PHYS 142 Formula Sheet
PHYS 142 Formula Sheet
Flux:
𝐶=(
By: Kareem Halabi
Flux on a surface: Φ𝑒 = 𝐸𝐴 cos 𝜃 = ∫𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝐸⃗⃗ ∙ 𝑑𝐴⃗ (theta is between normal of
series
Electric Fields:
plane and the electric field)
Permittivity constant: 𝜖0 = 8.85 ×
Gauss’s law Φ𝑒 =
C2 10−12 2 Nm
𝑄𝑖𝑛 𝜖0
(encompassing a net
Coulomb’s law: 𝐹𝑝𝑜𝑖𝑛𝑡 =
|𝑞1 ||𝑞2 |
4𝜋𝜖0
𝑟2
𝑈𝑒𝑙𝑒𝑐 = 𝑞𝐸𝑠 q is charge, E is elec-field and s is distance
Charge in a field: 𝐹⃗𝑜𝑛 𝑞 = 𝑞𝐸⃗⃗ Torque of Dipole in a uniform Field:
𝑈𝑒𝑙𝑒𝑐 =
1
𝑞1 𝑞2
4𝜋𝜖0
𝑟
Potential between two
𝜏 = 𝑝𝐸 sin 𝜃
point charges
Electric field of a point charge:
If there are more than two, potential is the sum of potential energies due to all pairs of charges
𝐸⃗⃗𝑝𝑜𝑖𝑛𝑡 =
1
𝑞
4𝜋𝜖0 𝑟 2
𝑟̂
Dipole moment: 𝑝⃗ = 𝑞𝑠 (from negative to positive charge) 𝐸⃗⃗𝑑𝑖𝑝𝑜𝑙𝑒 ≈
1 2𝑝⃗ 4𝜋𝜖0 𝑟 3
𝑉≡ (on the axis of an electric
dipole) 𝐸⃗⃗𝑑𝑖𝑝𝑜𝑙𝑒 ≈
−1 𝑝⃗ 4𝜋𝜖0 𝑟
3 (on bisecting plane, I.e the
plane that is perpendicular to the dipole moment and halfway between the charges r >> s) Charge densities: 𝜆 =
𝑄
Rod of charge: 𝐸𝑟𝑜𝑑 =
𝐿
| 𝜂=
𝑄 𝐴
| 𝜌=
𝑄 𝑉
4𝜋𝜖0 𝑟√𝑟 2 +(𝐿/2)2
Infinite line of charge: 𝐸𝑟𝑜𝑑 = Ring of charge: (𝐸𝑟𝑖𝑛𝑔 )𝑧 =
1
2|𝜆|
1
𝑄
charge [1V = J/C] 𝑉 = 𝐸𝑠 Electric potential inside a parallel plate capacitor 𝑉=
𝑞
4𝜋𝜖0 𝑟
Electric potential of a point
charge (or sphere) 𝐸=
Δ𝑉 𝑑
4𝜋𝜖0 𝑟
1
𝑧𝑄
4𝜋𝜖0 (𝑧 2 +𝑅 2 )3/2
𝜂 2𝜖0
[1 −
|𝑧| √𝑧 2 +𝑅
] 2
Plane of charge: (𝐸𝑝𝑙𝑎𝑛𝑒 )𝑧 = A sphere of charge 𝐸⃗⃗𝑠𝑝ℎ𝑒𝑟𝑒 =
𝜂
𝐶= 1
2 𝑟̂
(same as point charge but r >= R) Parallel plate capacitor: 𝐸⃗⃗𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑜𝑟 =
Relationship between electric
𝜖0 𝐴 𝑑 𝑄 Δ𝑉𝐶
𝜂 𝜖0
𝐶(Δ𝑉𝐶 )2 Potential energy
=
𝐸
𝜖0 2
𝐸 2 Energy density in [J/m3]
Δ𝑉0 Δ𝑉𝐶
=
𝐶
Dielectric Constant, the
𝐶0
quantities with subscript 0 are quantities with vacuum through 1
𝜂𝑖𝑛𝑑𝑢𝑐𝑒𝑑 = 𝜂0 (1 − ) Dielectric induced 𝜅
plane charge density
𝑖𝑒 = 𝑛𝑒 𝐴𝑣𝑑 Electron current, ne is electron density, A is cross sectional area, vd is electron drift speed 𝑛𝑒 𝑒𝜏𝐴
𝐸 Where e is elementary charge,
𝑚
tau is average collision time of electrons, and m is mass of an electron 𝐼=
𝑄
= 𝑒𝑖𝑒 Conventional current [1A = 1
Δ𝑡
C/s] 𝐽=
𝐼
= 𝑛𝑒 𝑒𝑣𝑑 = 𝜎𝐸 Current density
𝐴
∑ 𝐼𝑖𝑛 = ∑ 𝐼𝑜𝑢𝑡 Kirchhoff’s Junction Law: All input currents must equal output currents 𝑛𝑒 𝑒 2 𝜏
𝜎=
𝜌𝐿
𝑅= 𝐼=
Conductivity | 𝜌 =
𝑚
𝑅
Resistivity
Resistance of a wire [1Ω = 1V/A]
𝐴 Δ𝑉
1 𝜎
Ohm’s Law (for ohmic components)
𝑃 = 𝐼Δ𝑉 = 𝐼 2 𝑅 =
(Δ𝑉)2 𝑅
