Transcript Document

PHYS 222 SI Exam Review
3/31/2013
Answer: D
Answer: D,D
Answer: D,C
What to do to prepare
• Review all clicker questions, but more
importantly know WHY
• Review quizzes
• Make sure you know what all the equations
do, and when to use them
SI Leader Secrets!
Extra problems?
Visit the website below to get past exams all the
way back to 2001!!
(Note: the link below has stuff that you wouldn’t
otherwise see)
http://course.physastro.iastate.edu/phys222/ex
ams/ExamArchive222/exams/
𝑄 𝑡 =𝑄 ∞ 1−
𝑄 𝑡 =𝑄 0
𝜏 = 𝑅𝐶
I 𝑡 =𝐼 0
𝑡
−
𝑒 𝜏
𝑡
−
𝑒 𝜏
𝑡
−
𝑒 𝜏
• These are all equations used for an RC circuit.
𝑄 𝑡 =𝑄 ∞ 1−
I 𝑡 =𝐼 0
𝑡
−
𝑒 𝜏,
𝑡
−
𝑒 𝜏
𝑉
𝐼 0 =
𝑅
• Used to find the charge Q on the capacitor in
an RC circuit that initially has no charge and is
slowly brought to a maximum charge 𝑄 ∞ .
• What is 𝜏?
𝜏 = 𝑅𝐶
• 𝑄 ∞ = 𝐶𝑉
𝑄 𝑡
I 𝑡
𝑡
−
= 𝑄 0 𝑒 𝜏, 𝑄
𝑡
−
= 𝐼 0 𝑒 𝜏,
0 = 𝐶𝑉
• Used to find the charge Q on the capacitor in an RC circuit
that initially has charge Q(0) and has been disconnected
from the power source.
• I(t) is used to find the current in the resulting circuit. As
before, 𝜏 = 𝑅𝐶
𝑭 = 𝑞(𝑬 + 𝒗 × 𝑩)
• Used to find the force on a point charge of
charge q in an electric field E and magnetic
field B.
• Notice that the magnetic force is 𝒗 × 𝑩, and
only exists if the charge is moving.
𝑑𝑭 = 𝐼𝒅𝒍 × 𝑩
• This is the differential form of the magnetic
force on a length of wire carrying current.
• Probably more useful in this form:
– 𝑭 = 𝐼𝑳 × 𝑩
• Note that if the wire and B field are pointing in
the same direction, the force is zero.
Φ𝐵 = ∫ 𝑩 ⋅ 𝒅𝑨
Φ𝐵 is the magnetic flux through a closed
surface.
Ex: A uniform B field of 5 T goes through a
circular loop of wire of radius 10 m, What is the
magnetic flux?
Ans: Φ𝐵 = 5 ⋅ 𝜋 ⋅ 102 = 500𝜋 T ⋅ m2
𝑚𝑣
𝑅=
𝑞𝐵
• Here, a charge of magnitude q and mass m is
acted on by a constant B field. As a result, the
charge moves in a circle of radius R and its
tangential speed is v.
𝝁 = 𝐼𝑨
• This is the equation for the magnetic dipole
(𝝁) of a loop of current.
• 𝝁 is a vector
• As an example,
if the radius is
4 m and I=2,
then 𝝁 = 𝟑𝟐 up
𝝉=𝝁×𝑩
• This gives the torque on a magnetic dipole by
a magnetic field.
• Note that torque is zero if the magnetic dipole
and the B field point in the same direction.
𝑈 = −𝝁 ⋅ 𝑩
• This gives the potential energy of a magnetic
dipole in a magnetic field.
𝜇0 𝑞𝒗 × 𝒓
𝑩=
4𝜋 𝑟 2
• The equation for the magnetic field produced
by a moving charge q at a speed v.
• 𝑟 is just the distance away from the moving
charge.
• 𝒗 × 𝒓 just means to use the right hand rule to
determine which direction the magnetic field
points.
𝜇0 𝐼𝒅𝒍 × 𝒓
𝒅𝑩 =
4𝜋 𝑟 2
• Same equation as before, except that instead
of a single point charge moving, we have a
current I.
