Download Problem Set 5

Cornell University
Department of Physics
Physics 2213: Electromagnetism, Fall 2012
Problem Set # 5
(Due Friday, October 5 at 5:00pm sharp.)
Agenda and readings
Readings marked YF refer to sections from the text book, University Physics, 13th edition, volume 2,
by Young and Freedman.
• Lec 10 9/25: Resistors in parallel, circuit analysis by equivalent elements, power: YF 26.1,25.5
• Lec 11 9/27: General Kirkoff’s analysis, gradient of potential, electric potential energy: YF 26.2(review), 23.5, 23.1
• Lec 12 10/2: Electric Potential Energy, energy stored in capacitor, charging energy and conservation:
24.1,24.3
• Lec 13 10/4: RC circuits: 26.4
Contents
1 Computing resistances
1.1
1.2
1
Planar resistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.1
Insignificance of charge build up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1.2
Analysis of the resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Cylindrical Resistor
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2.1
Determination of resistance from basic definitions . . . . . . . . . . . . . . . . . . . . .
3
1.2.2
Determination of cylindrical resistance from equivalent circuit analysis . . . . . . . . .
4
2 Circuit analysis
4
3 Power and impedance matching
4
4 Electric field and potential gradient
5
1
1
1.1
Computing resistances
Planar resistor
Figure 1 reviews the derivation of resistance done in lecture, giving a little more detail. (Note that the
figure happens to show the particular case of a resistor whose length L is short relative to its cross-sectional
area A, ultimately this doesn’t affect the mathematics of the problem.) When a resistor is first connected,
the external circuit drives (positive) charge onto one face of the resistor and pulls charge from the other
face (leaving a negatively charged surface), as the figure shows. Both the incoming and outgoing faces
are generally coated with a conducting material so that the resulting charge distribution corresponds to
oppositely charged plates, resulting in a constant electric field E between the plates. As derived in lecture,
such an electric field results in a corresponding current density J = E/ρ flowing between the plates, where
ρ is the resistivity of the material.
L
Α
+σ
−σ
I
I
E, J
Figure 1: Planar resistor with length L and cross-sectional area A. At the left and right sides of the resistor
where the current enters and exits the resistor are conducting plates (shaded areas) to allow the current to
be uniformly distributed on the surfaces at the ends of the resistor.
1.1.1
Insignificance of charge build up
The total charge on each side of the resistor builds up very quickly and turns out to be negligible compared to
the current flowing through the circuit. For example, even in a high-speed electronic circuit, where currents
are on the order of milliamps (I =10−3 A) and times are on the order of nanoseconds (t = 10−9 s) so that
the relevant charges flowing through circuit elements are on the order of Q = It = 10−12 C=1 pC, the charge
build up on the sides of the resistors is much smaller.
Show this by estimating how much charge Q builds up on each end of a resistor which is 1 cm long and has
a cross-sectional area of 4 mm2 when a voltage of 10 V is applied to it.
Hint: You will want to determine the electric field from the length and the voltage, and then use the general
result for conductors that E = σ/0 .
2
1.1.2
Analysis of the resistance
Given that a charge density ±σ builds up on the left/right conducting faces of the resistor, what is (a) the
magnitude of electric field E in the resistive medium between the plates in terms of no quantities other than
σ, L, A, 0 and the resistivity ρ?
Given the above electric field, (b) what is the voltage V between the plates in in terms of no quantities other
than σ, L, A, 0 and the resistivity ρ? Also, given E, (c) what is the current density J and, (d) what is the
total current I flowing across each cross-sectional plane of the resistor, such as that shown in gray? (Note
that the total current flowing across each plane must be the same, so that there is no continual build-up of
charge in the resistor.)
Finally, (e) given the voltage V and current I that you found in parts (b) and (d), what is the resistance of
the resistor?
1.2
Cylindrical Resistor
Figure 2 shows the geometry for a cylindrical resistor, where current enters the resistor along an inner
cylindrical conducting electrode of radius r1 , then flows radially outward perpendicular to this electrode
through a medium of resistivity ρ, and finally is collected and exits from an outer cylindrical electrode of
radius r2 > r1 . The length of the cylindrical resistor is L.
r
r2
ρ
r1
I
−λ
+λ
I
E, J
L
Figure 2: Cylindrical resistor
This resistor functions much like the planar resistor, but with a slightly different geometry. Upon connection
to an external circuit, charges are driven onto the inner and pulled off the outer electrode, leaving a total
charge per unit length ±λ along the inner and outer electrodes, respectively.
1.2.1
Determination of resistance from basic definitions
The logic of computing the resistance for such a resistor is exactly the same, although the mathematics gets a
bit more complicated. First, given that a charge per unit length ±λ builds up on the inner/outer conducting
3
faces of the resistor, (a) what is the magnitude of the radial electric field E within the resistive medium at a
distance r between the inner and outer radii? Express your answer in terms of no quantities other than λ,
r, L, r1 , r2 , 0 and the resistivity ρ.