𝑅 = 𝑅1 + 𝑅2 + ⋯ + 𝑅𝑛 Resistors in series 𝑅=(
1 𝑅1
+
1 𝑅2
+ ⋯+
1 𝑅𝑛
)−1 Resistors in
parallel
RC Series Circuits:
Definition of Capacitance
Time constant 𝜏 = 𝑅𝐶 [s]
Capacitance is charge per volt [1F
Capacitor discharging through a resistor:
= 1C/V]
𝑄
4𝜋𝜖0 𝑟
𝑑𝑠
Capacitors: 𝐶=
2𝜖0
∙ 𝑑𝑠⃗
field and potential
(like a point
charge)
𝑓 − ∫𝑖 𝐸⃗⃗
Δ𝑉 = ∑𝑖 (Δ𝑉)𝑖 = 0 Kirchoff’s Loop Law: The sum of all the potential differences encountered while moving around a loop or closed path is zero 𝐸𝑠 = −
)−1 Capacitors in
[A/m ]
capacitor
𝑑𝑉
1 𝐶𝑛
2
Electric field inside a parallel plate
Δ𝑉 = 𝑉𝑓 − 𝑉𝑖 =
(along the axis that is perpendicular to & in the center of the disk) 4𝜋𝜖0 𝑧 2
Volt is defined as potential per
1
2
+ ⋯+
Current and Resistance
Electric potential obeys superposition principles
(along the axis that is perpendicular to & in the center of the ring) Disk of charge: (𝐸𝑑𝑖𝑠𝑘 )𝑧 =
𝐸0
𝑖𝑒 =
𝑞
1
=
=
𝐴𝑑
|𝑄|
1
(plane that bisects the rod)
If z>>R: (𝐸𝑑𝑖𝑠𝑘 )𝑧 ≈
𝑈𝐶
𝑢=
𝑈𝑑𝑖𝑝𝑜𝑙𝑒 = −𝑝𝐸𝑐𝑜𝑠𝜙 𝑈𝑞
2𝐶
𝐶2
stored in a capacitor
𝜅≡
Electric Potential:
1
+
𝑄2
𝑈𝐶 =
charge of Qin) 1
1 𝐶1
𝐶 = 𝐶1 + 𝐶2 + ⋯ + 𝐶𝑛 Capacitors in parallel
𝑡
𝑡
𝑡
𝑄 = 𝑄0 𝑒 −𝜏 | Δ𝑉 = Δ𝑉0 𝑒 −𝜏 | 𝐼 = 𝐼0 𝑒 −𝜏 Capacitor charging through a resistor: 𝑡
𝑄 = 𝑄𝑚𝑎𝑥 (1 − 𝑒 −𝜏 ) | Δ𝑉 = Δ𝑉𝑚𝑎𝑥 (1 − 𝑡
𝑒 −𝜏 )
Magnetism: Unit for magnetic field strength is Tesla: 1T ≡
1N A∙m
Permeability constant: 𝜇0 = 4𝜋 ×
T∙m 10−7 A
Magnetic field of a moving charge (aka Biot⃗⃗𝑝𝑜𝑖𝑛𝑡 𝑐ℎ𝑎𝑟𝑔𝑒 = 𝜇0 𝑞𝑣sin𝜃 Savart law): 𝐵 2 4𝜋
𝑟
[direction given by right hand rule] ϴ is the angle between v (velocity) and r (distance from charge) ⃗⃗𝑤𝑖𝑟𝑒 ∞ = 𝐵 𝜇0
𝐼𝑟 2
𝜇0 𝐼 2𝜋 𝑑
2 (𝑧 2 +𝑟 2 )3/2
⃗⃗𝑙𝑜𝑜𝑝 = |𝐵
Potential Energy of an inductor: 𝑈𝐿 =
⃗⃗ Torque of a current loop: 𝜏⃗ = 𝜇⃗ × 𝐵
Energy density of a magnetic field: 𝑢𝐵 =
𝐼
2
1
Induction:
2𝜇0
ℰ = 𝑣𝑙𝐵
𝐿𝑛
𝑅
where v is velocity,
Magnetic Field of a center of solenoid ⃗⃗𝑠𝑜𝑙𝑒𝑛𝑜𝑖𝑑 = 𝜇0𝑁𝐼 𝐵
Units of magnetic flux are Webber: 1 Wb =1 T 𝑚2
Magnetic dipole moment of a loop 𝜇 = 𝐴𝐼
(or in terms of vectors, where A is the area vector)
Ampère’s Law
⃗⃗ Φ𝑚 = 𝐴⃗ ∙ 𝐵
If B is everywhere perpendicular to a line, the 𝑓 ⃗⃗ ∙ 𝑑𝑠⃗ = 0 line integral of B is ∫ 𝐵
⃗⃗ ∙ 𝑑𝐴⃗ Non-uniform fields: Φ𝑚 = ∫𝑎𝑟𝑒𝑎 𝑜𝑓 𝑙𝑜𝑜𝑝 𝐵
Ampère’s Law: The total current through a closed curve is the line integral of the magnetic field around the curve
Lenz’s law: There is an induced current in a closed, conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field opposes the change in flux.