• This equation is probably easier to use in its
linear, non-differential form 𝑩 =
𝜇0 𝐼𝑳×𝒓
,
2
4𝜋 𝑟
• 𝐝𝐥 × 𝒓 just means to use the right hand rule to
determine which direction the magnetic field
points.
Right-hand rule
𝜇0 𝐼
𝑩=
2𝜋r
• This is the magnetic field a distance r away
from an infinite straight wire carrying current
I.
• The direction of the field is given by the right
hand rule.
𝐹 𝜇0 𝐼𝐼 ′
=
𝐿
2𝜋𝑟
• This gives the force between two parallel
wires. One wire carries current I, the other
wire carries current I’.
• If the currents are pointing in the same
direction, the force is attractive. If they are
opposite, the force is repulsive.
• Is the force attractive or repulsive?
• Answer: attractive.
1
2
𝑐 =
𝜇 0 𝜖0
• I doubt you’d find a practical use for this
equation in exam 2, because it really only says
that the speed of light squared is equal to the
inverse of the products of two constants. Cool,
but not really something testable.
𝐵𝑥 =
𝜇0
2
2
𝑥
2
𝑁𝐼𝑎
+
3
2
𝑎 2
• Let’s say you have a wire bent in a circle of radius
a (in the picture it’s shown as R), with N turns.
This equation gives the B field at the center of the
circle a distance x above the center (if the circle is
in the x-y plane, the variable x is the z
coordinate).
• The direction of the B field is given by the right
hand rule, as discussed earlier.
𝜇0 𝑁𝐼
𝐵𝑥 =
2𝑎
• This equation is really just a special case of the
previous one. This is the B field at the center
of the circle, in the plane.
Question:
• In the picture does the B field produced by the
current point into the page or out of the
page?
Question:
• In the picture does the B field produced by the
current point into the page or out of the
page?
• Answer: Into the page.
𝐵 = 𝜇0 𝑛𝐼
• This is the equation for the field inside of a
solenoid.
• Note that it is a uniform field (i.e. everywhere
inside of the solenoid it’s the same).
• Lowercase n is the turns per length.
𝑩 ⋅ 𝒅𝒍 = 𝜇0 𝐼𝒆𝒏𝒄
• This is sometimes known as Ampere’s law.
• Can be used to derive many magnetic fields,
𝜇0 𝐼
for example this one: 𝑩 =
. (Field away
2𝜋r
from any infinite straight wire)
𝑑Φ𝐵
𝜖 = −𝑁
𝑑𝑡
• This equation is known by many names,
including Faraday’s Law and Lenz’s Law,
depending on who you talk to.
• Basically it says that a current loop without a
voltage or current source can have an induced
voltage if there’s a changing magnetic flux
inside the loop.
• Note that the direction of the EMF is
OPPOSITE the change in flux.
𝜖=
𝒗 × 𝑩 ⋅ 𝒅𝒍
• This is just another way of expressing the EMF.
• Recall q(𝒗 × 𝑩) is the magnetic force, so here
we’re sort of (there’s no q up there) saying
that the path integral of the magnetic force is
equal to the emf.
𝑑Φ𝐵
𝑬 ⋅ 𝒅𝒍 = −
𝑑𝑡
• This just says that an induced E field is what
causes the induced EMF seen in the earlier
𝑑Φ𝐵
equation: 𝜖 = −𝑁
𝑑𝑡
• Notice how there’s an N missing in the
equation up top. That’s because ΦB includes
the N already, whereas in the bottom
equation it doesn’t.
𝑩 ⋅ 𝒅𝒍 = 𝜇0 𝑖𝐶 + 𝑖𝐷
𝑒𝑛𝑐
• This is a copy of an equation we saw earlier,
except that it includes the displacement
current.
• What is the displacement current? The
equation is on the next page, but the physical
meaning is that it’s not a true current, but
rather a mathematical construction to deal
with changes in electric flux.
𝑑Φ𝐸
𝑖𝐷 = 𝜖
𝑑𝑡
• Here’s the equation for displacement current.
