the limiting cases a o and z > a. [Answer: (o/2e0){(4/7) tan-1+ (a²/2z²) – 1}]arrow_forwardI have parts A & B. I just need help with c. The figure is attached. For the cylinder of uniform charge density in Fig. 2.26: (a) show that the expression there given for the field inside the cylinder follows from Gauss’s law My Answer: p = rho r<a: E = (p*r)/ (2 * epsilon not) r>a: E = (p * a^2)/(2 * epsilon * r) (b) find the potential φ as a function of r, both inside and outside the cylinder, taking φ = 0 at r = 0. My Answer: r<a: φ(r) = (-p * r^2)/(4 * epsilon not) r>a: φ(r) = (-p * a^2)/(4 * epsilon not) - (p * a^2)/(2 * epsilon not)(In(r/a)) c) Take the Laplacian in cylindrical coordinates and show that Poisson’s equation holds in this example.arrow_forward
- Use 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_forwardb) Given F(x, y, z) = (x³ + cosh z) i+ (2y³ – 3r²y)j – (x² + 4y²z) k. Use Gauss's theorem to calculate //F .n dS where n is the outward unit normal of o, the surface bounded by the planes, x = 0, z = 0 and x + z = 6, and the parabolic cylinder x = 4 – y².arrow_forwardD3.2. Calculate D in rectangular coordinates at point P(2, –3, 6) produced by: (a) a point charge QA charge PLB 55 mC at Q(-2, 3, –6); (b) a uniform line 20 mC/m on the x axis; (c) a uniform surface charge density || Ps : 120 µC/m² on the plane z = –5 m. Ans. 6.38a, – 9.57ay + 19.14a; µC/m²; –212a, + 424a, µC/m²; 60a, µC/m²arrow_forward
- Consider an infinitely long wire of charge carrying a positive charge density of A. The electric field due to λ this line of charge is given by E= 2kef= -, where is a unit vector directed radially outward Σπερμ from the infinitely long wire of charge. Hint #3 a. Letting the voltage be zero at some reference distance (V(ro) = 0), calculate the voltage due to this infinite line of charge at some distance r from the line of charge. Give your answer in terms of given quantities (A,ro,r) and physical constants (ke or Eo). Use underscore ("_") for subscripts and spell out Greek letters. Hint for V(r) calculation 3 V(r) = b. There is a reason we are not setting V(r → ∞o) = 0 as we normally do (in fact, in general, whenever you have an infinite charge distribution, this "universal reference" does not work; you need a localized charge distribution for this reference to work). Which of the following best describes what happens to potential as roo? (That is, what is V(ro), with our current…arrow_forwardIf A = 2yz i – (x + 3y – 2)j+ (x² + z)k, evaluate SS, (V × A)· dS over the surface of intersection of the cylinders x² +y? = 1, x? + z² = 1 which is included in the first octant.arrow_forwardThe electric field present in the region is given as È = (cosa + (1 + sinø) az)e-² in V/m. Determine how much flux, in pc, is passing through the rectangle defined by 4 < p < 6, 1arrow_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_forward2.1. Four 10nC positive charges are located in the z = 0 plane at the corners of a square 8cm on a side. A fifth 10nC positive charge is located at a point 8cm distant from the other charges. Calculate the magnitude of the total force on this fifth charge for e = €0: Arrange the charges in the xy plane at locations (4,4), (4,-4), (-4.4), and (-4,-4). Then the fifth charge will be on the z axis at location z = 4/2, which puts it at 8cm distance from the other four. By symmetry, the force on the fifth charge will be z-directed, and will be four times the z component of force produced by each of the four other charges.arrow_forwardCompute the flux of the vector field F = through the surface S where S is the surface bounded by r = 2, z = 0, z = 5 in the first octant. Round off answer to 2 decimal places. NOTE: if answer is 5pi, write 15.71 none of these above O 20pi O 25pi 5pi O 30pi Jattachments/766029175160176704/824757650461294642/image3.jpg O O Oarrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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