⃗⃗ ∙ 𝑑𝑠⃗ = 𝜇0 𝐼𝑡ℎ𝑟𝑜𝑢𝑔ℎ ∮𝐵
Faraday’s Law: an emf is induced around a closed loop if the magnetic flux through the loop changes. The magnitude of the emf is
Magnetic Force:
ℰ=|
Force on a moving point charge (𝛼 is angle between v and B):
The direction is given by Lenz’s law (or ℰ =
⃗⃗ = 𝑞𝑣𝐵 sin 𝛼 (Direction 𝐹⃗𝑜𝑛 𝑞 = 𝑞𝑣⃗ × 𝐵 given by right hand rule) Electric Field inside a metal moving perpendicular to a magnetic field: 𝐸 = 𝑣𝐵 Cyclotron frequency: 𝑓𝑐𝑦𝑐 =
𝑞𝐵 2𝜋𝑚
(m is
mass)
𝑑𝑡
𝑑Φ𝑚 𝑑𝑡
⃗⃗ ∙ | = |𝐵
𝑑𝐴⃗ 𝑑𝑡
+ 𝐴⃗ ∙
𝐼𝐵 𝑡𝑛𝑒
𝜇0 𝐼2 𝜇0 𝑙𝐼1 𝐼2 = 𝐼1 𝑙𝐵2 = 𝐼1 𝑙 = 2𝜋𝑑 2𝜋𝑑
In a uniform field two wires repel if they have currents in opposite directions and
𝑑𝑡
|
Faraday’s law for an N-turn coil: ℰ𝑐𝑜𝑖𝑙 = 𝑁|
𝑑Φ𝑝𝑒𝑟𝑐𝑜𝑖𝑙 𝑑𝑡
)−1
Transformers:
ℰ1 ℰ2
=
𝑁1 𝑁2
1 √𝐿𝐶
LR circuit: 𝐼 = 𝐼0 𝑒 −𝑡/(𝐿/𝑅) (L/R is known as time constant τ)
Instantaneous EMF & voltage and current across resistor ℰ = ℰ0 cos 𝜔𝑡 | 𝑣𝑅 = 𝑉𝑅 cos 𝜔𝑡 | 𝑖𝑅 = 𝐼𝑅 cos 𝜔𝑡 Instantaneous voltage, charge and current for capacitor 𝑣𝐶 = 𝑉𝐶 cos 𝜔𝑡 | 𝑞 = 𝜋 𝐶𝑉𝐶 cos 𝜔𝑡 | 𝑖𝐶 = 𝜔𝐶𝑉𝐶 cos(𝜔𝑡 + ) 2
Note: AC current of capacitor leads voltage by pi/2 or 90˚ Capacitive reactance: 𝑋𝐶 ≡
1 𝜔𝐶
(Units of
ohms) Relationship between peak voltage, peak current & capacitive reactance 𝑉𝐶 = 𝐼𝐶 𝑋𝐶 (this does NOT work for instantaneous voltage and instantaneous current) Peak Current of RC circuit with an AC EMF:
|
𝐼=
ℰ0 √𝑅 2 +𝑋 2
𝑟 𝑑𝐵
Induced E field inside solenoid: 𝐸 = | | 2 𝑑𝑡
More on Inductance:
(Where t is the
thickness of the metal plate and n is the charge carrier density C/m3)
⃗⃗ 𝑑𝐵
)
Inductance in Henrys 𝐿 =
Hall voltage: Δ𝑉𝐻 =
𝐹⃗𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑤𝑖𝑟𝑒𝑠
−
𝑑Φ𝑚
+⋯+
AC Circuits:
𝑖
𝑖
𝐿2
𝑄 = 𝑄0 cos 𝜔𝑡 | 𝐼 = 𝐼𝑚𝑎𝑥 sin 𝜔𝑡 | 𝜔 =
𝑙
If B is everywhere tangent to a line of length l and has the same magnitude B at 𝑓 ⃗⃗ ∙ 𝑑𝑠⃗ = 𝐵𝑙 every point, then ∫ 𝐵
1
LC electric oscillator: (omega is angular oscillation frequency) 𝜔 = 2𝜋𝑓
Magnetic Flux: Φ𝑚 = 𝐴𝐵 cos 𝜃
2 𝑟
1
𝐿1
Inductors in Parallel: 𝐿 = ( + 1
𝑣 2 𝑙2 𝐵2
𝐵2
Inductors in Series: 𝐿 = 𝐿1 + 𝐿2 + ⋯ + 𝐿𝑛
Motional electromotive force of a conductor moving perpendicular to a uniform magnetic field
Power: 𝑃 = 𝐹𝑣 =
1
𝐿 ∫0 𝐼 𝑑𝐼 = 𝐿𝐼 2
R is resistance
𝜇0 𝑁𝐼
⃗⃗𝑐𝑜𝑖𝑙 = |𝐵
they attract if they have currents in the same direction
1 henry = 1H ≡ 1
Wb A
=1
Φ𝑚
Instantaneous voltage across inductor: 𝑣𝐿 =
𝐼
𝐿
Tm2 A 𝜇0 𝑁 2 𝐴
Potential difference across an inductor: 𝑑𝐼 𝑑𝑡
1 𝑅𝐶
Inductor Circuits:
Inductance of a solenoid: 𝐿𝑠𝑜𝑙𝑒𝑛𝑜𝑖𝑑 =
Δ𝑉𝐿 = −𝐿
Crossover frequency when 𝑉𝐶 = 𝑉𝑅 : 𝜔𝐶 =
𝑙
𝑑𝑖𝐿 𝑑𝑡
Instantaneous current across inductor: 𝑖𝐿 = 𝜋
𝐼𝐿 cos (𝜔𝑡 − ) 2
Note: AC current of inductor lags voltage by pi/2 or 90˚ Inductive Reactance: 𝑋𝐿 ≡ 𝜔𝐿
𝑝𝜆 𝑎
Peak voltage across inductor: 𝑉𝐿 = 𝐼𝐿 𝑋𝐿
𝜃𝑝 =
Series RLC Circuits:
fringes (single slit)
Peak Current: 𝐼 =
ℰ0 √𝑅 2 +(𝑋𝐿 −𝑋𝐶 )2
=
ℰ0 𝑍
Peak Voltages: 𝑉𝑅 = 𝐼𝑅 | 𝑉𝐿 = 𝐼𝑋𝐿 | 𝑉𝐶 = 𝐼𝑋𝐶 Impedance: 𝑍 = √𝑅2 + (𝑋𝐿 − 𝑋𝐶 )2 Phase angle between emf and current: 𝜙 = arctan (
𝑋𝐿 −𝑋𝐶 𝑅
) 1
1
(𝑉𝑟𝑚𝑠 )2
2
𝑅
𝑃𝑅 = 𝐼𝑅 2 𝑅 = (𝐼𝑟𝑚𝑠 )2 𝑅 =
= 𝐼𝑟𝑚𝑠 𝑉𝑟𝑚𝑠
Average power of a resistor 𝐼𝑅
|𝑉𝑟𝑚𝑠 =
𝑉𝑅 √2
| ℰ𝑟𝑚𝑠 =
ℰ𝑅 √2
Root
mean square quantities (peak / sqrt(2)) 1
𝑝𝐶 = 𝑣𝐶 𝑖𝐶 = − 𝜔𝐶𝑉𝐶 2 sin 2𝜔𝑡 2
Instantaneous energy dissipation of a capacitor 1
𝑃𝑠𝑜𝑢𝑟𝑐𝑒 = 𝐼ℰ0 cos 𝜙 = 𝐼𝑟𝑚𝑠 ℰ𝑟𝑚𝑠 cos 𝜙 2
Average power supplied by EMF. cos 𝜙 is called the power factor
Optics: 𝜔 = 2𝜋𝑓 Angular frequency 𝑘=
2𝜋 𝜆
Wave number
𝐷𝑅 = 𝑎 sin(𝑘𝑥 − 𝜔𝑡) Right travelling wave 𝐷𝐿 = 𝑎 sin(𝑘𝑥 + 𝜔𝑡) Left travelling wave 𝐷𝑅 + 𝐷𝐿 = (2𝑎 sin 𝑘𝑥) cos 𝜔𝑡 Superposition 𝑥𝑚 =
𝑚𝜆 2
𝑚 = 0, 1, 2, 3 … Position of mth
node 𝜃𝑚 =
𝑚𝜆 𝑑
𝑚 = 0, 1, 2, 3 … angles of bright
fringes 𝑦𝑚 =
𝑚𝜆𝐿 𝑑
𝑚 = 0, 1, 2, 3 … position of bright
fringes 1 𝜆𝐿
𝑦′𝑚 = (𝑚 + ) 2
circular-aperture diffraction
(diameter d) 𝑤≈
2.44𝜆𝐿 𝐷
width of central maximum
𝜆
𝜃𝑚𝑖𝑛 = 𝑎 limit of resolution (a is width of Optics section may be incomplete
√𝐿𝐶
Power in AC Circuits:
√2
1.22𝜆 𝑑
slit)
Resonance frequency: 𝜔0 =
𝐼𝑟𝑚𝑠 =
𝜃1 =
𝑝 = 1, 2, 3 … angles of dark
𝑑
𝑚 = 0, 1, 2, 3 … position
of dark fringes 𝐼𝑑𝑜𝑢𝑏𝑙𝑒 = 4𝐼1 cos 2 (
𝜋𝑑 𝜆𝐿
𝑦) Intensity of double
slit interference pattern
Flux:
𝐶=(
By: Kareem Halabi
Flux on a surface: Φ𝑒 = 𝐸𝐴 cos 𝜃 = ∫𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝐸⃗⃗ ∙ 𝑑𝐴⃗ (theta is between normal of
series
Electric Fields:
plane and the electric field)
Permittivity constant: 𝜖0 = 8.85 ×
Gauss’s law Φ𝑒 =
C2 10−12 2 Nm
𝑄𝑖𝑛 𝜖0
(encompassing a net
Coulomb’s law: 𝐹𝑝𝑜𝑖𝑛𝑡 =
|𝑞1 ||𝑞2 |
4𝜋𝜖0
𝑟2
𝑈𝑒𝑙𝑒𝑐 = 𝑞𝐸𝑠 q is charge, E is elec-field and s is distance
