Download Magnetic Field

Physical Sciences 3: All Concepts
TF: Widagdo Setiawan, [email protected]
Notes:
 I will put more stuff in appropriate sections (not at the bottom) as the lecture progresses. The
numbering and ordering will change considerably
 Do not attempt to memorize the concepts. Instead try to understand them.
General
1) Vector has direction (electric field, magnetic field, force, torque)
a) You have to specify the direction on all forces that you calculate. Direction could be: up, down,
left, right, out of the page, into the page, 𝑥̂, 𝑦̂, 𝑧̂ , or anything that describe a direction
2) Scalar does not have direction (potential, potential energy, energy, pressure, temperature, density,
concentration). Never specify direction on scalar quantities. It does not make any sense.
3) Field: something that exist at every point in space (electric field, magnetic field, gravitational field)
4) Integrals (you can either blindly memorize these or put them on the exam formula sheet):
a) ∫ 𝑑𝑟 = 𝑟
1
b) ∫ 𝑟 𝑑𝑟 = 2 𝑟 2
1
c) ∫ 𝑑𝑟 = ln(𝑟)
𝑟
1
1
d) ∫ 2 𝑑𝑟 = −
𝑟
𝑟
1
e) ∫ 𝑥+𝑎 𝑑𝑥 = ln(𝑥 + 𝑎)
f)
∫
g) ∫
1
√𝑥 2 +𝑦2
𝑥
√𝑥 2 +𝑦2
𝑑𝑥 = ln(𝑥 + √𝑥 2 + 𝑦 2 )
𝑑𝑥 = √𝑥 2 + 𝑦 2
5) Approximations for 𝑥 ≪ 𝑎 (you can blindly memorize these also):
a)
b)
1
1
𝑥
𝑥2
≈
−
+
+⋯
2
𝑎+𝑥
𝑎
𝑎
𝑎3
1
1
2𝑥
3𝑥 2
≈
−
+
+
2
2
3
(𝑎+𝑥)
𝑎
𝑎
𝑎4
⋯
6) Trig identities
𝜔1 +𝜔2
𝜔 −𝜔
𝑡) cos ( 1 2 2 𝑡)
2
a) sin(𝜔1 𝑡) + sin(𝜔2 𝑡) = 2 sin (
b) sin(𝑎 ± 𝑏) = sin 𝑎 cos 𝑏 ± cos 𝑎 sin 𝑏
Mechanics
1
7) Kinetic Energy: 𝐸𝑘 = 2 𝑚𝑣 2
8) Mechanical Energy: 𝑈 = 𝐸𝑘 + 𝐸𝑣 where 𝐸𝑣 is the potential energy
9) Conservation of mechanical energy: Δ𝑈 = 0
10) Potential Energy (𝑈𝑔𝑟𝑎𝑣 , 𝑈𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 ) vs Potential (𝜙𝑔𝑟𝑎𝑣 (𝑟⃑), 𝑉(𝑟⃑)):
a) Potential is a field. There is a potential for every single point in space.
b) Potential energy is a number associated with an object configuration. A ball at some height has a
gravitational potential energy.
c) Absolute value of potential is not measureable and thus has no physical meaning. You can
define any point that you want (but just one point) as the zero of the potential.
d) Since 𝑈𝑔𝑟𝑎𝑣 = 𝑚𝜙𝑔𝑟𝑎𝑣 and 𝑈𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 = 𝑄𝑉𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 and since the absolute value of potential is
not measureable, the absolute value of potential energy is also not measureable.1
𝑓𝑖𝑛𝑎𝑙
11) Mechanical work: 𝑊 = ∫𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝐹⃑ . 𝑑𝑟⃑
a) Be really careful with work. You have to define who is performing the work
b) Example: a human lift a ball from the floor (𝑦 = 0) to height (𝑦 = 𝑦0 )
𝑦
i) 𝑊𝑔𝑟𝑎𝑣 = ∫ 𝐹⃑𝑔𝑟𝑎𝑣 . 𝑑𝑟⃑ = ∫ 0 (−𝑚𝑔)𝑑𝑦 = −𝑚𝑔𝑦0 (negative because the force on the ball by
0
gravity points down)
𝑦
ii) 𝑊ℎ𝑢𝑚𝑎𝑛 = ∫ 𝐹⃑ℎ𝑢𝑚𝑎𝑛 . 𝑑𝑟⃑ = ∫0 0 𝑚𝑔 𝑑𝑦 = 𝑚𝑔𝑦0 (positive because the force from your
hand on the ball points up)
iii) 𝑊𝑔𝑟𝑎𝑣 = −𝑊ℎ𝑢𝑚𝑎𝑛
c) On exam, you should always specify which work you are calculating because the answer could
differ by a negative sign.
