Chapter 4.2, Problem 4.11P

A charged particle moves along a straight line in a uniform electric field E with a speed v. If the motion and the electric field are both in the x direction, (a) show that the magnitude of the acceleration of the charge q is given by a=dvdt=qEm(1v2c2)3/2 (b) Discuss the significance of the dependence of the acceleration on the speed. (c) If the particle starts from rest at x = 0 at t = 0, find the speed of the particle and its position after a time t has elapsed. Comment on the limiting values of v and x as t .

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

The 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?

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

Consider an infinitesimal segment located at an angular position θ on the semicircle, measured from the lower right corner of the semicircle at x=a, y=0. (Thus θ=π2 at x=0, y=a and θ=π at x=−a, y=0.) What are the x- and y- components of the electric field at point P (dEx and dEy) produced by just this segment? Express your answers separated by a comma in terms of some, all, or none of the variables Q, a, θ, dθ, and the constants k and π.

Obtain the required relation please

Compute the force on charge A

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.

Evaluate the commutator [Â,B̂] of the following operators.