Assignment 9

Problem 1:

In the LHC, there are actually two circular tubes, side-by-side, that protons travel through.  The protons go one way in one tube, and the other way in the other tube. ( And then eventually the two paths are made to cross to smash the protons together.)  The diagram below shows a short segment of the two tubes of radius r and separation d (black), with coils of wire on them (red) that carry a current I to generate the magnetic fields.  The wires are essentially long straight wires that run right along the sides of the tubes for a length L, and then wrap over the top to the other side.  The drawing just shows one red line, but this is supposed to represent N loops of wire forming a bundle with that shape.  The wires are coated with an insulator so that current flows along the length of wire only, and not from one wire to another in the bundle.

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(a)  If we want the protons in the upper tube in the figure to go counterclockwise, and the protons in the lower tube to go clockwise, find the direction of current flow in the upper and lower coils (clockwise or counterclockwise for each).
(b)  Near the midpoint of the straight segment of the wire, we can neglect the magnetic fields from the ends, and consider the wires to be just infinite straight, parallel wires.  Find the magnetic field (magnitude and direction) at the center of each tube, near the midpoint of the straight sections.  Your answer should be in terms of r, d, I, and N.
(c)  Given the numbers B = 8 T, r = 50 mm, d = 250 mm, and N = 80, evaluate the current I in the wires.
(d)  Ignoring the semicircles at the ends, calculate the force (magnitude and direction) on the bundle of wires on one side of a pipe due to the bundle of wires on the other side of the pipe.  First find a symbolic equation in terms of N, I, L, and r, then plug in the numbers above and L = 15 m.

Solution:

Problem 2:

Consider a particle of charge q and mass m in the presence of a constant, uniform magnetic field B = B0 k, and of a uniform electric field of amplitude E0, rotating with
frequency ω in the (x,y) plane, either in the clockwise or in counterclockwise direction.
Let E = E0cos(ωt) i ± E0sin(ωt) j.
(a)  Write down the equation of motion for the particle and solve for the Cartesian velocity components vi(t) in terms of  B0, E0, and ω if ω ≠ ωc = qB0/m (the cyclotron frequency).
Show that, if ω = ωc, a resonance is observed for the appropriate sign of ω.
Hint:  Let ζ = vx + ivy, and solve for ζ(t).
(b)  Solve for the Cartesian velocity components vi(t) at resonance.
(c)  Now assume the presence of a frictional force f = -mγv, where v is the velocity of the particle.  Find the general solution for ζ(t) for clockwise rotation of the electric field, and find the steady state solution (t >> 0)  when ω = ωc.

Solution:

Problem 3:

imageRefer to the figure.  One end of a conducting rod rotates with angular velocity ω in a circle of radius a making contact with a horizontal, conducting ring of the same radius.  The other end of the rod is fixed.  Stationary conducting wires connect the fixed end of the rod (A) and a fixed point on the ring (C) to either end of a resistance R.  A uniform vertical magnetic field B passes through the ring.

(a)  Find the current I flowing through the resistor and the rate at which heat is generated in the resistor.
(b)  What is the sign of the current, if positive I corresponds to flow in the direction of the arrow in the figure?
(c)  What torque must be applied to the rod to maintain its rotation at the constant angular rate ω? 
What is the rate at which mechanical work must be done?

Solution:

Problem 4:

imageA square loop made of wire with negligible resistance is placed on a horizontal frictionless table as shown (top view).  The mass of the loop is m and the length of each side is b.  A non-uniform vertical magnetic field exists in the region; its magnitude is given by the formula B = B0 (1 + kx), where B0 and k are known constants.
The loop is given a quick push with an initial velocity v along the x-axis as shown.  The loop stops after a time interval t.  Find the self-inductance L of the loop.

Solution:

Problem 5:

A current I flows in the circuit shown below.  Calculate the magnetic field at P as a function of the current I and the distances a and b.  Segments BC and AD are arcs of concentric circles.  Segments AB and DC are straight-line segments.

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Solution: