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8.1
Electricity and Magnetism II
Griffiths Chapter 8 Conservation Laws
Clicker Questions
8.3
Local conservation of electric charge is expressed mathematically by:
¶r
= -Ñ × J
¶t
where J is “current density”
J = ρv has units of (charge/sec)/m2
We are trying to come up with a “conservation of energy” expression:
¶(energy density)
= -Ñ ×(something)
¶t
What sort of beast is this “something” ?
- Is it a scalar, vector, something else?
- How would you interpret it, what words would you use to try to
describe it?
- What are its UNITS?
A) J
E) Other!
B) J/s
C) J/m2
D) J/(s m2)
8.4
CT 29.18
A + and - charge are held a distance R apart and released.
The two particles accelerate toward each other as a result of the Coulomb
attraction.
As the particles approach each other, the energy contained in the electric
field surrounding the two charges...
+Q
-Q
A: increases
B: decreases
C: stays the same
D: ??
8.5
Uem (outside V) = EM energy in field outside volume V
S
V
(
)
d
U em outside V = ?
dt
ò
A)
Ñ ×S dt
B)
outside V
C)
ò S× da
S
ò
Ñ ×S dt
inside V
D) - ò S× da
S
E) Ñ ×S
8.6
Can the force from a magnetic field
do work on a charge?
B
v
+
A) Yes, sure
B) Yes, but only if other forces are
moving the charge
C) Yes, but only if no other forces are
moving the charge
D) No, never
8.7
A steady current I flows along a long straight wire.
E=?
I
The parallel component of the E-field, E|| , just outside the
surface …
A) must be zero
B) must be non-zero
C) might be zero or non-zero depending on details
8.8
FAB = F_onA_fromB
FBA = F_onB_fromA
B
A
FAB = FBA
Newton’s 3rd Law is equivalent to..
A) Conservation of energy
B) Conservation of linear momentum
C) Conservation of angular momentum
D) None of these. NIII is a separate law of physics.
8.9
Two short lengths of wire carry currents as shown.
(The current is supplied by discharging a capacitor.)
The diagram shows the direction of the force on wire 1
due to wire 2.
What is the direction of the force on wire 2 due
to wire 1?
A)
B)
C)
D)
E) None of these
I2
B2
I1
Fon 1 from 2
8.10
B
Feynman’s Paradox:
Two charged balls are attached to a horizontal ring
that can rotate about a vertical axis without friction.
A solenoid with current I is on the axis. Initially,
everything is at rest.
The current in the solenoid is turned off.
What happens to the charges?
A) They remain at rest
B) They rotate CW.
C) They rotate CCW.
Does this device violate Conservation of
Angular Momentum?
A) Yes
B) No
C) Neither, Cons of Ang Mom does not apply
in this case.
n, I
Q +
R
B
+
Q
C
I
8.11
Two charged capacitors discharge through wires. The magnetic
field forces are not equal and opposite. After the discharge the
momentum of the capacitors is to the lower right. What’s the
resolution of this Newton’s Third Law paradox?
I2
Answer:
Momentum in the EM radiation.
When all momentum is included, the
center-of-mass of the system
remains stationary.
B1
Fon 2 from 1
B2
I1
Fon 1 from 2
8.12
¶
¶ e0 2
1 2
uq = - ( E +
B ) - Ñ×S
¶t
¶t 2
2 m0
(Where S=E x B /μ0)
How do you interpret this equation? In particular:
Does the – sign on the first term on the right seem OK?
A) Yup B) It’s disconcerting, did we make a mistake? C) ??
¶
e0 2 1 2
(uq + E +
B ) = -Ñ × S
¶t
2
2 m0
¶
(uq + uEM ) = -Ñ × S
¶t
8.13
d
(uq + uEM )dt = - òòò Ñ × S dt
òòò
dt
Would you interpret S as the
A) OUTFLOW of energy/area/time or
B) INFLOW of energy/area/time
C) OUTFLOW of energy/volume/time
D) INFLOW of energy/volume/time
E) ???
(Where S=E x B /μ0)
8.14
d
(uq + uEM )dt = - òòò Ñ × S dt
òòò
dt
= - òò S × da
Would you interpret S as the
A) OUTFLOW of energy/area/time or
B) INFLOW of energy/area/time
C) OUTFLOW of energy/volume/time
D) INFLOW of energy/volume/time
E) ???
(Where S=E x B /μ0)
8.15
What are the SI metric units of the combination of fields
E x B /μ0 ?
A) J
E) Other!
B) J/s
C) J/m2
D) J/(m2 s)
8.16
The fields can change the total energies of charged
particles by:
A)
B)
C)
D)
Doing work on the particles
Changing the potential energies only
Changing the kinetic energies only
Applying forces only perpendicular to the particle
motion.
E) None of the above.
8.17
Consider the cylindrical volume of space bounded by the capacitor
plates. Compute S = E x B /μ0 at the outside (cylindrical, curved)
surface of that volume.
Which WAY does it point?
A) Always inward B) Always outward
C) ???
I
I
8.18
Given a quantity with units of (Joules/m3),
you can convert it to a quantity with units of
Joules/(m2 * seconds) by multiplying by:
A)
B)
C)
D)
E)
a length
a frequency
a speed
an acceleration
None of the above
8.19
Consider a current I flowing through a cylindrical resistor of length
L and radius a with voltage V applied. What is the E field inside
the resistor?
L
a
J
z
ϕ ×
A.
B.
C.
D.
E.