12) Work and Potential Energy:
a) 𝑊𝑔𝑟𝑎𝑣 = −Δ𝑈𝑔𝑟𝑎𝑣
b) 𝑊𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 = −Δ𝑈𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐
⃑⃑𝑈, which is just a shorthand notation for:
13) Force: 𝐹⃑ = −∇
a) 𝐹𝑥 = −
𝜕
𝑈,
𝜕𝑥
𝐹𝑦 = −
𝜕
𝑈,
𝜕𝑦
𝐹𝑧 = −
𝜕
𝑈
𝜕𝑧
b) The symbol ⃑∇⃑ (upside down triangle with an arrow on top) is pronounced “gradient”
14) Centripetal acceleration
a) 𝑎 =
𝑣2
𝑅
Superposition
15) You can use superposition on the following
a) The potential at a point caused by multiple charges
i) 𝑉𝑡𝑜𝑡𝑎𝑙 = 𝑉1 + 𝑉2 + 𝑉3 + ⋯
b) The electric field at a point caused by multiple charges
i) 𝐸⃑⃑𝑡𝑜𝑡𝑎𝑙 = ⃑⃑⃑⃑⃑
𝐸1 + ⃑⃑⃑⃑⃑
𝐸2 + ⃑⃑⃑⃑⃑
𝐸3 + ⋯
Technically, we can measure absolute potential energy thanks to Einstein equation 𝐸 = 𝑚𝑐 2 . (𝐸 on that equation
is the total absolute potential energy for stationary object). In fact, the only thing that you have to do is measure
the mass of the object. However, this number is usually really really large and since the total mass rarely changes
(except in Nuclear reaction), this potential energy is almost always constant. For the purpose of this class, we just
set this total energy to 0, thus making the absolute value of potential energy meaningless.
1
c) The force on a charge 𝑞0 caused by other charges 𝑞1 , 𝑞2 , 𝑞3 , …
i) 𝐹⃑𝑡𝑜𝑡𝑎𝑙 = 𝐹⃑1→0 + 𝐹⃑2→0 + 𝐹⃑3→0 + ⋯
Gravity
16) For object closes to the surface of the earth:
a) Gravitational potential: 𝜙𝑔𝑟𝑎𝑣 = 𝑔ℎ where ℎ is the height location of the object
b) Gravitational potential energy: 𝑈𝑔𝑟𝑎𝑣 = 𝑚𝜙𝑔𝑟𝑎𝑣 = 𝑚𝑔ℎ
⃑⃑𝑈𝑔𝑟𝑎𝑣
c) Gravitational force: 𝐹⃑𝑔𝑟𝑎𝑣 = −∇
Electric Potential
17) Charge: 𝑄
a) Unit: Coulombs
18) Electric Potential: 𝑉(𝑥, 𝑦, 𝑧)  potential as a function of 𝑥, 𝑦, 𝑧
a) Scalar field
b) Unit: volt =
Joule
Coulomb
c) Only Δ𝑉 is meaningful. Can set 𝑉 = 0 wherever you want
d) Voltage = Δ𝑉 = potential difference. (voltage is NOT the same potential, it’s potential difference)
e) Example on 9 Volt battery:
i) The voltage of a 9 V battery is nine volt
ii) If we set 𝑉− = 0 at negative terminal, then the potential at the positive terminal is 𝑉+ = 9 V
iii) If we set 𝑉− = −1000 V at negative terminal, then the potential at the positive terminal is
𝑉+ = −1000 V + 9 V = −991 V
19) Electric Potential Energy
a) 𝑈𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 = 𝑄𝑉
b) This equation means that if you put a charge 𝑄 on a location with potential 𝑉, the electrical
potential energy on that charge is 𝑈𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐
c) This equation can also be used for battery. A battery with potential difference 𝑉 has to spend an
energy 𝑈𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 in order to deliver 𝑄 amount of charge
d) This equation DOES NOT mean that if you have a capacitor with a charge 𝑄 and potential
difference 𝑉, the energy stored in the capacitor is 𝑈𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 . Again, do NOT use this equation for
capacitor.
20) Equipotential surface/line
a) Surface/line where the potential is the same/constant
b) Never crosses one another
c) The direction of the gradient/electric field is always perpendicular to the equipotential
surface/line
21) The following is true only for point charge (assume the point charge has charge 𝑞1 )
1
a) Potential: 𝑉(𝑟) = 4𝜋𝜖
0
𝑞1
𝑟
b) Force on another particle 𝑞2 , located at distance 𝑟:
i)
Magnitude: 𝐹1→2 =
1 |𝑞1 𝑞2 |
4𝜋𝜖0 𝑟 2
ii) Direction: either repulsive or attractive, or radial from 𝑞1
22) Continuous charge distribution:
a) 𝑉𝑡𝑜𝑡𝑎𝑙 = ∫𝑜𝑣𝑒𝑟−𝑎𝑙𝑙−𝑐ℎ𝑎𝑟𝑔𝑒𝑠 𝑑𝑉
b) 𝑑𝑞 is tiny bit of charge
1
𝑑𝑞
0 𝑟
1
𝑑𝑞
∫ 𝑟
4𝜋𝜖0
c) 𝑑𝑉 = 4𝜋𝜖
d) 𝑉 =
e) 𝑑𝑞 = 𝜆 𝑑𝑠, 𝑑𝑠 is a tiny bit of length, 𝜆 is linear charge density
f) 𝑑𝑞 = 𝜎 𝑑𝐴, 𝑑𝐴 is a tiny bit of area, 𝜎 is area charge density
g) 𝑑𝑞 = 𝜌 𝑑𝑉, 𝑑𝑉 is a tiny bit of volume, 𝜌 is volume charge density
Electric Field and Gauss Law
23) Electric Field and potential
⃑⃑ 𝑉
a) 𝐸⃑⃑ = −∇
𝑏
b) 𝑉(𝑏) − 𝑉(𝑎) = − ∫𝑎 𝐸⃑⃑ (𝑟). 𝑑𝑟⃑
c) 𝐹⃑ = 𝑞𝐸⃑⃑
24) Electric field pushes positive charges and attract negative charges
25) Gauss’s Law:
a) Draw a Gaussian surface such that the electric field on that surface is constant
b) Electric flux: Φ = ∮ 𝐸⃑⃑ . 𝑑𝐴⃑
c) Electric flux: Φ =
26)
27)
28)
29)
𝑞𝑖𝑛𝑠𝑖𝑑𝑒
𝜖0
d) Equate the two electric flux calculations to deduce the value of 𝐸⃑⃑
Gaussian surface:
a) Normal vector is perpendicular to the surface
b) The direction of normal is pointing outwards
3 different possible symmetries for Gauss’s Law problems:
a) Spherical: Φ = 𝐸 4𝜋𝑟 2
b) Cylindrical: Φ = 𝐸 2𝜋𝑟𝐿
c) Plane: Φ = 𝐸𝐴
Properties of electric field lines:
a) Start from + charge, end on – charge, but can also start / end at infinity
b) Number of field lines on a charge is proportional to the amount of charge
c) Cannot split or merge
d) Tend to stay far away from each other
e) The electric field is stronger where the lines are close together
Examples of potentials and electric field
a) Spherically symmetric object, outside the object
i) Such as: point charge, charged sphere
1
ii) 𝑉(𝑟) = 4𝜋𝜖
1
iii) 𝐸⃑⃑ = 4𝜋𝜖
0
𝑞
0𝑟
𝑞
𝑟̂
𝑟2
b) A plate located at 𝑥 = 0, outside the plate
𝜎
i) 𝑉(𝑥) = − 2𝜖 |𝑥|
ii) 𝐸 =
𝜎
,
2𝜖0
0
away and perpendicular from the plate
c) A cylindrically symmetric object, outside the object
1
𝑟
i) 𝑉(𝑟) = − 4𝜋𝜖 2 𝜆 ln (𝑟 )
0
0
ii) 𝑟0 is an arbitrary radius where you define the potential to be zero
iii) 𝐸 =
2𝜆 1
,
4𝜋𝜖0 𝑟
radially outward
d) Ring of charge located at 𝑥 = 0
i)
2𝜋𝑅
1
𝑉(𝑥) = 4𝜋𝜖 ∫0
0
𝜆𝑑𝑠
√𝑥 2 +𝑅2
1
= 4𝜋𝜖
2𝜋𝑅𝜆
0
√𝑥 2 +𝑅2
e) Disk of charge located at 𝑥 = 0
𝑅=𝑅 𝜎2𝜋𝑅𝑑𝑅
1
i) 𝑉(𝑥) = 4𝜋𝜖 ∫𝑅=0 0 2 2 =
√𝑥 +𝑅
0
𝜎
2𝜖0
[√𝑥 2 + 𝑅 2 − |𝑥|]
Dipole and Dielectric
30) Electric dipole:
a) 𝑝⃑ = 𝑞𝑑⃑, direction from – to +
b) Torque such that 𝑝⃑ and 𝑑⃑ will align with 𝐸⃑⃑
c) The direction of the electric field produced by the dipole is the opposite of the direction of the
external electric field
d) In uniform field, the net force on the dipole is zero
e) Torque: 𝜏⃑ = 𝑝⃑ × 𝐸⃑⃑ ,
i) Magnitude: 𝜏 = 𝑝𝐸 sin 𝜃, 𝜃 is the angle between 𝑝⃑ and 𝐸⃑⃑
ii) Direction: use intuition
Potential energy: 𝑈 = −𝑝⃑. 𝐸⃑⃑
f)
31) Dielectric:
a) Reduces the total electric field: 𝐸𝑡𝑜𝑡𝑎𝑙 =
𝐸0
𝜅
b) 𝜖 = 𝜅𝜖0
Conductor, Resistor, Capacitor
32) Conductor:
a) Potential drop across conductor is zero
b) Charges can only be located at the surface of a conductor. Could be inner surface or outer
surface, but never on the conductor body itself
c) Equivalent to a resistor with zero resistance
d) Electric field is always perpendicular to the conductor surface
e) Electric field inside a hollow conductor is zero
33) Conductivity(𝜎) and Resistivity(𝜌)
a) Describe a material (not an object, just the material)
1
b) 𝜌 = 𝜎
c) For ions in solutions only:
1
𝑓
i) 𝜌 = 𝜎 = 𝑐𝑞2
ii) 𝑐: concentration of ion
iii) 𝑞: charge of the ion
iv) 𝑓: drag coefficient of that ion
v) 𝐷 =
𝑘𝐵 𝑇
: diffusion constant
𝑓
vi) Only works when the friction is linear in velocity
d) Metal: high temperature  more resistivity
e) Glass: high temperature  less resistivity
34) Resistance(𝑅)
a) Describes a property of an object (which is material + geometry)
𝜌
b) 𝑅 = ∫ 𝐴 𝑑𝐿
i) 𝐴 is the cross section area perpendicular to the direction of the current
c) For constant 𝐴 and 𝜌:
i) 𝑅 =
𝜌𝐿
𝐴
35) Power dissipation in resistor
a) 𝑃 = 𝐼 Δ𝑉 = 𝐼 2 𝑅 =
(Δ𝑉)2
𝑅
b) 𝑃 is the power dissipated on the resistor
c) 𝐼 is the current through the resistor
d) Δ𝑉 is the potential drop across the resistor
e) 𝑅 is the resistance of the resistor
36) Capacitance(𝐶)
a) Describes a property of an object (material + geometry)
b) A proportionality factor between Δ𝑉 and 𝑄
𝑄
c) 𝐶 = Δ𝑉
d) Examples:
i) Single spherical conductor: 𝐶 = 4𝜋𝜅𝜖0 𝑅
ii) Parallel plates: 𝐶 =
𝑘𝜖0 𝐴
𝑑
iii) Cylindrical double plates: 𝐶 =
2𝜋𝜅𝜖0 𝐿
ln(
𝑅+𝑎
)
𝑅
37) Energy stored in capacitor
1 𝑄2
𝐶
a) 𝑈𝑒𝑙𝑒𝑐 = 2
1
1
= 2 𝑄Δ𝑉 = 2 𝐶(Δ𝑉)2
b) Change of energy per unit time (power): 𝑃 =
𝑑𝑈
𝑑𝑡
𝑑𝑉
𝑄 𝑑𝑄
𝑑𝑡
= 𝐶𝑉 𝑑𝑡 = 𝐶
𝑄
= 𝐶 𝐼 = 𝑉𝐼
Circuits
38) Potential in circuits:
a) Pick one point, and exactly one point in the circuit as the ground (𝑉 = 0)
b) Every single point in the circuit has a potential
c) Points connected by wire has the same potential
d) Note that the equations only specify Δ𝑉 not the absolute potential. For example, it is possible
that both sides of a resistor have electrical potential of 1000 𝑉 without any current flowing
since the potential difference is zero
39) Series and parallel resistor
a) Series: 𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + 𝑅3
b) Parallel:
1
𝑅𝑒𝑞
=
1
𝑅1
+
1
𝑅2
+
1
𝑅3
40) Series and parallel capacitor
a) Parallel: 𝐶𝑒𝑞 = 𝐶1 + 𝐶2 + 𝐶3
b) For series capacitor, the formulas below only apply if all of the capacitors are uncharged in the
beginning (or every capacitor has identical amount of charge in the beginning). The formula will
NOT apply if the series configuration is applied after the capacitors are charged. See practice
problems for examples. And yes, these weird cases are all time favorite on exams.
i)
1
1
1
1
1
2
3
Series: 𝐶 = 𝐶 + 𝐶 + 𝐶
𝑒𝑞
41) Kirchoff algorithm:
a) Define direction of current for every single section in the circuit, and label the current
(𝐼1 , 𝐼2 , 𝐼3 , etc). Stick to it.
b) Apply junction rule: Sum of current into the junction = Sum of current out of the junction
c) Define several loops until every single section is covered by at least one loop. Remember that
the loop has direction.
d) Sum of potential drop across each loop is ZERO. Be careful with the sign of the potential drop
e) The capacitor rule: 𝐼 =
𝑑𝑞
𝑑𝑡
for each capacitor. The 𝐼 is the current going to a capacitor, and the 𝑞
is the charge on that same capacitor.
42) Battery:
a) Δ𝑉 = 𝑉𝑒𝑚𝑓 = 𝜀
b) The positive plate of the battery has higher potential
43) Resistor in circuit:
a) Δ𝑉 = 𝐼𝑅
b) Current flows from high potential to low potential
44) Capacitor in circuit:
a) 𝐼 is going to the “positive plate” of the capacitor by convention.
b) 𝑄 is the charge on the positive plate. (−𝑄 is the charge on the negative plate)
𝑄
c) Δ𝑉 = 𝐶
d) The positive plate has higher potential (same as battery)
e) 𝐼 =
𝑑𝑄
,
𝑑𝑡
where
i) 𝑑𝑄 is the amount of charge increase on the positive plate of the capacitor
ii) 𝑑𝑡 is the change in time
iii)
𝑑𝑄
𝑑𝑡
is the amount of charge increase on the positive plate per unit time
f)
After a very short amount of time
i) Charge on the capacitor is the same as a short amount of time ago
ii) Since charge is the same, 𝑉 = 𝑄/𝐶 implies that voltage across the capacitor is the same
iii) If the capacitor is empty (non-charged), for a short moment, the capacitor can be replaced
by wire
g) On steady state / after a long time:
i) Current to / out of capacitor is zero
ii) Charge on the capacitor is constant
iii) Voltage on the capacitor is constant
45) 𝑅, 𝐶, 𝜀 circuit. A circuit with a resistor, a capacitor, and an optional battery in series
a) General solution:
𝑡
𝑄(𝑡) = 𝑄𝑓𝑖𝑛𝑎𝑙 + (𝑄𝑖𝑛𝑖𝑡𝑖𝑎𝑙 − 𝑄𝑓𝑖𝑛𝑎𝑙 )𝑒 − 𝑅𝐶
b) 𝑄𝑖𝑛𝑖𝑡𝑖𝑎𝑙 is the initial charge on the positive plate of the capacitor. This quantity is usually
specified in the problem
c) 𝑄𝑓𝑖𝑛𝑎𝑙 is the final charge after a long time on the positive plate of the capacitor. It’s probably
either +𝐶𝜀 or – 𝐶𝜀, depending on how the positive plate of the capacitor is connected to the
battery. Check the practice problem where 𝑄𝑓𝑖𝑛𝑎𝑙 is some other expression
d) If there is no battery, 𝑄𝑓𝑖𝑛𝑎𝑙 = 0
Logic
46) Mosfet rules:
a) The current through the Gate pin is zero
b) If the gate is high, the mosfet is ON / conducting, potential across drain to source is zero
c) If the gate is low, the mosfet is OFF / insulating / no current from drain to source
47) Not so obvious Boolean algebra relations:
a) 𝑥 ∧ (𝑦 ∨ 𝑧) = (𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)
b) 𝑥 ∨ (𝑦 ∧ 𝑧) = (𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)
c) 𝑥 ∧ (𝑥 ∨ 𝑦) = 𝑥
d) 𝑥 ∨ (𝑥 ∧ 𝑦) = 𝑥
e) 𝑥 ∨ ¬𝑥 = 1
f) 𝑥 ∧ ¬𝑥 = 0
g) ¬(𝑥 ∧ 𝑦) = (¬𝑥) ∨ (¬𝑦)
h) ¬(𝑥 ∨ 𝑦) = (¬𝑥) ∧ (¬𝑦)
i) ¬¬𝑥 = 𝑥
Thermodynamics
𝑝1
𝑝2
48) Boltzmann distribution:
= 𝑒−
𝐸1 −𝐸2
𝑘𝑇
Δ𝑈
= 𝑒 − 𝑘𝑇
a) The probability of a particle to be in state 1 compared to the probability of a particle to be in
state 2
𝑐
𝑐1
49) Work against concentration gradient from 1 to 2: 𝑊 = 𝑘𝑇 ln ( 2 )
Nernst, GHK, Neuron
50) 𝑅𝐶 cable model
a)
51)
𝜕𝑉
𝜕𝑡
𝑎𝑑 𝜕2 𝑉
= ( 𝜌𝜖 ) 𝜕𝑥 2
𝑘𝑇
ln(𝑋)
𝑧𝑒
=
60mV
log10 (𝑋)
𝑧
52) Work required to go against concentration gradient:
[𝐶𝑓𝑖𝑛𝑎𝑙 ]
a) 𝑊 = 𝑘𝑇 ln ([𝐶
)
𝑖𝑛𝑖𝑡𝑖𝑎𝑙 ]
53) Nernst: Potential difference on membrane without active transport:
a) 𝑉 =
[𝐶 ]
𝑘𝑇
ln ( [𝐶𝑜𝑢𝑡] )
𝑧𝑒
𝑖𝑛
b) True for each ion type that does not have active transport
54) GHK equation
a) 𝑉𝑚𝑒𝑚𝑏 =
]+𝑃𝐶𝑙 [𝐶𝑙𝑖𝑛 ]
𝑘𝑇
𝑃 [𝐾 ]+𝑃 [𝑁𝑎
ln ( 𝑃𝐾 [𝐾𝑜𝑢𝑡]+𝑃 𝑁𝑎[𝑁𝑎 𝑜𝑢𝑡
)
𝑒
𝐾 𝑖𝑛
𝑁𝑎
𝑖𝑛 ]+𝑃𝐶𝑙 [𝐶𝑙𝑜𝑢𝑡 ]
b) 𝑃𝐾 is the permeability of 𝐾 ions
c) Given the ion concentrations, what is the potential across the membrane?
d) The equation assumes that the charge flux is zero (no current)
𝐽𝑁𝑎 + 𝐽𝐾 − 𝐽𝐶𝑙 = 0
Magnetic Field
55) Properties of magnetic dipole
a) North is positive. South is negative
b) North is pushed by magnetic field, South is pulled by magnetic field
c) Magnetic dipole direction goes from S to N
d) Inside the magnet, magnetic field goes from S to N
e) Outside the magnet, magnetic field goes from N to S
f) Magnetic force on a magnetic dipole in a uniform magnetic field is zero
g) Magnetic force on a magnetic dipole in a non-uniform magnetic field is not zero
h) Magnetic field lines always form closed loops
56) Magnitude of magnetic field generated by objects
𝜇 𝐼
0
a) Long wire: 𝐵 = 2𝜋𝑟
b) You have to figure out the direction using right hand rule
57) Magnetic force
⃑⃑
a) Force on a moving charge: 𝐹⃑ = 𝑞 𝑣⃑ × 𝐵
⃑⃑
b) Force on a wire: 𝐹⃑ = 𝐼 𝐿⃑⃑ × 𝐵
58) Current loop dipole moment
a) 𝜇⃑ = 𝐼 𝐴⃑
b) The direction of 𝐴⃑ is related to the direction of the current via right hand rule
⃑⃑
c) Magnetic torque: 𝜏⃑ = 𝜇⃑ × 𝐵
59) Magnetic Flux:
⃑⃑. 𝐴⃑
a) Φ = 𝐵
b) The direction of 𝐴⃑ is related to the loop direction (current direction) via right hand rule
60) Faraday’s Law
a) Change in magnetic flux induces EMF
b) 𝜀induced = −
𝑑Φ
𝑑𝑡
c) Emf induced around the loop is minus the time rate of change of magnetic flux through the loop
d) Lenz’s law: the induced current flows in the direction that is needed to counteract the change in
flux
61) Some electrical rules that has to be modified when there is varying B fields
a) Kirchoff loop rule does not apply. The potential across a loop is now given by Faraday’s Law
b) Electric field lines can now go in circles too in addition to going from + charge to - charge
MRI
62) Spin of particles
1
a) Angular momentum 𝐿 = ± 2 ℏ
1
b) Magnetic dipole moment:𝜇 = ± 2 𝛾ℏ
c) 𝛾: gyromagnetic ratio. Different number for different particles.
d) Spin up means the angular momentum / dipole moment align with the magnetic field. Low
energy state.
e) Spin down means the angular momentum / dipole moment anti-align with the magnetic field.
High energy state.
