Introduction to Electrodynamics
4th Edition
ISBN: 9781108420419
Author: David J. Griffiths
Publisher: Cambridge University Press
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Chapter 3.4, Problem 3.28P
To determine
The first three terms
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For each of the following vector fields F , decide whether it is conservative or not by computing curl F . Type in a potential function f (that is, ∇f=F).
F(x,y)=(−3siny)i+(10y−3xcosy)j
Consider a uniformly charged ring in the xy plane, centered at the origin. The ring has radius a and positive charge q distributed evenly along its circumference.
Imagine a small metal ball of mass m and negative charge −q0. The ball is released from rest at the point (0,0,d) and constrained to move along the z axis, with no damping. If 0<d≪a, what will be the ball's subsequent trajectory?
repelled from the origin
attracted toward the origin and coming to rest
oscillating along the z axis between z=d and z=−d
circling around the z axis at z=d
calculate the flux of the vector field through the given surface.
~F = z ~k through a square of side length 5 in the plane z = 2. The square is centered on the z-axis, has sides parallel to the axes, and is oriented in the positive z-direction.
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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- Determine whether the vector field is conservative and, if so, find the general potential function. F=⟨cos(z),2y^3,−xsin(z)⟩arrow_forwardThe figure here shows a Gaussian cube of face area Aimmersed in a uniform electric field that has the positivedirection of the z axis. In terms of E and A, what is the fluxthrough (a) the front face (which is in the xy plane), (b) therear face, (c) the top face, and (d) the whole cube?arrow_forwardA thin, circular plate of radius B, uniform surface charge density, and total charge Q, is centered on the origin and lies in the x-y plane. What is the electric flux ΦE through a sphere of radius r, also centered on the origin, as a function of r? Consider both (a.) r > B and (b.) r < B:arrow_forward
- An ideal uniformly charged ring is situated on the xy-plane withits center at the origin. Assuming that the charge of the ring is negative,a proton moving in the +z -axis going to the origin willarrow_forwardProve that the vector field F(x,y,z) = (x^2 + yz)i − 2y(x + z)j + (xy + z^2)k is incompressible, and find its vector potential function.arrow_forwardFind the vector c from an arbitary element of the surface charge to Qarrow_forward
- Consider a uniformly charged ring in the xy plane, centered at the origin. The ring has radius a and positive charge q distributed evenly along its circumference. A small metal ball of mass m and negative charge −q0 is released from rest at the point (0,0,d) and constrained to move along the z axis, with no damping. At 0<d≪a, the ball's subsequent trajectory is oscillating along the z axis between z=d and z=−d. The ball will oscillate along the z axis between z=d and z=−d in simple harmonic motion. What will be the angular frequency ω of these oscillations? Use the approximation d≪a to simplify your calculation; that is, assume that d2+a2≈a2. Express your answer in terms of given charges, dimensions, and constants.arrow_forwardSuppose a piece of infinite length. We have the surface density of Sigma, which has width a, find the field at distance z؟arrow_forwardFind the electric field a distance z above the midpoint of a straight line segment of length 2L, which carries a uniform line charge A.arrow_forward
- If Force B on the x-z plane is equal to 300N and h = 4m and v = 10m, then what is the i and k components of Force B?arrow_forwardA large non-conducting slab of area A and thickness d has a charge density rho=Cx^4. The origin is through the center of the slab. That is to say, it bisects the slab into two equal volumes of d/2 thickness and with an area of A, with -d/2 to the left of x=0, and d/2 to the right of x=0. Express all answers in terms of C, x, and any known constants. Gaussian surface 1 (cylinder) is located such that its volume encompasses the charge contained within the slab. Apply Gauss's Law to cylinder 1 to determine the electric field to the left and to the right of the slab. Make sure you incude the domains over which the field is valid.arrow_forwardSuppose a twodimensional force field is everywhere directed outward from the origin, and C is a circle centered at the origin.What is the angle between the field and the unit vectors tangent to C ?arrow_forward
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