CP A thin disk with a circular hole at its center, called an annulus , has inner radius R 1 and outer radius R 2 ( Fig. P21.91 ). The disk has a uniform positive surface charge density σ on its surface. (a) Determine the total electric charge on the annulus. (b) The annulus lies in the yz -plane, with its center at the origin. For an arbitrary point the x -axis (the axis of the annulus), find the magnitude and direction of the electric field E → . Consider points both above and below the annulus. (c) Show that at points on the x -axis that are sufficiently close to the origin, the magnitude of the electric field is approximately proportional to the distance between the center of the annulus and the point. How close is “sufficiently close”? (d) A point particle with mass m and negative charge − q is free to move along the x -axis (but cannot move off the axis). The particle is originally placed at rest at x = 0.01 R 1 and released. Find the frequency of oscillation of the particle. ( Hint: Review Section 14.2. The annulus is held stationary.) Figure P21.91
CP A thin disk with a circular hole at its center, called an annulus , has inner radius R 1 and outer radius R 2 ( Fig. P21.91 ). The disk has a uniform positive surface charge density σ on its surface. (a) Determine the total electric charge on the annulus. (b) The annulus lies in the yz -plane, with its center at the origin. For an arbitrary point the x -axis (the axis of the annulus), find the magnitude and direction of the electric field E → . Consider points both above and below the annulus. (c) Show that at points on the x -axis that are sufficiently close to the origin, the magnitude of the electric field is approximately proportional to the distance between the center of the annulus and the point. How close is “sufficiently close”? (d) A point particle with mass m and negative charge − q is free to move along the x -axis (but cannot move off the axis). The particle is originally placed at rest at x = 0.01 R 1 and released. Find the frequency of oscillation of the particle. ( Hint: Review Section 14.2. The annulus is held stationary.) Figure P21.91
CP A thin disk with a circular hole at its center, called an annulus, has inner radius R1 and outer radius R2 (Fig. P21.91). The disk has a uniform positive surface charge density σ on its surface. (a) Determine the total electric charge on the annulus. (b) The annulus lies in the yz-plane, with its center at the origin. For an arbitrary point the x-axis (the axis of the annulus), find the magnitude and direction of the electric field
E
→
. Consider points both above and below the annulus. (c) Show that at points on the x-axis that are sufficiently close to the origin, the magnitude of the electric field is approximately proportional to the distance between the center of the annulus and the point. How close is “sufficiently close”? (d) A point particle with mass m and negative charge −q is free to move along the x-axis (but cannot move off the axis). The particle is originally placed at rest at x = 0.01 R1 and released. Find the frequency of oscillation of the particle. (Hint: Review Section 14.2. The annulus is held stationary.)
An infinitely long rod lies along the x-axis and carries a uniform linear charge density λ = 5 μC/m. A hollow cone segment of height H = 27 cm lies concentric with the x-axis. The end around the origin has a radius R1 = 8 cm and the far end has a radius R2 = 16 cm. Refer to the figure.
a. Consider the conic surface to be sliced vertically into an infinite number of rings, each of radius r and infinitesimal thickness dx. Enter an expression for the electric flux differential through one of these infinitesimal rings in terms of λ, x, and the Coulomb constant k.
b. Integrate the electric flux over the length of the cone to find an expression for the total flux through the curved part of the cone (not including the top and bottom) in terms of λ, H, and the Coulomb constant k. Enter the expression you find.
c. Calculate the electric flux, in N•m2/C, through the circular end of the cone at x = 0.
d. Calculate the electric flux, in N•m2/C, through the circular end of the cone at x = H.
e.…
A thin rod of length ℓ and uniform charge per unit length λ lies along the x axis, as shown in Figure P23.35. Show that the electric field at P, a distance y from the rod along its perpendicular bisector, has no x component and is given by E = 2ke λ sin θ0/y.
ring-shaped conductor with radius a = 2.50 cm has a total positive charge Q = +0.125 nC uniformly distributed around it. The center of the ring is at the origin of coordinates O. (a) What is the electric field (magnitude and direction) at point P, which is on the x-axis at x = 40.0 cm? (b) A point charge Q = -2.50 ?C is placed at point P. What are the magnitude and direction of the force exerted by the charge q on the ring?
Chapter 21 Solutions
University Physics, Volume 2 - Technology Update Custom Edition for Texas A&M - College Station, 2/e
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