𝑄𝑒𝑛𝑐
𝑬 ⋅ 𝒅𝑨 =
𝜖0
• One of the so-called “Maxwell’s Equations”
• Also known as Gauss’s law.
• Used to calculate the E fields for many
common charge shapes, such as spheres and
cylinders. (Theoretically can be used for
complicated ones too, but that requires fancy
mathematical software)
𝑩 ⋅ 𝒅𝑨 = 0
• One of the so-called “Maxwell’s Equations”
• Says that the magnetic flux through a closed,
3-D surface is always zero.
𝑑Φ𝐵
𝑬 ⋅ 𝒅𝒍 = −
𝑑𝑡
• One of the so-called “Maxwell’s Equations”
• This is basically the same as the induced EMF
equation.
𝑩 ⋅ 𝒅𝒍 = 𝜇0
𝑑Φ𝐸
𝑖 𝐶 + 𝜖0
𝑑𝑡
𝑒𝑛𝑐𝑙
• One of the so-called “Maxwell’s Equations”
• This equation basically appears twice on the
equation sheet.
𝑑𝑖2
𝜖1 = −𝑀
𝑑𝑡
• If you have two loops of current with mutual
inductance M, and a current i2 is going
through one of them, then an emf 𝜖1 (voltage)
is produced through the other one, which
excites a current in that one.
𝑑𝑖1
𝜖2 = −𝑀
𝑑𝑡
• If you have two loops of current with mutual
inductance M, and a current i1 is going
through one of them, then an emf 𝜖2 (voltage)
is produced through the other one, which
excites a current in that one.
• Basically the same idea as the last equation.
𝑁1 Φ𝐵2 𝑁2 Φ𝐵2
𝑀=
=
𝑖1
𝑖1
• The definition of mutual inductance M. Use
the side of the equation that is relevant.
• Note that although it appears that M depends
on current i, the fact of the matter is that M
never depends on i because the i in the
numerator cancels with the i in the
denominator.
• There is an i in the numerator because flux
depends on B, and B depends on i.
𝑑𝑖
𝜖 = −𝐿
𝑑𝑡
• This is the induced emf across an inductor.
Note that the induced emf occurs opposite
the change in current.
𝑁Φ𝐵
𝐿=
𝑖
• Definition of self-inductance L.
𝑖 = 𝐼∞ (1 −
𝑡
−
𝑒 𝜏)
• Current across an inductor in an LR circuit
when you just start flowing current in the
circuit.
𝑖=
𝑡
−
𝐼0 𝑒 𝜏
• Current across an inductor in an LR circuit
when you just stop flowing current in the
circuit.
𝐿
𝜏=
𝑅
• The time constant in LR circuits.
2
𝐿 = 𝜇0 𝑛 𝐿𝐴
• Self inductance of a solenoid of n turns per
length, of length L, and cross sectional area A.
1 2
𝑈 = 𝐿𝐼
2
• Energy contained within an inductor (i.e.
solenoid).
2
𝐵
𝑢=
2𝜇0
• Energy density for a point with a magnetic
field B.
• Not really covered in lecture as far as I recall
𝑞 = 𝑄0 cos(𝜔𝑡 + 𝜙)
𝜔=
1
𝐿𝐶
• The equation that tells you the charge q on a
capacitor in an LC circuit.
• Notice that it’s oscillatory- Simple Harmonic
Motion!
• The frequency 𝜔 depends on L and C
𝑞=
𝑅
− 𝑡
𝑄0 𝑒 2𝐿
′
𝜔 =
′
cos(𝜔 𝑡 + 𝜙)
1
√(
𝐿𝐶
−
𝑅2
)
2
4𝐿
• This is an RLC circuit.
• The idea is similar to the LC circuit, except that
now the charge q is also exponentially
decreasing as it oscillates.
• The oscillation frequency depends on L, C, and
R.
Past exam problems….
Answer: C
Answer: A
Answer: D
Answer: B
Answers: D, B
Answers: A,D
Answer: D
Answer: B, D
Answer: D
Answers: C, B
Answers: C, B
Answers: E, B
Answers: A,D
Answer: E
C, D, E
D, B
D, D
D