(V/L) z-hat
(V/L) ϕ-hat
(V/L) s-hat
(Vs/a2) z-hat
None of the above
s
8.20
Consider a current I flowing through a cylindrical resistor of length
L and radius a with voltage V applied. What is the B field inside
the resistor?
J
J
z
ϕ ×
A.
B.
C.
D.
E.
(Iμ0/2πs) ϕ-hat
(Iμ0s/2πa2) ϕ-hat
(Iμ0/2πa) ϕ-hat
-(Iμ0/2πa) ϕ-hat
None of the above
s
8.21
Consider a current I flowing through a cylindrical resistor of length
L and radius a with voltage V applied.
What is the direction of the S vector on the outer curved surface of
L
the resistor?
a
J
J
z
ϕ ×
A.
B.
C.
D.
±ϕ-hat
± s-hat
± z-hat
???
And, is it + or -?
s
8.22
The energies stored in the electric and magnetic fields are:
A) individually conserved for both E and B, and cannot
change.
B) conserved only if you sum the E and B energies
together.
C) are not conserved at all.
D) ???
8.23
The fields can change the total momentum of
charged particles by:
A)
B)
C)
D)
E)
Fields cannot change particle momentum
Applying a net force to the particles
Changing only the potential energy
Only if they do net work on the particles.
None of the above.
8.24
Consider two point charges, each moving with constant velocity
v, charge 1 along the +x axis and charge 2 along the +y axis.
They are equidistant from the origin.
What is the direction of the magnetic force on charge 1 from
charge 2? (You’ll need to sketch this! Don’t do it in your head!)
A.
B.
C.
D.
E.
+x
+y
+z
More than one of the above
None of the above
8.25
Consider two point charges, each moving with constant velocity v,
charge 1 along the +x axis and charge 2 along the +y axis.
They are equidistant from the origin.
What is the direction of the magnetic force on charge 2 from
charge 1? (You’ll need to sketch this! Don’t do it in your head!)
A. Equal to the answer of the previous question
B. Equal but opposite to the answer of the previous question
C. Something different than either of the above.
8.26
We seek a local conservation law that relates
the time change in momentum density (units
of momentum/m3), to the divergence of a
current density, “T”, with units of:
A) Newtons/m2
B) kg*m/(m2*second2)
C) Joules/m3
D) More than one of the above
E) None of the above
8.27
Momentum in the fields:
pEM / volume = m0e 0 S
Consider a charged capacitor placed in a uniform B field
in the +y direction
x
y
------------------
+++++++++++++++++++
Which way does the stored field
Momentum in this system point?
A) +/- x
B) +/- y
C) +/- z
D) Zero!
E) Other/???
z
8.28
Momentum in the fields:
pEM / volume = m0e 0 S
Now “short out” this capacitor with a small wire.
As the current flows, (while the capacitor is discharging)…
x
y
------------------
+++++++++++++++++++
which way does the magnetic force
push the wire (and thus, the system)?
A) +/- x
B) +/- y
C) +/- z
D) Zero!
E) Other/???
Is your answer consistent with
“conservation of momentum”?
z
8.29
What units should a momentum density have?
A.
B.
C.
D.
E.
N s/m3
J s/m3
kg/(s m2)
More than one of the above
None of the above
8.30
What units should a momentum flux density have?
A.
B.
C.
D.
E.
N/m3
N/m2
kg/(s m)
More than one of the above
None of the above
8.31
The Maxwell stress tensor is given by:
Tij = e 0 (E i E j - d ij E ) +
1
2
2
1
m0
(Bi B j - 12 d ij B 2 )
What is the E field part of the Tzx term?
A.
B.
C.
D.
E.
ε0(EzEx-½(Ex2+Ez2))
ε0(EzEx-½Ey2)
ε0(EzEx-½(Ex2+Ey2+Ez2))
ε0(EzEx)
None of the above
8.32
The Maxwell stress tensor is given by:
Tij = e 0 (E i E j - d ij E ) +
1
2
2
1
m0
(Bi B j - 12 d ij B 2 )
What is the E field part of the Tzz term?
½
A.
B.
C.
D.
E.
ε0(Ez2-(Ex2+Ey2))/2
ε0(Ez2-½(Ex2 +Ey2)
-ε0(Ex2+Ey2)
ε0(Ez2)
None of the above
8.33
Given a general Maxwell Stress tensor with all elements non-zero,
what is the net force on a small isolated area element
da = (dx dy) z ?
Txx
Txy
Txz
Tyx
Tzx
Tyy
Tzy
Tyz
Tzz
A. Txz dx dy z
B. Tyz dx dy z
C. Txz dx dy z
D. (Txz x + Tyz y + Tzz z) dx dy
E. Something else!
F = òò T × dA
8.34
An infinite plane of surface charge σ lies in the xz plane. In the
region y > 0, which element(s) of the stress tensor is(are) nonzero?
z
σ
y
x
A.
B.
C.
D.
E.
Tyy
Txx, Tyy, Tzz
Txy,Tyx,Tyy,Tyz,Tzy
Txy,Tyx,Tyy,Tyz,Tzy,Txx,Tyy
None of the above
8.35
Suppose we have a plane of surface charge σ with electric field Ea
above the plane and Eb below the plane.
Ea
++++++++++++++++++++++++++++++ σ
Eb
What is the net force per area on the plane?
A.
B.
C.
D.
E.
(Ea + Eb) σ
(Ea - Eb) σ
(Ea + Eb) σ/2
(Ea - Eb) σ/2
None of the above