Time Dilation
 Light moves at speed 𝑐 in all frames
 Two events that occur simultaneously in one
reference frame may not be simultaneous in
another frame
 Two events that occur at the same location
in frame 𝐴, separated by a time 𝑡𝐴 , will take
a greater time 𝑡𝐵 = 𝛾 𝑡𝐴 in frame 𝐵 moving
at velocity 𝑣 with respect to frame 𝐴
 𝛾=
1
2
√1−𝑣2
(with 𝛾 ≥ 1)
𝑐
Head Start Effect



Two clocks are positioned at the end of a train of length 𝐿 (as
measured in its own frame. The clocks are synchronized in the
train’s reference frame
The train travels past an observer at speed 𝑣. The observer will
see the rear clock showing a higher reading than the front clock
𝐿𝑣
by Δ𝑡 = 𝑐 2
The rear clock does not tick faster than the front clock, it simply
remains a fixed time ahead of the front clock
Length Contraction
 An object in frame 𝐴 moves with velocity 𝑣 relative to frame 𝐵.
The object has length 𝑙′ along this direction of motion in 𝐴’s
frame. The object has a shorter length 𝑙 =
𝑙′
𝛾
in 𝐵’s frame
 Length contraction does not apply in the directions transverse to
the direction of motion (set by 𝑣) between two reference frames
Example
A muon moves from a height ℎ straight down towards the Earth
with a large velocity 𝑣. It decays after time 𝑇 in its own frame.
Can the muon reach the ground?


In the Earth’s frame, time dilation applies. The muon
has to travel a distance ℎ in time 𝛾 𝑇
In the muon’s frame, length contraction kicks in. The
ℎ
muon must travel a distance 𝛾 in time 𝑇
Lorentz Transformation


Frame 𝑆′ moves at velocity 𝑣 relative to frame 𝑆
The length and time between two events (an event is anything that
has space and time coordinates) in 𝑆′ are related to those in 𝑆 by
Δ𝑥 = 𝛾(Δ𝑥 ′ + 𝑣 Δ𝑡 ′ )
𝑣
Δ𝑡 = 𝛾 (Δ𝑡 ′ + 2 Δ𝑥 ′ )
𝑐


Δ𝑥′ = 𝛾(Δ𝑥 − 𝑣 Δ𝑡)
𝑣
Δ𝑡′ = 𝛾 (Δ𝑡 − 2 Δ𝑥)
𝑐
Since there is no transverse length contraction, Δ𝑦 = Δ𝑦 ′ and Δ𝑧 = Δ𝑧 ′
Δ𝑥 2 − 𝑐 2 Δ𝑡 2 has the same value in all reference frames
Velocity Addition

An object moves at speed 𝑣1 with respect to frame 𝑆′. Frame 𝑆′ moves
at speed 𝑣2 with respect to frame 𝑆 in the same direction of motion as
the object. The speed 𝑢 of the object in frame 𝑆 is
𝑢=


𝑣1 + 𝑣2
𝑣 𝑣
1 + 122
𝑐
An object moving at speed 𝑐 in one reference frame moves at speed 𝑐
in all reference frames
An object moving slower than 𝑐 in one reference frame moves slower
than 𝑐 in all reference frames
Momentum and Energy

The relativistic energy and momentum of a particle moving with velocity 𝑣⃗ are given by
𝑝⃗ = 𝛾𝑚𝑣⃗
𝐸 = 𝛾𝑚𝑐 2

𝑝⃗ ≈ 𝑚𝑣⃗
1
𝐸 ≈ 𝑚𝑐 2 + 𝑚 𝑣 2
2
A particle moves at speed 𝑢′ in frame 𝑆′, which moves at speed 𝑣
relative to frame 𝑆. Momentum (𝛾𝑢′ 𝑚𝑢′) and energy (𝛾𝑢′ 𝑚𝑐 2 ) in 𝑆′ is
related to that in 𝑆 by the same Lorentz transformation as space and
time (with 𝛾 ≡ 𝛾𝑣 )
𝐸 = 𝛾(𝐸 ′ + 𝑣 𝑝′ )
𝑣
𝑝 = 𝛾 (𝑝′ + 2 𝐸 ′ )
𝑐


