Assignment 11

Problem 1:

Find an expression for the radius of the trajectory of a particle of charge q moving at a speed v at right angles to a uniform magnetic field of magnitude B, if v << c.  Estimate the radius of motion for an electron and also that for a proton assuming that they move at velocity ~0.1 c at right angles to a magnetic field of 5*10−6 G, a typical interstellar magnetic field.  Will such protons and electrons be confined to a galaxy of size 105 light years?
(1G = 10-4 T)

Problem 2:

An infinitely long cylinder with radius a and permeability μ is placed into an initially uniform magnetic field B0 with its axis perpendicular to B0.  Find the resultant field inside and outside of the cylinder.

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The space between a pair of coaxial cylindrical conductors is evacuated.  The radius of the inner cylinder is a, and the inner radius of the outer cylinder is b as shown in the figure.

The outer cylinder, called the anode, may be given a positive potential V relative to the inner cylinder.  A static homogeneous magnetic field B parallel to the cylinder axis, directed out of the plane of the figure, is also present.  Induced charges in the conductors are neglected.
We study the dynamics of electrons with mass m and charge -qe.  The electrons are released at the surface of the inner cylinder.
(a)   First the potential V is turned on, but B = 0.  An electron is set free with negligible velocity at the surface of the inner cylinder.  Determine its speed v when it hits the anode.  Give the answer both when a non-relativistic treatment is sufficient and when it is not.

For the remaining parts of the problem a non-relativistic treatment suffices.
(b)   Now V = 0 but the homogeneous magnetic field B is present.  An electron starts out with an initial velocity v0 in the radial direction.  For magnetic field magnitudes larger than a critical value Bc, the electron will not reach the anode.  Make a sketch of the trajectory of the electron when B is slightly larger than Bc.  Determine Bc.

From now on both the potential V and the homogeneous magnetic field B arc present.
(c)  The magnetic field will give the electron a non-zero angular momentum L with respect to the cylinder axis.  Write down an equation for the rate of change dL/dt of the angular momentum.  Show chat this equation implies that L - kqeBr2 is constant during the motion, where k is a definite pure number.  Here r is the distance front the cylinder axis.  Determine the value of k.
(d)  Consider an electron released front the inner cylinder with negligible velocity, that does not reach the anode, but has maximum distance from the cylinder axis equal to rm.  Determine the speed vrm at that point rm where radial distance is maximum, in terms of rm.
(e)  We are interested in using the magnetic field to regulate the electron current to the anode.  For B larger than a critical magnetic field Bc, an electron released with negligible velocity will not reach the anode.  Determine Bc.

Problem 4:

In Bohr's 1913 model of the hydrogen atom, the electron is in a circular orbit of radius 5.29*10-11 m and its speed is 2.19*106 m/s.
(a)  What is the magnitude of the magnetic moment due to the electron's motion?
(b)  If the electron orbits counterclockwise in a horizontal circle, what is the direction of this magnetic moment vector?

imageProblem 5:

Find the magnetic field B at the center of a flat spiral with current I.  The spiral is contained between two radii r and R and has N turns.  Do not consider the effect of the connecting wires.