Preparing for the Physics GRE
Day 4
Circuits and Optics Daniel T. Citron
dtc65@cornell.edu
Cornell University
3/15/16
Circuits •  Kirchoff’s rules •  Basic circuit elements –  R, L, C –  Units and 9me scales –  Addi9on Rules •  Differen9al equa9on framework •  Common Circuit Types •  AC Circuits –  Inductance –  RLC circuit equa9on and solu9ons •  Other topics –  Logic Gates Kirchoff’s Laws •  Conserva9on of Charge ó No net current flux at any point •  Conserva9on of Energy ó No net voltage over any loop •  Ohm’s law –  Most general: –  More useful for circuits: V = IR V = IZ Basic Circuit Elements •  Remember how they affect voltages Basic Circuit Elements •  Voltage drops ó units –  [R] = [V/I] = JS/C2 –  [C] = C/[V] = C2/J –  [L] = [V/(I/S)] = JS2/C2 •  Time scales –  [RC] = S –  [L/R] = S –  [LC] = S2 •  (Will be useful for circuits with 9me constants) Addi9on Rules •  R: Add in series •  C: Add in parallel •  L: Add in series output voltage. If the angular frequency
input voltage is varied, which of the follow
gives the ratio Vo Vi G as a function of
V
(B) 40
m s
(C) 1 1012
(D) 4
1012
(E) 2
1013
V
m s
V
m s
V
m s
Example (A)
68. The circuit shown
(B)in the figure above consists
of eight resistors, each with resistance R, and
a battery with terminal voltage V and negligible
internal resistance. What is the current flowing
through the battery?
(A)
(B)
(C)
(D)
68. The circuit shown in the figure above consists (E)
of eight resistors, each with resistance R, and
a battery with terminal voltage V and negligible
internal resistance. What is the current flowing
through the battery?
1V
3R
1V
2R
V
R
3V
2R
V
3
R
(C)
(D)
(E)
(B) 40
V
m s
(C) 1 1012
(D) 4
1012
(E) 2
1013
V
m s
V
m s
V
m s
input voltage is varied, which of the fo
gives the ratio Vo Vi G as a functio
Example •  What i(B)
s the voltage drop across each of the resistors in parallel? •  What d(C)oes this tell us about the voltage drop across the two center resistors (D)
•  (Related problem: symmetric resistor cube) 68. The circuit shown in the figure above consists
of eight resistors, each with resistance R, and
a battery with terminal voltage V and negligible
internal resistance. What is the current flowing
through the battery?
1V
(A)
(E)
Differen9al Equa9on Framework •  For a single loop in a circuit, we can use Kirchoff’s loop rule to write an ODE rela9ng voltage drops across each element •  Example: LC Series Circuit 





h]ps://upload.wikimedia.org/wikipedia/commons/b/b2/LC_parallel_simple.svg Differen9al Equa9on Framework •  Voltage drop across capacitor vs 9me: •  Voltage drop across inductor vs 9me: 

•  Combine using loop rule: 



al
s
ws
y the
then be
(A)
(B)
(C)
(D)
(E)
deflected in the
deflected in the
deflected in the
deflected in the
undeflected
+x-direction
x-direction
+y-direction
y-direction
LC Circuit ODE Example 59. For an inductor and capacitor connected in series,
the equation describing the motion of charge is
d 2Q
1
L 2 + Q = 0,
C
dt
where L is the inductance, C is the capacitance,
and Q is the charge. An analogous equation can
be written for a simple harmonic oscillator with
position x, mass m, and spring constant k.
Which of the following correctly lists the
mechanical analogs of L, C, and Q ?
(A)
(B)
(C)
(D)
(E)
L
m
m
k
1/k
x
C
k
1/k
x
1/m
1/k
Q
x
x
m
x
1/m






LR Circuit Example •  (Ignore numerical values for R, L for now) •  What is the voltage drop across R? •  What is the voltage drop across L? 40. In the circuit shown above, the switch S is closed at t = 0.
Which of the following best represents the voltage across the
inductor, as seen on an oscilloscope?
(A)
(B)
(C)
(D)
Time scale of solu9on? (E)
40. In the circuit shown above, the switch S is closed at
RLC Circuits •  Most general solu9on gives us damped oscilla9ons •  Important to pay a]en9on to boundary condi9ons •  Can remove terms based on other circuits Impedance •  Like resistance, but with a phase factor •  Affects signal amplitude Another Example •  What kind of behavior do circuit elements allow? Another Example •  What is the star9ng charge on the capacitor? Geometric Op9cs •  Reflec9on •  Refrac9on -­‐ Snell’s Law •  Mirrors and Lenses –  Thin Lens Equa9on –  Magnifica9on –  Lensmaker’s Formula •  Other topics –  Telescopes –  Apertures Reflec9on •  Angle of incidence equals angle of reflec9on θin = θout  
Snell’s Law •  Relates angle of incidence to angle of refrac9on –  n represents refrac&on index of material –  Can change depending on light wavelength 