Charge in a field: 𝐹⃗𝑜𝑛 𝑞 = 𝑞𝐸⃗⃗ Torque of Dipole in a uniform Field:
𝑈𝑒𝑙𝑒𝑐 =
1
𝑞1 𝑞2
4𝜋𝜖0
𝑟
Potential between two
𝜏 = 𝑝𝐸 sin 𝜃
point charges
Electric field of a point charge:
If there are more than two, potential is the sum of potential energies due to all pairs of charges
𝐸⃗⃗𝑝𝑜𝑖𝑛𝑡 =
1
𝑞
4𝜋𝜖0 𝑟 2
𝑟̂
Dipole moment: 𝑝⃗ = 𝑞𝑠 (from negative to positive charge) 𝐸⃗⃗𝑑𝑖𝑝𝑜𝑙𝑒 ≈
1 2𝑝⃗ 4𝜋𝜖0 𝑟 3
𝑉≡ (on the axis of an electric
dipole) 𝐸⃗⃗𝑑𝑖𝑝𝑜𝑙𝑒 ≈
−1 𝑝⃗ 4𝜋𝜖0 𝑟
3 (on bisecting plane, I.e the
plane that is perpendicular to the dipole moment and halfway between the charges r >> s) Charge densities: 𝜆 =
𝑄
Rod of charge: 𝐸𝑟𝑜𝑑 =
𝐿
| 𝜂=
𝑄 𝐴
| 𝜌=
𝑄 𝑉
4𝜋𝜖0 𝑟√𝑟 2 +(𝐿/2)2
Infinite line of charge: 𝐸𝑟𝑜𝑑 = Ring of charge: (𝐸𝑟𝑖𝑛𝑔 )𝑧 =
1
2|𝜆|
1
𝑄
charge [1V = J/C] 𝑉 = 𝐸𝑠 Electric potential inside a parallel plate capacitor 𝑉=
𝑞
4𝜋𝜖0 𝑟
Electric potential of a point
charge (or sphere) 𝐸=
Δ𝑉 𝑑
4𝜋𝜖0 𝑟
1
𝑧𝑄
4𝜋𝜖0 (𝑧 2 +𝑅 2 )3/2
𝜂 2𝜖0
[1 −
|𝑧| √𝑧 2 +𝑅
] 2
Plane of charge: (𝐸𝑝𝑙𝑎𝑛𝑒 )𝑧 = A sphere of charge 𝐸⃗⃗𝑠𝑝ℎ𝑒𝑟𝑒 =
𝜂
𝐶= 1
2 𝑟̂
(same as point charge but r >= R) Parallel plate capacitor: 𝐸⃗⃗𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑜𝑟 =
Relationship between electric
𝜖0 𝐴 𝑑 𝑄 Δ𝑉𝐶
𝜂 𝜖0
𝐶(Δ𝑉𝐶 )2 Potential energy
=
𝐸
𝜖0 2
𝐸 2 Energy density in [J/m3]
Δ𝑉0 Δ𝑉𝐶
=
𝐶
Dielectric Constant, the
𝐶0
quantities with subscript 0 are quantities with vacuum through 1
𝜂𝑖𝑛𝑑𝑢𝑐𝑒𝑑 = 𝜂0 (1 − ) Dielectric induced 𝜅
plane charge density
𝑖𝑒 = 𝑛𝑒 𝐴𝑣𝑑 Electron current, ne is electron density, A is cross sectional area, vd is electron drift speed 𝑛𝑒 𝑒𝜏𝐴
𝐸 Where e is elementary charge,
𝑚
tau is average collision time of electrons, and m is mass of an electron 𝐼=
𝑄
= 𝑒𝑖𝑒 Conventional current [1A = 1
Δ𝑡
C/s] 𝐽=
𝐼
= 𝑛𝑒 𝑒𝑣𝑑 = 𝜎𝐸 Current density
𝐴
∑ 𝐼𝑖𝑛 = ∑ 𝐼𝑜𝑢𝑡 Kirchhoff’s Junction Law: All input currents must equal output currents 𝑛𝑒 𝑒 2 𝜏
𝜎=
𝜌𝐿
𝑅= 𝐼=
Conductivity | 𝜌 =
𝑚
𝑅
Resistivity
Resistance of a wire [1Ω = 1V/A]
𝐴 Δ𝑉
1 𝜎
Ohm’s Law (for ohmic components)
𝑃 = 𝐼Δ𝑉 = 𝐼 2 𝑅 =
(Δ𝑉)2 𝑅
𝑅 = 𝑅1 + 𝑅2 + ⋯ + 𝑅𝑛 Resistors in series 𝑅=(
1 𝑅1
+
1 𝑅2
+ ⋯+
1 𝑅𝑛
)−1 Resistors in
parallel
RC Series Circuits:
Definition of Capacitance
Time constant 𝜏 = 𝑅𝐶 [s]
Capacitance is charge per volt [1F