⃑⃑ = ∓ 1 𝛾ℏ𝐵0
f) 𝑈 = −𝜇⃑ . 𝐵
2
g) Δ𝑈 = 𝛾ℏ𝐵0
h) Larmor frequency: 𝜔 = 𝛾𝐵0 ,
i) The frequency needed to induce transition between the two spin-states (between up and
down)
ii) Also the precession frequency when the spin points sideways
63) Energy of photon: 𝑈 = ℏ𝜔 = ℎ𝑓
64) Rotating magnetization
𝜃
a) Length of pulse: 𝑡 = 𝛾𝐵
1
65) Magnetization relaxation:

𝑇1 : the z component returns to positive z. Property of single particle
𝑀𝑧 = 𝑀initial 𝑒

−
𝑡
𝑇1
+ 𝑀final (1 − 𝑒
−
𝑡
𝑇1 )
𝑇2 : decay of xy component because the xy component become out of phase. Property of many
particles
𝑀𝑥𝑦 = 𝑀initial 𝑒
−
𝑡
𝑇2
66) Magnetic Resonance Imaging
a) Apply strong constant magnetic field, say in the z direction. Now there is a (small) net
magnetization in the z direction (spin up) due to Boltzmann distribution. This process can be
modeled like an RC circuit. The time constant is 𝑇1 . 𝑀𝑡 = 𝑀0 (1 − 𝑒
−
𝑡
𝑇1
)
b) We then apply magnetic field gradient in the z direction. Since the Larmor frequency depends
on the magnetic fields, protons at different z location have different Larmor frequencies.
𝜋
c) When then apply 2 pulse, which is a radio frequency pulse running at the Larmor frequency for
𝑡=
𝜃
𝛾𝐵1
time period. After this step, the magnetization is pointing somewhere on xy plane. Note
that only the magnetization of spins located at a particular z coordinate rotates to the xy plane.
The magnetization at other z coordinate still point in the same direction as before (positive z).
d) We then remove the magnetic field gradient on the z axis, and apply magnetic field gradient on
y axis.
e) The magnetization on the xy plane now precesses at different Larmor frequency depending on
the y coordinate.
f) The previous step is then repeated with gradient applied at different orientation on xy plane
until the entire image can be reconstructed
g) Repeat the entire process for different xy plane to reconstruct the full 3D image
Photoelectric
67) Photoelectric effect:
a) Energy of photon: 𝐸 = ℎ𝑓
b) You need each individual photon to have enough energy to knock off an electron from metal
c) Kinetic energy 𝐾𝐸 = ℎ𝑓 − 𝑊𝑎 , where 𝑊𝑎 is the binding energy
ℎ
d) De Broglie wavelength 𝜆 = 𝑝
ℎ
e) Photon momentum 𝑝 = 𝜆 =
𝐸
𝑣
Geometric Optics
68) Reflection: 𝜃𝑖 = 𝜃𝑟 . The angles are measured relative to a line perpendicular to the surface of the
mirror.
69) Shared Mirror / Lens Properties:
a) 𝑑0 : distance of the object from the mirror/lens
b) 𝑑𝑖 : distance of the image from the mirror/lens
c) 𝑓: distance of the focus from the mirror/lens
1
1
1
d) Imaging equation: 𝑓 = 𝑑 + 𝑑
e) Magnification: 𝑚 =
ℎ𝑖
ℎ𝑜
𝑜
𝑖
=
𝑑
− 𝑑𝑖
𝑜
f)
Upright – inverted:
i) Upright: ℎ0 and ℎ𝑖 have the same signs
ii) Inverted: ℎ0 and ℎ𝑖 have different signs
g) All rays are reversible
Mirror
70) Mirror Only Properties:
𝑅
2
a) 𝑓 = , focal length of the mirror. 𝑅 is the radius of curvature
i) Concave: 𝑓 > 0
ii) Convex: 𝑓 < 0
b) Mirror positive - negative:
i) Positive if in front of the mirror (real)
ii) Negative if behind the mirror (virtual)
c) Ray tracing rules:
i) Parallel ray: ray goes out parallel to the axis, reflects through the focus
ii) Focal ray: ray goes through the focus is reflected parallel to the axis
iii) Perpendicular ray: ray goes through C reflects back on itself
71) Summary of ALL possible object and image locations for positive (=concave=converging) and
negative (=convex=diverging) mirrors. This is useful for double checking your answer, or just to get
some intuitive guess on where the image would be like in multiple choice questions. It is NOT useful
for finding quantitative answers. This method is NOT a valid reasoning method on the exam when
they asked you to provide reasoning. Some graders are not aware of this method.
Rules:



Image and Object have the same zone numbers
Image Zone Number + Object Zone Number = 5
Larger and Smaller rules:
o If OZ > IZ, then the Object is LARGER than the Image (|ℎ0 | > |ℎ𝑖 |)
o If OZ < IZ, then the Object is SMALLER than the Image (|ℎ0 | < |ℎ𝑖 |)
o If OZ = IZ, (can only be true for OZ = 2.5 and IZ = 2.5), Object and Image have the
same size
Refraction
72) Refraction: 𝑛1 sin 𝜃1 = 𝑛2 sin 𝜃2
𝑐
73) Index of refraction: 𝑛 = 𝑣 ≥ 1
74) Total internal reflection
a) From high index to low index if the angle is larger than critical angle
b) Used in fiber optics
c) Critical angle: sin 𝜃2 = 1
Lens
1
75) Focusing power of lens: 𝑃 = 𝑓 (Nothing to do with energy per unit time)
76) Ray tracing rules for lens
a) Parallel ray: enters parallel to the axis and emerges through the focal point
i) Positive lens: use far focal point
ii) Negative lens: use near focal point
b) Focal ray: ray through focal point emerges parallel
i) Positive lens: use near focal point
ii) Negative lens: use far focal point
c) Central ray: straight through the center of the lens
77) Ray diagram for positive (converging) and negative (diverging) lenses
78) Summary of ALL possible object and image locations for positive(=convex=converging) and negative
(=concave=diverging) lenses. This is useful for double checking your answer, or just to get some
intuitive guess on where the image would be like in multiple choice question. It is NOT useful for
finding quantitative answers. This method is NOT a valid reasoning method on the exam when they
asked you to provide reasoning.
