Problem 1
a) According to the Formula Sheet, the magnitude of the magnetic field inside the solenoid will be B = µonI, with n = 500 wdg/m => B = 0.01257 T (126 G). The electron will move around a circular orbit with the cyclotron frequency w = (e/m)B = 2.21.109 rad/s. Since w = 2pf, we find that the "repetition" frequency is 3.518.108 orbits/s = 0.352 GHz.
b) In order not to bump into the coils, the radius of the circular orbit of the electron has to be less or equal to the radius of the solenoid, i.e. R = 0.03 m. According to the formula pperp = qBR, we find a maximum pperp of pperp = eBR = 6.04.10-23 mkg/s, corresponding to a maximum velocity of 6.63.107 m/s (22% of the speed of light; this number wasn't required by the question).
Problem 2
If one increases the current through
the solenoid, the magnetic field inside would also increase. This means
that the magnetic flux FB
through any closed loop (e.g. , the electron orbit) inside the solenoid
would also increase. According to Faraday's Law, this means that there
would be a (non-static) electric field circling the axis of the solenoid,
which in turn would accelerate the electron due to the electric force F
= qE. In case you're interested whether the electron speeds up or
slows down, let's consider the case where the field in the solenoid points
vertically up (the axis is vertical). Following the Lorentz force law F
= qvxB
= -evxB,
the electron would experience a centripetal force (as it must) if it is
circling counterclockwise (as seen from above), due to its negative charge.
If the field increases, then the induced electric field would have to be
clockwise (the increasing current flows counterclockwise, and according
to Lenz' law, the induced electric field will oppose it). However, again
because of the negative electron charge, a clockwise field will lead to
a tangential force in the opposite direction, which will make the electron
go faster.
Problem 3
The correct answer is in boldface, with a short explanation added.
a) A superconducting (R=0) solenoid of 1 m length and
0.2 m diameter (inductance L = 62.5 H) is ramped up to produce a field
of 5 T throughout its volume. If it suddenly "quenches" (becomes normal-conducting:
R increases to several 100 W),
what could possibly happen?
i) Nothing much; the energy stored
is too small.
ii) The current in the solenoid will
disappear instantaneously.
iii) The released energy would
be enough to heat up the solenoid significantly. The energy stored
is B2/2µo times volume, which is 312.5 kJ.
This energy will be released over a few seconds (time constant L/R), which
generates an enormous amount of heat in a short time.
iv) The current in the solenoid will
increase exponentially.
b) The LC circuit of a radio receiver is tuned to 94.9 MHz. By which factor (Cnew/Cold) to I have to change the capacitance C of the circuit to tune it to 89.4 MHz instead (assume L stays the same)?
i) 0.89 ii) 1.0 iii) 1.06 iv) 1.127
Since w2 = 1/LC, we have C = 1/w2L, so if w is to be reduced by a factor of 94.9/89.4 = 1.062, C must be increased by the square of that.
c) An inductor L = 0.07 H and a resistor
R = 1.4 W form a closed
circuit with an initial current Io = 1.2 A. How long will it
take until the current has fallen to 1/e of its initial value (i.e. to
0.44 A)?
i) 50 ms ii) 98 ms iii) 0.37
s iv) 20 s (t=L/R)
d) What is the force between two parallel
wires, both running 15 A and separated by 3 mm, for each 1 m of their length?
i) 15 µN ii) 0.015 N
iii) 15 N iv) zero
The field generated by the first wire at 3 mm distance is 0.001 T, and the force on the second wire is F=ILB = 0.015N.