Snell’s Law 97. A beam of light has a small wavelength spread
d l about a central wavelength l . The beam
travels in vacuum until it enters a glass plate at an
Suppose
a system
in quantum
stat
99
angle q98.
relative
to the that
normal
to the plate,
as
has
energy
E
.
In
thermal
equilibrium
shown in the figure above. The
i index of refraction
of the glass isexpression
given by n ( l ) . The angular spread
dq of the refracted beam is given by
(A) dq
(B) dq
(C) dq
(D) dq
97. A beam of light has a small wavelength spread
d l about a central wavelength l . The beam
travels in vacuum until it enters a glass plate at an
angle q relative to the normal to the plate, as
(E) dq
Ei e
1
dl
n
Ei / kT
i
dn ( l )
dl
dl
e
Ei / kT
i
represents
which of the following?
1 dl
dl
l dnThe average energy of the system
(A)
(B)
The partition function
sin q d l
(C)
l
sin q Unity
(D) The probability to find the system
E
dn ( l )
tan q energy
dl i
(E)n Thedlentropy of the system
99. A photon strikes an electron of mass m
Thin Lens Equa9on •  This is the only equa9on you need, provided you can interpret it correctly •  O = object distance from lens •  I = image distance from lens •  F = focal point distance from lens Thin Lens – Geeng Signs Right •  Sign conventions (why this is nontrivial)
–  A is where light comes from, B is where light
passes to
–  Note side B is different for mirrors and lenses
Thin Lens – Geeng Signs Right •  Recommend picking on case to memorize –  O > 0 –  F > 0 –  I = ? 74. The figure above shows an object O placed at a distance R
Thin Lens Example 74. The figure above shows an object O placed at a distance R
to the left of a convex spherical mirror that has a radius of
curvature R. Point C is the center of curvature of the mirror.
The image formed by the mirror is at
(A) infinity
(B) a distance R to the left of the mirror and inverted
(C) a distance R to the right of the mirror and upright
R
(D) a distance
to the left of the mirror and inverted
3
R
(E) a distance
to the right of the mirror and upright
3
Thin Lens Example – Ray Diagram  
•  Draw rays to get qualita9ve sense of image –  Rays from object to focus reflect parallel –  Parallel rays from object reflect from focus –  Rays from object to center reflect at equal angle •  Same deal for lenses, but with passing through Example Ray Diagrams -­‐ Lenses h]p://hyperphysics.phy-­‐astr.gsu.edu/%E2%80%8Chbase/geoopt/raydiag.html Magnifica9on •  Magnifica9on is image size rela9ve to object –  Image is “imaginary” if not in real space (I < 0) –  Image is “real” if projected into real space One more example: Mul9ple Lenses •  Treat the image from the
first lens as a virtual object
8.
–  1. Find the image from the
first lens
A positive charge Q is located at a distance L
2. Use
geometry
to find O
above an–
infinite
grounded
conducting plane,
as shown in the figure above. What is the total
foronthe
second lens
charge induced
the plane?
(A) 2Q –  3. Apply lens equation a
(B) Q
second time
(C) 0
(D) Q
(E) 2Q
9. Five positive charges of magnitude q are
arranged symmetrically around the circumference
of a circle of radius r. What is the magnitude of
the electric field at the center of the circle?
( k = 1 4 p!0 )
11. An object is located 40 centimeters from the
first of two thin converging lenses of focal lengths
20 centimeters and 10 centimeters, respectively,
as shown in the figure above. The lenses are
separated by 30 centimeters. The final image
formed by the two-lens system is located
(A)
(B)
(C)
(D)
(E)
5.0 cm to the right of the second lens
13.3 cm to the right of the second lens
infinitely far to the right of the second lens
13.3 cm to the left of the second lens
100 cm to the left of the second lens
Mul9ple lenses Q is located at a distance L
ounded conducting plane,
re above. What is the total
he plane?
11. An object is located 40 centimeters from the
first of two thin converging lenses of focal lengths
20 centimeters and 10 centimeters, respectively,
as shown in the figure above. The lenses are
separated by 30 centimeters. The final image
formed by the two-lens system is located
(A)
(B)
(C)
(D)
(E)
5.0 cm to the right of the second lens
13.3 cm to the right of the second lens
infinitely far to the right of the second lens
13.3 cm to the left of the second lens
100 cm to the left of the second lens
•  First lens: O = 40 cm, F = 20 cm, I = ?
•  Second lens: F = 10 cm, O = ?, I = ?
s of magnitude q are
ally around the circumference
r. What is the magnitude of
he center of the circle?
Lensmaker’s formula 14. Two experimental techniques determine the
mass of an object to be 11 1 kg and 10 2 kg.
These two measurements can be combined to
give a weighted average. The uncertainty of the
weighted average is equal to which of the
following?
•  Find the focal point, given the radii of the two faces of the lens (A)
(B)
(C)
1
kg
2
2
kg
5
2
kg
3
(D)
2 kg
(E)
5 kg
15. If the five lenses shown below are made of the
same material, which lens has the shortest positive
focal length?
(A)
(B)
(C)
(D)
(E)
h]p://hyperphysics.phy-­‐astr.gsu.edu/hbase/hframe.html Telescopes •  Telescope angular magnification
–  fE: Focal length of eyepiece
–  fO: Focal length of objective
–  Note: The two lenses share a focal point