Induced emf and energy

Problem:

A vertically oriented square loop of wire falls from a region where the magnetic field B is horizontal, uniform and perpendicular to the plane of the loop, into a region where the field is zero.  Let the length of each side be s and the diameter of the wire be d.  The resistivity of the wire is ρR and the density of the wire is ρm.  If the loop reaches terminal velocity while its upper segment is still in the magnetic field region, find an expression for the terminal speed.

Solution:

Problem:

A conducting circular loop made of wire of diameter d, resistivity ρ, and mass density ρm is falling from a great height h in a magnetic field with a component Bz = B0(1 + kz), where k is some constant.  The loop of diameter D is always parallel to the x-y plane.  Disregard air resistance, and find the terminal velocity of the loop.

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Solution:

Problem:

A thin disk of ordinary metal with electrical conductivity σ has radius R and thickness d.  It is held fixed in a perpendicular magnetic field B(t)k = (B0 + αt)k where B0 and α are positive constant quantities.  In the following, neglect the self inductance of the disk.  See the sketch.

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(a)  Find the current density vector j(r) at distance r from the axis of the disk.
(b)  Determine the energy/time delivered to the disk by the field.  What becomes of this energy?
(c)  Suppose that the constant a becomes negative, i.e., that α --> —α.  How do your results in parts (a) and (b) change?

Solution:

Problem:

A uniform magnetic field B = B0k points in the z-direction.  A particle with mass m and charge q moves with kinetic energy E0 in the x-y plane in a circular orbit centered at the origin as shown.

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At t = 0 the magnetic field strength starts changing slowly, so that at t = t1 it is B = B1k.  Neglect radiation.
(a)  What is the radius of the orbit R0 of the particle for t < 0 in terms of B0, and E0?
(b)  Assuming that the radius R of the orbit of the particle does not change appreciably while the particle completes one revolution, what is the kinetic energy E1 of the particle at time t1 in terms of B0, B1, and E0?

Now assume that the magnetic field stays constant (B = B0k), but that the particle is subject to a drag force Fd = -mv/τ, where τ is a constant.  At time t = 0 the position and velocity of the particle are R0 = (0, R0, 0) and v0 = (v0, 0, 0).
(c)  Write Newton's equations of motion for the velocity components vx, vy, and vz.
(d)  Construct an equation of motion for z = vx + ivy, and solve it. 
(e)  Find expressions for vx(t), vy(t), x(t), and y(t).  Describe the trajectory of the particle in words.

Solution:

Problem:

A thin metallic square frame of mass m, electrical resistance R, and side a is rotating about an axis perpendicular to a uniform magnetic field B as shown in the figure.  Initially the square frame rotates with a frequency ω0.
(a)  Determine the average energy loss per cycle due to Joule heating.
(b)  Determine the time it takes for the frequency of the rotation to slow down to 1/e of its initial value.  (Assume that the fractional change in the frame's rotation frequency per cycle is small.)

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Solution: