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.

(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.

Consider a particle of charge q and mass m in the presence of a constant,
uniform magnetic field B = B_{0 }**k**, and of a uniform electric
field of amplitude E_{0}, rotating with

frequency ω in the (x,y) plane, either in the clockwise or in counterclockwise
direction.

Let **E** = E_{0}cos(ωt) **i** ± E_{0}sin(ωt) **
j**.

(a) Write down the equation of motion for the particle and solve for
the Cartesian velocity components v_{i}(t) in terms of B_{0}, E_{0},
and ω if ω ≠ ω_{c} = qB_{0}/m (the cyclotron
frequency).

Show that, if ω = ω_{c}, a resonance is observed for the appropriate sign of ω.

Hint: Let ζ = v_{x} + iv_{y}, and solve for ζ(t).

(b) Solve for the Cartesian velocity components v_{i}(t) at
resonance.

(c) Now assume the presence of a frictional force **f** = -mγ**v**, 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 = ω_{c}.

Refer 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?

A^{ }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 = B_{0} (1 + kx), where B_{0}
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.^{ }

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.