Rules











OZ = Object Zone
IZ = Image Zone
Zone 1 is somewhere from the lens to the focus (0 < 𝑑 < 𝑓)
Zone 1.5 is at the focus (𝑑 = 𝑓)
Zone 2 is somewhere between 𝑓 and 2𝑓. (𝑓 < 𝑑 < 2𝑓 )
Zone 2.5 is at twice the focus (𝑑 = 2𝑓)
Zone 3 is somewhere between 2𝑓 and ∞. (2𝑓 < 𝑑 < ∞)
Zone 3.5 is at infinity (𝑑 = ∞)
Zone 4 is at the opposite side of the lens. (𝑑 < 0), (𝑑 is negative) (also means Virtual)
Note that for negative lens, the image and object zones are flipped as shown in the figure
Object-Image location rule: OZ + IZ = 5, that is the sum of the object zone number and the
image zone number is five.
 Examples:
o Object in OZ 1.5 (focus), then Image must be in IZ 3.5 (infinity)
o Object in OZ 2.5 (twice focus), then Image must be at IZ 2.5 (twice focus)
o Object is somewhere inside OZ 2, Image is somewhere in IZ 3 (from 2f to ∞)
 Object-Image size relation rules:
o If OZ > IZ , then the Object is LARGER than the Image (|ℎ0 | > |ℎ𝑖 |)
o If OZ < IZ , then the Object is SMALLER than the Image (|ℎ0 | < |ℎ𝑖 |)
o If OZ = IZ , (can only be true for OZ = 2.5 and IZ = 2.5), Object and Image have the
same size
79) Lens positive - negative madness:
a) Light rays travel from the near side to the far side
b) Positive quantities are “real”
c) Negative quantities are “virtual”
Quantity
Positive if
Negative if
𝑓 focal point
Converging lens
Diverging lens
(thick in the middle)
(thin in the middle)
𝑑0 object distance Object on near side (real)
Object on far side (virtual)
𝑑𝑖 image distance
ℎ height
𝑚 magnigication
Image on the far side (real)
Image is right-side up
Image is upright with
respect to the object
Image on the near side (virtual)
Image is upside-down
Image is upside-down with
respect to the object
80) Human eye:
a) The light sensitive cells are located behind the nerve wiring, blocking some part of the light
coming to the sensors
b) Blind spot is a spot where the nerves bundles and passes through the light sensitive are towards
the brain. There is no light sensitive cells on the blind spot
c) Curved surface of cornea is the most important focusing element, while the lens is just
secondary focusing element
d) Normal vision: can focus between 25 cm to ∞
e) Nearsighted: can only see near things
f) Farsighted: can only see far things
g) Rod cells: for seeing black and white. Max sensitivity: 498 nm.
Wave
81) The speed of light in vacuum, 𝑐 =
1
√𝜀0 𝜇0
= 3 × 108 m/s
82) Visible wavelength (violet to red): 390 nm − 750 nm
83) General equation for wave: 𝑦(𝑥, 𝑡) = 𝐸0 sin(𝑘𝑥 ± 𝜔𝑡 + 𝜙)
a) A wave that propagates on the positive x direction
2𝜋
𝑘
𝜔
𝜆
= =
𝑘
𝑇
b) Wavelength: 𝜆 =
c) Velocity: 𝑣 =
d)
1
𝑇
=𝑓=
𝜆𝑓
𝜔
2𝜋
e) 𝜙 is the phase that can be calculated from boundary condition. I recommend using the
maximum of the wave to calculate the phase.
84) Waves:
𝑇
a) Wave in string: 𝑣 = √𝜇. 𝑇 is the tension, and 𝜇 =
𝑚
𝐿
is mass per unit length
𝐵
𝜌
b) Sound waves: 𝑣 = √ . 𝐵 is the bulk modules, and 𝜌 is the mass density
85) Power: 𝑃 =
𝑑𝐸
𝑑𝑡
∝ 𝐴2
𝑃
86) Intensity: 𝐼 = 𝐴𝑟𝑒𝑎
a) Plane wave: Intensity is constant
1
1
b) Spherical wave: Intensity goes like ∝ 𝑟2 in 3D. Amplitude goes like ∝ 𝑟
c) Spherical wave in 2D (like water on the surface of a pond), also called circular wave: intensity
1
goes like ∝ 𝑟 in 2D. Amplitude goes like ∝
1
√𝑟
87) Loudness:
𝐼
a) Loudness (in decibel): 𝛽 = 10 log10 (𝐼 )
0
b) 𝐼0 = 10−12 𝑊/𝑚2  𝛽 = 0 dB
88) Longitudinal wave
a) The displacement is parallel to the direction of propagation
b) Examples: sound wave
89) Transverse wave:
a) The displacement is perpendicular to the direction of propagation
b) Examples: electromagnetic wave, wave on a string
Standing Wave
90) Standing wave: 𝑦(𝑥, 𝑡) = 𝐴 sin(𝑘𝑥) cos(𝜔𝑡)
a) More generally: 𝑦(𝑥, 𝑡) = 𝐴 sin(𝑘𝑥 + 𝜙) cos(𝜔𝑡 + 𝜓)
b) Superposition of two counter-propagating waves with the same frequency
91) String standing wave: both ends are displacement node.
