Hand-In Problem Set 6
(due Thursday, 08/04/2015, 5p.m.)
*Note: these are not arranged in order of difficulty!
Two questions out of the full homework set will be graded. I expect you to do the whole
assignment but, for those who for some reason don’t manage to do this, I have indicated in the
homework one of the questions that will be graded. You will not receive full credit for a
question unless you show all your working.
1. The circuit below has been connected to the 10.0 V battery
for a long time.
(a) What is the voltage across the 1.00 μF capacitor?
(b) If the battery is disconnected, how long does it take the
capacitor to discharge to one tenth of its initial voltage?
a) Can you solve a circuit using equivalent resistance and
Kirchhoff’s laws? Can you use the principle of Kirchhoff’s
loop law to calculate the potential difference between two points in a circuit? Do you understand
that no current flows to a fully charged capacitor? i.e. the capacitor acts like a break in the circuit
after a long time.
b) Can you redraw a circuit to make something easier to solve? Are you able to make use of the
decay time constant in a discharging capacitor?
2. A portable defibrillator stores energy in a capacitor that is charged to a high
potential difference. When the capacitor is discharged through the patient’s
body, short high-powered pulse (an electric shock) is applied to the heart over a
time interval of a few milliseconds. Electrodes, coated with conducting paste,
are held against the chest on both sides of the heart. Their handles are insulated
to prevent injury to the operator, who calls “Clear!” and pushes a button on one
paddle to discharge the capacitor through the patient’s chest. On the right there
is a diagram of a simplified circuit for a
defibrillator. When the switch is at 1, the capacitor is
charging. When the switch is at 2, the defibrillator is
discharging through the body. Ignore the inductor, and treat
it like a bare wire. You will learn in week 7 what role
inductors play in a circuit.
a. If a defibrillator is charged up to 4500V and stores
an energy of 550J, what is the capacitance of the
capacitor in μF?
Did you recall that the energy in a capacitor is
Picture adjusted from one found at
U=1/2 CV2?
b. The power supply has an internal resistance of 15Ω, http://www.frca.co.uk/article.aspx?articleid=100392
and supplies an emf of 5000V to the circuit. How
long will a fully discharged defibrillator (i.e.
uncharged) take to charge up to 4500V? Checking you know what to do with a charging
RC circuit. Of course, you can just use
V C (t)=V max (1−e−t / τ ) .
The switch is turned to 2 when the capacitor reaches 4500V. The defibrillator takes 4.5ms to
deliver 320J to the heart.
c. What is the potential difference across the capacitor plates 4.5ms after switch 2 is closed?
Now you have a discharging capacitor. The values were chosen to be close to a real-life
situation.
d. What is the resistance of the path through the heart? (Hint: What is the time constant of
the circuit?) Using τ=RC from a different perspective. Again, the resistance
between the paddles should be the same order-of-magnitude to a real-life situation.
3.
A parallel-plate, air-gap capacitor has a capacitance of 0.14 μF. The plates are 0.50 mm apart.
a. What is the stored energy if the capacitor is charged to 3.2 μC?
b. How much charge can the capacitor carry before dielectric breakdown of the air between
the plates occurs?
c. What is the energy density (energy per unit volume) in the case of b)?
Do you know what the dielectric breakdown of air means? How is the electric field in a capacitor related
to the energy density of an electric field. Note, there are two ways to solve part c).
4. (This one WILL be graded.) The coaxial cable (another one!) shown in
the figure consists of a very solid inner conductor of radius R 1,
surrounded by a hollow, very thin outer conductor of radius R 2. The two
carry equal currents I, but in opposite directions. The current density is
uniformly distributed in the outer conductor, but non-uniformly
distributed, with function
J (r )=Ce
−r / R 1
, in the inner conductor.
a. Write C as a function of I. You had a similar question in
Midterm 2. Do you understand the implications of a variable current density. Do you
know to start with dI=JdA?
b. Find expressions for the three magnetic fields in the space: outside the outer conductor,
between the conductors, and inside the inner conductor. (Your answer should be
expressed in terms of I, R1, R2 and known universal constants such as μ0).
Did you know to use Ampere's law here? High symmetry! Did you find the cirrect
through each loop correctly. Integral limits are important here!
c. Draw a graph of B vs r from r=0m to r=2R2, assuming R2=3 R1 .
Can you interpret the functions from part b) into a graph?
5. As you move further north, the angle of the earth’s magnetic field with respect to the horizontal
changes? This angle is referred to as the dip angle.
a. Assume you are in the northern hemisphere. Is the dip angle below or above the
horizontal? Does it increase or decrease as you go further north?
Do you understand what the dip angle is. Think back to the exercise you did in lab.
b. Green turtles seem to use this dip angle to determine their latitude. Suppose you are a
researcher wanting to test this idea. You have gathered green turtle hatchlings from Islas
Marías, Mexico which has a total magnetic field of 41.3μT, and dip angle of 48.2°. You
then set up a temporary lab on these islands, and put the turtles in a 1.2-m-diameter
circular tank and monitor the direction in which they swim as you vary the magnetic field
in the tank. You change the magnetic field felt by the turtles by passing a current through
a 100-turn coil wrapped around the tank. This creates a field that adds to that
of the earth. What current should you pass through the coil, and in what
direction, to produce a net field in the center of the tank that has a dip angle of
62°? Can you determine what magnetic field vector you need to add to the
earth's magnetic field vector to get a resultant vector with a dip angle of 62°? This
problem was also testing if you could pull out relevant information from a text.
6. The loop in the figure has current I=0.50A running through it.
a) Use the Biot-Savart law to calculate the magnetic field strength
and direction at point P in the figure in terms of L, I and known
universal constants. Can you apply the Biot-Savart Law?
b) If L=25cm, what is the magnitude of the magnetic field at point
P? Add numbers to your problems – you should have clearly
defined your variables in part a.