Download GRE Review: Lab Methods

GRE Physics Review
GRE Review: Lab Methods
Officially, this section contains: “data and error analysis, electronics, instrumentation, radiation
detection, counting statistics, interaction of charged particles with matter, lasers and optical
interferometers, dimensional analysis, fundamental applications of probability and statistics”
After looking through the practice tests, the key things to review are how to calculate errors, how
lasers work, how to read data charts or instrumentation output, and how to calculate I, R, and V
given a circuit diagram (i.e. Kirchoff’s Law, series vs. parallel, capacitance, etc.).
A good textbook to skim over for the basics is Giancoli “Physics for Scientists and Engineers:
Volume II”.
(And I noticed a few practice problems involving the Michelson Interferometer – check out
http://scienceworld.wolfram.com/physics/MichelsonInterferometer.html .
B. CAPACITORS
The Capacitance of two oppositely charged conductors in a uniform dielectric medium is
C 
Q
V0
Units: Farads = coul

where Q = the total charge in either conductor
V0 = the potential difference between the two conductors.
EXAMPLE:
Capacitance of the parallel-plate capacitor:
–p - - - - - - s
z=d
conductor
E
surface
+ps + + + + + + +z=0
s
az

 is the permittivity of the homogeneous dielectric
D  s.az
E
On lower plate:
Dn  Dz  s
Dn is the normal value of D.
On upper plate:
Dn = –Dz
V0 = The potential difference


lower
upper
E.dL
s
d
dz  s
d 

s
Q
C 

V0
d

1

0
GRE Physics Review
s d

considering conductor planes of area S are of linear dimensions much greater than d.
Total energy stored in the capacitor:
1
1 s d s
WE 
E 2dv 
dz ds
2 vol
2 0 0 2
Q = sS and V0 =

WE 
 
1
1
1 Q2
CV 2  QV0 
2
2
2 C
Multiple dielectric capacitors
conducting
plates
Area S
d2
d
2
d1
1
2
c2

1
A parallel-plate capacitor containing two dielectrics with
the dielectric interface parallel to the conducting plates;
C = 1/{(d1/1S) + (d2/2S)}.
1
C 
 1   1 



 C1   C2 
where C1 
1S
d1
C2 
1S
d2
V0 = A potential difference between the plates
= E1d1 + E2d2
 
V
E1 = 0   1  d2
d1  2 
s1 = The surface charge density = D1 = 1E1
V0
 D2
=
 d1   d2 
 

 1   2 
C 
 .S
Q
1
1
 s


V0
V0
 d1   d2   1   1 


 


 1s   2 s   C1   C2 
C. CURRENT AND RESISTANCE
DEFINITIONS
Current:
2
i 
dq
amperes
dt
c1
GRE Physics Review
where
i = Electric Current
q = Net Charge
t = Time
Current Density and Current:
j
where
i
Amperes/m2
A
j = Current Density
i = Current
A = Cross-sectional Area
Mean Drift Speed:
j
ne
where v0 = Mean Drift Speed
j = Current Density
n = Number of atoms per unit volume.
vD 
Resistance:
V
Ohms ()
i
where R = Resistance
V = Potential Difference
i = Current
R
Resistivity:
E
Ohm – meters ( m)
j
where  = Resistivity
E = Electric Field
j = Current Density

Power:
P  VI  I 2R 
V2
Watts (w)
R
where P = Power
I = Current
V = Potential Difference
R = Resistance
D. CIRCUITS
Electromotive Force, EMF()

3
dw
dq
GRE Physics Review
where  = Electromotive Force
w = Work done on Charge
q = Electric Charge
Current in a Simple Circuit
i 

R
where i = Current
 = Electromotive Force
R = Resistance
Resistances:
+
–
RTotal = (R1+R2+R3)  (in series)
R1
R2
–
+
R3
1
RTotal
 1
1
1
 


 R1 R2 R3

 (in parallel)

The Loop Theorem
V1 + V2 + V3 ……. = 0
For a complete circuit loop
EXAMPLE
i
a
r
i
b
Then
4
l

i
Simple circuit with resistor
Vab =  – iR = + ir
 – iR – ir = 0
R
GRE Physics Review
NOTE: If a resistor is traversed in the direction of the current, the voltage change is represented as
a voltage drop, –iR. A change in voltage while traversing the EMF (or battery) in the direction of the
EMF is a voltage rise +.
Circuit With Several Loops
i
0
n
n
EXAMPLE
i1 + i2 + i3 = 0
1
2
B
A
C
i1
i3
R1
R3
F
i2
R2
D
E
Multiloop circuit
RC CIRCUITS (RESISTORS AND CAPACITORS)
RC charging and discharging
Differential Equations
dq q