Capacitor discharging through a resistor:
= 1C/V]
𝑄
4𝜋𝜖0 𝑟
𝑑𝑠
Capacitors: 𝐶=
2𝜖0
∙ 𝑑𝑠⃗
field and potential
(like a point
charge)
𝑓 − ∫𝑖 𝐸⃗⃗
Δ𝑉 = ∑𝑖 (Δ𝑉)𝑖 = 0 Kirchoff’s Loop Law: The sum of all the potential differences encountered while moving around a loop or closed path is zero 𝐸𝑠 = −
)−1 Capacitors in
[A/m ]
capacitor
𝑑𝑉
1 𝐶𝑛
2
Electric field inside a parallel plate
Δ𝑉 = 𝑉𝑓 − 𝑉𝑖 =
(along the axis that is perpendicular to & in the center of the disk) 4𝜋𝜖0 𝑧 2
Volt is defined as potential per
1
2
+ ⋯+
Current and Resistance
Electric potential obeys superposition principles
(along the axis that is perpendicular to & in the center of the ring) Disk of charge: (𝐸𝑑𝑖𝑠𝑘 )𝑧 =
𝐸0
𝑖𝑒 =
𝑞
1
=
=
𝐴𝑑
|𝑄|
1
(plane that bisects the rod)
If z>>R: (𝐸𝑑𝑖𝑠𝑘 )𝑧 ≈
𝑈𝐶
𝑢=
𝑈𝑑𝑖𝑝𝑜𝑙𝑒 = −𝑝𝐸𝑐𝑜𝑠𝜙 𝑈𝑞
2𝐶
𝐶2
stored in a capacitor
𝜅≡
Electric Potential:
1
+
𝑄2
𝑈𝐶 =
charge of Qin) 1
1 𝐶1
𝐶 = 𝐶1 + 𝐶2 + ⋯ + 𝐶𝑛 Capacitors in parallel
𝑡
𝑡
𝑡
𝑄 = 𝑄0 𝑒 −𝜏 | Δ𝑉 = Δ𝑉0 𝑒 −𝜏 | 𝐼 = 𝐼0 𝑒 −𝜏 Capacitor charging through a resistor: 𝑡
𝑄 = 𝑄𝑚𝑎𝑥 (1 − 𝑒 −𝜏 ) | Δ𝑉 = Δ𝑉𝑚𝑎𝑥 (1 − 𝑡
𝑒 −𝜏 )
Magnetism: Unit for magnetic field strength is Tesla: 1T ≡
1N A∙m
Permeability constant: 𝜇0 = 4𝜋 ×
T∙m 10−7 A
Magnetic field of a moving charge (aka Biot⃗⃗𝑝𝑜𝑖𝑛𝑡 𝑐ℎ𝑎𝑟𝑔𝑒 = 𝜇0 𝑞𝑣sin𝜃 Savart law): 𝐵 2 4𝜋
𝑟
[direction given by right hand rule] ϴ is the angle between v (velocity) and r (distance from charge) ⃗⃗𝑤𝑖𝑟𝑒 ∞ = 𝐵 𝜇0
𝐼𝑟 2
𝜇0 𝐼 2𝜋 𝑑
2 (𝑧 2 +𝑟 2 )3/2
⃗⃗𝑙𝑜𝑜𝑝 = |𝐵
Potential Energy of an inductor: 𝑈𝐿 =
⃗⃗ Torque of a current loop: 𝜏⃗ = 𝜇⃗ × 𝐵
Energy density of a magnetic field: 𝑢𝐵 =
𝐼
2
1
Induction:
2𝜇0
ℰ = 𝑣𝑙𝐵
𝐿𝑛
𝑅
where v is velocity,
Magnetic Field of a center of solenoid ⃗⃗𝑠𝑜𝑙𝑒𝑛𝑜𝑖𝑑 = 𝜇0𝑁𝐼 𝐵
Units of magnetic flux are Webber: 1 Wb =1 T 𝑚2
Magnetic dipole moment of a loop 𝜇 = 𝐴𝐼
(or in terms of vectors, where A is the area vector)
Ampère’s Law
⃗⃗ Φ𝑚 = 𝐴⃗ ∙ 𝐵
If B is everywhere perpendicular to a line, the 𝑓 ⃗⃗ ∙ 𝑑𝑠⃗ = 0 line integral of B is ∫ 𝐵
⃗⃗ ∙ 𝑑𝐴⃗ Non-uniform fields: Φ𝑚 = ∫𝑎𝑟𝑒𝑎 𝑜𝑓 𝑙𝑜𝑜𝑝 𝐵
Ampère’s Law: The total current through a closed curve is the line integral of the magnetic field around the curve
Lenz’s law: There is an induced current in a closed, conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field opposes the change in flux.
⃗⃗ ∙ 𝑑𝑠⃗ = 𝜇0 𝐼𝑡ℎ𝑟𝑜𝑢𝑔ℎ ∮𝐵
Faraday’s Law: an emf is induced around a closed loop if the magnetic flux through the loop changes. The magnitude of the emf is
Magnetic Force:
ℰ=|
Force on a moving point charge (𝛼 is angle between v and B):
The direction is given by Lenz’s law (or ℰ =
⃗⃗ = 𝑞𝑣𝐵 sin 𝛼 (Direction 𝐹⃗𝑜𝑛 𝑞 = 𝑞𝑣⃗ × 𝐵 given by right hand rule) Electric Field inside a metal moving perpendicular to a magnetic field: 𝐸 = 𝑣𝐵 Cyclotron frequency: 𝑓𝑐𝑦𝑐 =
𝑞𝐵 2𝜋𝑚
(m is
mass)
𝑑𝑡
𝑑Φ𝑚 𝑑𝑡
⃗⃗ ∙ | = |𝐵
𝑑𝐴⃗ 𝑑𝑡
+ 𝐴⃗ ∙
𝐼𝐵 𝑡𝑛𝑒
𝜇0 𝐼2 𝜇0 𝑙𝐼1 𝐼2 = 𝐼1 𝑙𝐵2 = 𝐼1 𝑙 = 2𝜋𝑑 2𝜋𝑑
In a uniform field two wires repel if they have currents in opposite directions and
𝑑𝑡
|
Faraday’s law for an N-turn coil: ℰ𝑐𝑜𝑖𝑙 = 𝑁|
𝑑Φ𝑝𝑒𝑟𝑐𝑜𝑖𝑙 𝑑𝑡
)−1
Transformers:
ℰ1 ℰ2
=
𝑁1 𝑁2
1 √𝐿𝐶
LR circuit: 𝐼 = 𝐼0 𝑒 −𝑡/(𝐿/𝑅) (L/R is known as time constant τ)
Instantaneous EMF & voltage and current across resistor ℰ = ℰ0 cos 𝜔𝑡 | 𝑣𝑅 = 𝑉𝑅 cos 𝜔𝑡 | 𝑖𝑅 = 𝐼𝑅 cos 𝜔𝑡 Instantaneous voltage, charge and current for capacitor 𝑣𝐶 = 𝑉𝐶 cos 𝜔𝑡 | 𝑞 = 𝜋 𝐶𝑉𝐶 cos 𝜔𝑡 | 𝑖𝐶 = 𝜔𝐶𝑉𝐶 cos(𝜔𝑡 + ) 2
Note: AC current of capacitor leads voltage by pi/2 or 90˚ Capacitive reactance: 𝑋𝐶 ≡
1 𝜔𝐶
(Units of
ohms) Relationship between peak voltage, peak current & capacitive reactance 𝑉𝐶 = 𝐼𝐶 𝑋𝐶 (this does NOT work for instantaneous voltage and instantaneous current) Peak Current of RC circuit with an AC EMF:
|
𝐼=
ℰ0 √𝑅 2 +𝑋 2
𝑟 𝑑𝐵
Induced E field inside solenoid: 𝐸 = | | 2 𝑑𝑡
More on Inductance:
(Where t is the
thickness of the metal plate and n is the charge carrier density C/m3)
⃗⃗ 𝑑𝐵
)
Inductance in Henrys 𝐿 =
Hall voltage: Δ𝑉𝐻 =
𝐹⃗𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑤𝑖𝑟𝑒𝑠
−
𝑑Φ𝑚
+⋯+
AC Circuits:
𝑖
𝑖
𝐿2
𝑄 = 𝑄0 cos 𝜔𝑡 | 𝐼 = 𝐼𝑚𝑎𝑥 sin 𝜔𝑡 | 𝜔 =
𝑙
If B is everywhere tangent to a line of length l and has the same magnitude B at 𝑓 ⃗⃗ ∙ 𝑑𝑠⃗ = 𝐵𝑙 every point, then ∫ 𝐵
1
LC electric oscillator: (omega is angular oscillation frequency) 𝜔 = 2𝜋𝑓
Magnetic Flux: Φ𝑚 = 𝐴𝐵 cos 𝜃
2 𝑟
1
𝐿1
Inductors in Parallel: 𝐿 = ( + 1
𝑣 2 𝑙2 𝐵2
𝐵2
Inductors in Series: 𝐿 = 𝐿1 + 𝐿2 + ⋯ + 𝐿𝑛