Given the above electric field, (b) what is the upstream voltage V (up) = V1←2 between the plates, in terms
of no quantities other than λ, L, r1 , r2 , 0 and the resistivity ρ?
Hint: Your answer will involve an integral that results in a factor of − ln rr12 = ln rr21 .
Again, given the electric field, (c) what is the total current I flowing across the dashed cylindrical surface of
radius r shown in the figure, in terms of no quantities other than λ, L, r1 , r2 , 0 and the resistivity ρ?
Hint: As in the case of the planar resistor, you should find that the current crossing each surface is exactly
the same, regardless of the value of r, so that charge does not build up with time within the resistor.
Finally, given the voltage V (up) = V1←2 and the current I flowing from r1 to r2 , (d) determine the resistance
of the cylindrical resistor in terms of no quantities other than λ, L, r1 , r2 , 0 and the resistivity ρ.
1.2.2
Determination of cylindrical resistance from equivalent circuit analysis
Imagine the cylindrical resistor divided into a series of thin layers much like an onion, with one slice consisting
of the material between radii r1 and r1 + dr, the next the material between r1 + dr and r1 + 2dr, all the way
until the last layer between r2 − dr and r2 .
Given that the current flows radially outward, with the same current flowing through each of these segments
and the voltage equaling the sum of voltages across each shell, (a) should we regard these slices as resistors
in series or in parallel?
Next, each of these slices is a thin shell that acts much like the planar resistor in Figure 1. Given that the
current flows radially outward, for the layer between r and r + dr, (b) what is the cross sectional area A of
the corresponding resistor, and (c) what is the length L of the corresponding resistor? Given the standard
formula for such a resistor, R = ρL/A, (d) what is the resistance of the shell between r and r + dr?
Finally, combining each of the above resistances, either in parallel or in series in accord with your answer in
(a), compute (e) the total resistance of the cylindrical resistor between r1 and r2 , and verify that you get
the same answer as in Problem 1.2.1.
2
Circuit analysis
(a) Do Exercise 26.14 from YF. (Practice with reductions by equivalent elements.)
(b) Do Exercise 26.64 from YF. (Practice with Kirchhoff analysis.)
Followup: (c) What value of the emf E would be needed to pump current “backwards” through the 24!volt
battery?
(d) Do Exercise 26.68 from YF, part (a) only!!
Hint: First reduce the mass of resistors on the top down to a single resistor using equivalents, then then
apply Kirchhoff’s laws.
(e) Do Exercise 26.77 from YF. (Example of equivalent resistance to for a non-reducible combination of
resistors.)
Hint: You’ll want to use Kirchhoff’s laws.
4
3
Power and impedance matching
A typical 9 V battery has an internal resistance of 1.5 Ω. This question concerns the design of the brightest
possible bulb to go along with this battery; i.e., the choice of a bulb that will draw the maximum power.
First, for a bulb with resistance R, (a) compute the power P that the bulb receives from the battery when
wired directly to the battery with ideal wires. Then, (b) sketch the resulting power as a function of resistance
for resistances in the range R = 0 to 5 Ω. Now, (c) determine the value of resistance that the bulb should
have for it to receive the maximum amount of power from the battery.
Finally, for the general case of a battery of EMF E and internal resistance r, (d) compute what the bulb
resistance should be for it to receive the maximum power from the battery?
Hint: You should find a nice, simple result.
4
Electric field and potential gradient
~
In lecture we learned that the gradient
of the electric
potential V (r) gives the electric field; i.e., E(~x) =
∂
∂
∂
−∇V (~x). (Recall that, here, ∇ ≡ î ∂x + ĵ ∂y + k̂ ∂z .) This means, in practice, that to get any component
of the electric field, we just take the corresponding derivative,
Ex
Ey
Ez
∂
V (x, y, z)
∂x
∂
= − V (x, y, z)
∂y
∂
= − V (x, y, z).
∂z
= −
Often, this provides a much more convenient way to calculate the electric field because one can compute
~
V (x, y, z) without all the use of vector components, as are needed in calculating the electric field E.
As one example of this, (a) do Exercise 23.68 part (b) only.
Note: You already did the first part of this problem on the last problem set.
Also, (b) compute the potential V (z) at a point a distance z along the z-axis above a ring of charge of radius
R and total charge Q which lies in the xy-plane, centered on the origin. (This is the same geometry as the
disc in part (a), just now we deal with a narrow ring, charges arranged on the circumference of a circle. See
Figure 3.) Then, (c) use your result for V (z) to determine the the z-component
of the electric
R field Ez (z)
P
at that same point. Finally, (d) use the traditional method of summing n (kC qn /r2 )r̂n ≡ (kC /r2 )r̂ dq
the contributions to the electric field to determine Ez (z), and (e) compare your results. Which method was
easier?
5
z
y
R
Figure 3: Ring of charge
6
x