92) Pipe standing wave:
a) Open end, pressure node, pressure is not changing because it’s connected to the outside world
b) Closed end, pressure antinode. Maximum amount of pressure change
93) Pipe with one open end and one closed end. This is the official convention in this class
a) 𝑓1: Fundamental = 1st harmonic
b) 3𝑓1, 3rd harmonic
c) 5𝑓1, 5th harmonic
d) Frequencies: 𝑓 =
𝑛𝑣
,
4𝐿
(𝑛 = 1,3,5,7, . . )
94) Harmonics in any other cases (both opened or both closed or string):
a) 𝑓1 : Fundamental = 1st harmonic
b) 2𝑓1: 1st overtone = 2nd harmonic
c) 3𝑓1 : 2nd overtone = 3rd harmonic
d) Frequencies: 𝑓 =
𝑛𝑣
,
2𝐿
(𝑛 = 1,2,3,4, . . )
95) Constraints on standing waves:
a) Hard constraint: pressing a piece of string tightly
Effectively, the instrument is getting shorter.
b) Soft constraint: holding a piece of string loosely, hole on a pipe
The length of the instrument is the same, but only the modes that satisfy the constraints are
allowed
Doppler Effect
𝑣
96) Doppler effect: 𝑓
′
=𝑓
1± 𝑣0
𝑣
1∓ 𝑣𝑠
=𝑓
𝑣±𝑣0
𝑣∓𝑣𝑠
a) 𝑣: speed of sound
b) 𝑣0 : observer velocity. Negative when moving AWAY from the source
c) 𝑣𝑠 : source velocity. Negative when moving TOWARDS the observer
d) 𝑣𝑜 and 𝑣𝑠 are measured relative to the air. If the air itself is moving, the 𝑣𝑜 and 𝑣𝑠 must be
adjusted accordingly.
Interference
97) Small angle approximation: sin 𝜃 ≈ tan 𝜃 ≈ 𝜃. Small means the angle is less than 100 . 𝜃 has to be in
radians, not in degrees.
98) Interference
a) Δϕ = ϕ2 − 𝜙1 = 0, ±2𝜋, ±4𝜋 = 2𝜋𝑚 means two waves are in phase, constructive
interference
1
2
b) Δϕ = ϕ2 − 𝜙1 = 𝜋, ±3𝜋, ±5𝜋 = 2𝜋 (𝑚 + ) means two waves are out of phase, destructive
interference
c) 𝑚 is integers
d) Δ𝜙 = 2𝜋
Δ𝑠
𝜆
= 𝑘 Δ𝑠
e) Δ𝑠 = difference in path length
f)
Note that 𝜆 changes in different medium. 𝜆′ =
𝜆𝑣𝑎𝑐𝑢𝑢𝑚
,
𝑛
where 𝑛 is the index of refraction
99) Additional phase change from reflection
a) From fast speed medium to low speed medium: 𝜋 phase change
b) From low speed medium to high speed medium: no phase change
c) String to a fix point at the end: reflection: opposite phase
d) String to a free point at the end: reflection: same phase
100) Interference pattern from double slits
1
a) Location of dark bands: 𝑑 sin 𝜃 = (𝑚 + 2) 𝜆
b)
c)
d)
101)
Location of bright bands: 𝑑 sin 𝜃 = 𝑚𝜆
𝑚 is integer
𝑑 is the distance between the two slits
Diffraction from single slit
a) Minima are at 𝐷 sin 𝜃 = 𝑚𝜆. 𝑚 is integer. The exception is at 𝑚 = 0, which is NOT a minimum.
b) At 𝜃 = 0 (or 𝑚 = 0) for the minima location, there is the Primary maximum. This is the only
exception for the formula for the minima
1
c) The location for the maxima is at 𝐷 sin 𝜃 = (𝑚 + 2) 𝜆. The exception is at 𝑚 = 0 and 𝑚 = −1.
102)
Those two locations are neither maxima nor minima.
Diffraction from a circular aperture:
𝜆
a) Location of first dark ring: 𝜃 = 1.22 𝐷
103) Rayleigh Criterion: can distinguish two point sources if the maximum of one is at the first
minimum of the other. In other words, the resolution is the same as the location of the first dark
ring/band
a) sin 𝜃 > 1.22
𝜆
𝐷
104) Bragg diffraction (X-Ray)
a) Maxima at 2𝑑 sin 𝜙 = 𝑚𝜆
b) 𝜙 is defined with respect to the horizontal NOT the normal
Microscope
105) Numerical aperture. 𝑁𝐴 = 𝑛𝑜𝑖𝑙 sin 𝛼
a) 𝛼 is the half angle of the objective lens opening size
106) The specimen is located at the focus of the objective
a) Resolving power of microscope 𝑆 = 0.61
𝜆
𝑁𝐴
i) 𝜆 is the wavelength
ii) 𝑓 is the focal length
iii) 𝐷 is the diameter of the lens
b) 𝑆 is the smallest feature that can be resolved (the smaller the better)
c) For small angle 𝑆 ≈ 1.22
𝜆𝑓
𝐷
𝜆
d) For a good microscope, diffraction limit is 𝑆 ≈ 2
e) The higher the index of refraction, the higher the NA, the smaller S you get.
Polarization
107)
a)
b)
108)
a)
Un-polarized light passing through a polarizer:
Output intensity/power: half of the incoming power
Output polarization: as always, polarized in the same direction as the polarizer
Polarized light passing through a polarizer
Output intensity/power: 𝐼 = 𝐼0 cos 2 𝜃. Where 𝜃 = angle between the light polarization (E field
vector) and the axis of the polarizer.
b) Output polarization: as always, polarized in the same direction as the polarizer