(Charging)
dt C
dq q (Discharging)
OR

dt C
R
i
a
+

–
s R
b
C
An RC circuit
Charge in the Capacitor

q  C   1  e
q  C   e
5
t
 RC
t
 RC

(Charging)
(Discharging)
GRE Physics Review
Charge in the Resistor
 t
i    e RC
R
 t
i  –   e RC
R
(Charging)
(Discharging)
where e = 2.71828 (Exponential Constant)
KIRCHOFF’S CURRENT LAW
The algebraic sum of all currents entering a node equals the algebraic sum of all current
leaving it.
N
i
n
0
n 1
KIRCHOFF’S VOLTAGE LAW (SAME AS LOOP THEOREM)
The algebraic sum of all voltages around a closed loop is zero.
THEVENIN’S THEOREM
In any linear network, it is possible to replace everything except the load resistor by an
equivalent circuit containing only a single voltage source in series with a resistor (Rth Thevenin
resistance), where the response measured at the load resistor will not be affected.
Linear
Active
Network
x
y
+
x
–
y
Procedures to Find Thevenin Equivalent:
1)
Solve for the open circuit voltage Voc across the output terminals.
Voc = Vth
2)
Place this voltage Voc in series with the Thevenin resistance which is the resistance
across the terminals found by setting all independent voltage and current sources to
zero. (i.e., short circuits and open circuits, respectively.)
RLC CIRCUITS AND OSCILLATIONS
These oscillations are analogous to, and mathematically identical to, the case of mechanical
harmonic motion in its various forms. (AC current is sinusoidal.)
SIMPLE RL AND RC CIRCUITS
+
Source Free RL Circuit
i(t)
R
–
+
L
–
6
VL
GRE Physics Review
Properties: Assume initially i(0) = I0.
di
0
dt
A)
 R   L  Ri  L
B)
i(t )  I0e
C)
Power dissipated in the resistor =
 Rt
L
 I0e
PR  i 2R  I02 Re
D)
t
2 Rt
L

,  = time constant =
L
R
.
Total energy in terms of heat in the resistor =
WR = 12 LI02.
i(t)
Source Free RC Circuit
+
C
R
v(t)
–
Properties: Assume initially (0) = V0
dv 
C

0.
A)
dt
R
B)
(t )  (0) et / RC  V0 et / RC .
C)
1
C

t

i()d  i(t )R  0
i(t )  i(0)e
t
RC
THE RLC CIRCUITS
Parallel RLC Circuit (source free)
V
Circuit Diagram:
i
R
L
C
KCL equation for parallel RLC circuit:
 1

R L

t
t
dt – i(t0 )  C
d
 0;
dt
and the corresponding linear, second-order homogeneous differential equation is
d 2  1 dv 
C 2 
 0
T dt
L
dt
General Solution:
V  A1eS1t  A2eS2t ;
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GRE Physics Review
where
2
S1,2 
1
1
 1 
 


2RC
LC
 2RC 
S1,2    2  02 ;
or
where  = exponential damping coefficient never frequency =
and
0 = resonant frequency =
1
2RC
1
LC
critically damped
v(t)
underdamped
overdamped
t(a)
COMPLETE RESPONSE OF RLC CIRCUIT
The general equation of a complete response of a second order system in terms of voltage for
an RLC circuit is given by.
(t) = Vf + AeS1t  BeS2t
forced
response
Natural response
(i.e., constant for DC excitation)
NOTE: A and B can be obtained by
1)
Substituting  at t = 0+
2)
Taking the derivative of the response, i.e.,
dv
 0  S1 AeS1t  S2 BeS2t
dt
dv
where
at t = 0+ is known.
dt
D. THE LASER
Light Amplification by Stimulated Emission of Radiation
Properties
1. The light is coherent. (Waves are in phase)
2. Light is nearly Monochromatic. (One wavelength)
3. Minimal divergence
4. Highest intensity of any light source.
Many atoms have excited energy levels which have relatively long life-times. (10-3s instead of
10 s). These levels are known as metastable.
-8
8
GRE Physics Review
Through a process known as population inversion, the majority of an assembly of atoms is
brought to an excited state.
Population inversion can be accomplished through a process known as optical pumping, where
atoms of a specific substance, such as ruby, are exposed to a given wavelength of light. This
wavelength is enough to excite the ruby atoms just above metastable level. The atoms rapidly lose
energy and fall to the metastable level.
Ruby
2.25 eV
Transition (energy loss
without radiation)
1.79 eV
Laser Transmission
Ground
State
Optical
Pumping
Once population inversion has been obtained, induced emission can occur from photons
dropping from an excited metastable state to ground state. The photons have a wavelength equal to
the wavelength of photons produced by each individual atom. The radiated light waves will be
exactly in phase with the incident waves, resulting in an enhanced beam of coherent light. Hence the
familiar Laser effect.
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