Motional electromotive force of a conductor moving perpendicular to a uniform magnetic field
Power: 𝑃 = 𝐹𝑣 =
1
𝐿 ∫0 𝐼 𝑑𝐼 = 𝐿𝐼 2
R is resistance
𝜇0 𝑁𝐼
⃗⃗𝑐𝑜𝑖𝑙 = |𝐵
they attract if they have currents in the same direction
1 henry = 1H ≡ 1
Wb A
=1
Φ𝑚
Instantaneous voltage across inductor: 𝑣𝐿 =
𝐼
𝐿
Tm2 A 𝜇0 𝑁 2 𝐴
Potential difference across an inductor: 𝑑𝐼 𝑑𝑡
1 𝑅𝐶
Inductor Circuits:
Inductance of a solenoid: 𝐿𝑠𝑜𝑙𝑒𝑛𝑜𝑖𝑑 =
Δ𝑉𝐿 = −𝐿
Crossover frequency when 𝑉𝐶 = 𝑉𝑅 : 𝜔𝐶 =
𝑙
𝑑𝑖𝐿 𝑑𝑡
Instantaneous current across inductor: 𝑖𝐿 = 𝜋
𝐼𝐿 cos (𝜔𝑡 − ) 2
Note: AC current of inductor lags voltage by pi/2 or 90˚ Inductive Reactance: 𝑋𝐿 ≡ 𝜔𝐿
𝑝𝜆 𝑎
Peak voltage across inductor: 𝑉𝐿 = 𝐼𝐿 𝑋𝐿
𝜃𝑝 =
Series RLC Circuits:
fringes (single slit)
Peak Current: 𝐼 =
ℰ0 √𝑅 2 +(𝑋𝐿 −𝑋𝐶 )2
=
ℰ0 𝑍
Peak Voltages: 𝑉𝑅 = 𝐼𝑅 | 𝑉𝐿 = 𝐼𝑋𝐿 | 𝑉𝐶 = 𝐼𝑋𝐶 Impedance: 𝑍 = √𝑅2 + (𝑋𝐿 − 𝑋𝐶 )2 Phase angle between emf and current: 𝜙 = arctan (
𝑋𝐿 −𝑋𝐶 𝑅
) 1
1
(𝑉𝑟𝑚𝑠 )2
2
𝑅
𝑃𝑅 = 𝐼𝑅 2 𝑅 = (𝐼𝑟𝑚𝑠 )2 𝑅 =
= 𝐼𝑟𝑚𝑠 𝑉𝑟𝑚𝑠
Average power of a resistor 𝐼𝑅
|𝑉𝑟𝑚𝑠 =
𝑉𝑅 √2
| ℰ𝑟𝑚𝑠 =
ℰ𝑅 √2
Root
mean square quantities (peak / sqrt(2)) 1
𝑝𝐶 = 𝑣𝐶 𝑖𝐶 = − 𝜔𝐶𝑉𝐶 2 sin 2𝜔𝑡 2
Instantaneous energy dissipation of a capacitor 1
𝑃𝑠𝑜𝑢𝑟𝑐𝑒 = 𝐼ℰ0 cos 𝜙 = 𝐼𝑟𝑚𝑠 ℰ𝑟𝑚𝑠 cos 𝜙 2
Average power supplied by EMF. cos 𝜙 is called the power factor
Optics: 𝜔 = 2𝜋𝑓 Angular frequency 𝑘=
2𝜋 𝜆
Wave number
𝐷𝑅 = 𝑎 sin(𝑘𝑥 − 𝜔𝑡) Right travelling wave 𝐷𝐿 = 𝑎 sin(𝑘𝑥 + 𝜔𝑡) Left travelling wave 𝐷𝑅 + 𝐷𝐿 = (2𝑎 sin 𝑘𝑥) cos 𝜔𝑡 Superposition 𝑥𝑚 =
𝑚𝜆 2
𝑚 = 0, 1, 2, 3 … Position of mth
node 𝜃𝑚 =
𝑚𝜆 𝑑
𝑚 = 0, 1, 2, 3 … angles of bright
fringes 𝑦𝑚 =
𝑚𝜆𝐿 𝑑
𝑚 = 0, 1, 2, 3 … position of bright
fringes 1 𝜆𝐿
𝑦′𝑚 = (𝑚 + ) 2
circular-aperture diffraction
(diameter d) 𝑤≈
2.44𝜆𝐿 𝐷
width of central maximum
𝜆
𝜃𝑚𝑖𝑛 = 𝑎 limit of resolution (a is width of Optics section may be incomplete
√𝐿𝐶
Power in AC Circuits:
√2
1.22𝜆 𝑑
slit)
Resonance frequency: 𝜔0 =
𝐼𝑟𝑚𝑠 =
𝜃1 =
𝑝 = 1, 2, 3 … angles of dark
𝑑
𝑚 = 0, 1, 2, 3 … position
of dark fringes 𝐼𝑑𝑜𝑢𝑏𝑙𝑒 = 4𝐼1 cos 2 (
𝜋𝑑 𝜆𝐿
𝑦) Intensity of double
slit interference pattern