Slow speeds
(𝑣 ≪ 𝑐)
𝐸′ = 𝛾(𝐸 − 𝑣 𝑝)
𝑣
𝑝′ = 𝛾 (𝑝 − 2 𝐸)
𝑐
Energy and momentum are conserved in collisions within the same reference frame
The invariant mass formula 𝐸 2 − 𝑝2 𝑐 2 = 𝑚2 𝑐 4 applies any mass in any reference frame
Cartesian Coordinates
Polar Coordinates
An oldie but a goodie, yet not always the best choice!
Area of a circle in Cartesian coordinates
√𝑅2 −𝑥 2
𝑅
∫ ∫
𝑝𝑎𝑖𝑛
𝑑𝑥 𝑑𝑦 ⇒
𝑥 = 𝑟 Cos[𝜃]
𝑟 = √𝑥 2 + 𝑦 2
𝑦 = 𝑟 Sin[𝜃]
𝜃 = ArcTan [
𝜋𝑅 2
−√𝑅2 −𝑥 2
−𝑅
Polar
to
Cartesian
𝑦
]
𝑥
Cartesian
to
Polar
Unit Vectors
Area of a circle in Polar coordinates
2𝜋
∫
𝑅
𝑒𝑎𝑠𝑦
∫ 𝑟 𝑑𝑟 𝑑𝜃 ⇒
0
𝜋𝑅 2
0
𝑥̂ = Cos[𝜃] 𝜌̂ − Sin[𝜃] 𝜃̂
𝑟̂ = Cos[𝜃] 𝑥̂ + Sin[𝜃] 𝑦̂
𝑦̂ = Sin[𝜃] 𝜌̂ + Cos[𝜃] 𝜃̂
𝜃̂ = −Sin[𝜃] 𝑥̂ + Cos[𝜃] 𝑦̂
Area Element
Area Element
𝑑𝑉 = 𝑑𝑥 𝑑𝑦
𝑑𝑉 = 𝑟 𝑑𝑟 𝑑𝜃
Cylindrical Coordinates
Spherical Coordinates
𝑥 = 𝜌 Cos[𝜃]
𝜌 = √𝑥 2 + 𝑦 2
𝑦 = 𝜌 Sin[𝜃]
𝜃 = ArcTan [
𝑧=𝑧
𝑧=𝑧
Volume Element
𝑑𝑉 = 𝑟 2 Sin[𝜃] 𝑑𝑟 𝑑𝜃 𝑑𝜙
𝑦
]
𝑥
𝑟 = √𝑥 2 + 𝑦 2 + 𝑧 2
(𝑥 2 + 𝑦 2 )1/2
𝜃
=
ArcTan
[
]
𝑦 = 𝑟 Sin[𝜃] Sin[𝜙]
𝑧
𝑦
𝜙 = ArcTan [ ]
𝑧 = 𝑟 Cos[𝜃]
𝑥
Volume Element
𝑥 = 𝑟 Sin[𝜃] Cos[𝜙]
𝑑𝑉 = 𝑟 2 Sin[𝜃] 𝑑𝑟 𝑑𝜃 𝑑𝜙
Dot Product
Cross Product
ሬ⃗ = 𝐴𝑥 𝐵𝑥 + 𝐴𝑦 𝐵𝑦 + 𝐴𝑧 𝐵𝑧
𝐴⃗ ∙ 𝐵
ሬ⃗หCos[𝜃]
= ห𝐴⃗หห𝐵
𝐴⃗ = 𝐴𝑥 𝑥̂ + 𝐴𝑦 𝑦̂ + 𝐴𝑧 𝑧̂
ሬ⃗ = 𝐵𝑥 𝑥̂ + 𝐵𝑦 𝑦̂ + 𝐵𝑧 𝑧̂
𝐵



ሬ⃗ is a scalar
𝐴⃗ ∙ 𝐵
2
ሬ⃗ = 𝐵
ሬ⃗ ∙ 𝐴⃗
𝐴⃗ ∙ 𝐵
𝐴⃗ ∙ 𝐴⃗ = ห𝐴⃗ห
For any unit vector 𝑛̂, 𝐴⃗ ∙ 𝑛̂ represents the length
of 𝐴⃗ along the direction 𝑛̂
ሬ⃗ = ൫𝐴𝑦 𝐵𝑧 − 𝐴𝑧 𝐵𝑦 ൯𝑥̂
𝐴⃗ × 𝐵
+ + (𝐴𝑧 𝐵𝑥 − 𝐴𝑥 𝐵𝑧 )𝑦̂
+
+ ൫𝐴𝑥 𝐵𝑦 − 𝐴𝑦 𝐵𝑥 ൯𝑧̂

ሬ⃗ is a vector
𝐴⃗ × 𝐵

ሬ⃗ = −𝐵
ሬ⃗ × 𝐴⃗
𝐴⃗ × 𝐵

Direction given by right-hand rule and magnitude
ሬ⃗
𝐴⃗ × 𝐴⃗ = 0
ሬ⃗ห = ห𝐴⃗หห𝐵
ሬ⃗หSin[𝜃] equal to parallelogram area
ห𝐴⃗ × 𝐵
Law of Cosines


ሬ⃗
Triangle defined by vectors 𝐴⃗ and 𝐵
ሬ⃗
Thirds leg given by 𝐶⃗ = 𝐴⃗ − 𝐵
𝟐
𝟐
𝟐
ሬ⃗ห = ห𝑨
ሬ⃗ห + ห𝑩
ሬሬ⃗ห − 𝟐ห𝑨
ሬ⃗หห𝑩
ሬሬ⃗ห𝐂𝐨𝐬[𝜽]
ห𝑪
Proof
ሬ⃗൯ ⋅ ൫𝐴⃗ − 𝐵
ሬ⃗൯
𝐶⃗ ⋅ 𝐶⃗ = ൫𝐴⃗ − 𝐵
2
2
ሬ⃗ห − 2 𝐴⃗ ⋅ 𝐵
ሬ⃗
= ห𝐴⃗ห + ห𝐵
Trig Functions
Cos[0] = 1
Sin[0] = 0
𝜋
1
𝜋
√3
Sin [ ] =
Cos [ ] =
6
2
6
2
𝜋
√2
𝜋
√2
Sin [ ] =
Cos [ ] =
4
2
4
2
𝜋
𝜋
1
√3
Sin [ ] =
Cos [ ] =
3
2
3
2
𝜋
𝜋
Cos [ ] = 0
Sin [ ] = 1
2
2
Cos[𝜋 − 𝑥] = −Cos[𝑥] Sin[𝜋 − 𝑥] = Sin[𝑥]
∞
𝑥3 𝑥5
𝑥 2𝑗+1
𝑒 𝑖𝑥 − 𝑒 −𝑖𝑥
Sin[𝑥] = 𝑥 − + − ⋯ = ∑(−1)𝑗
=
(2𝑗 + 1)!
3! 5!
2
𝑗=0
∞
Cos[𝑥] = 1 −
𝑥2 𝑥4
𝑥 2𝑗
𝑒 𝑖𝑥 + 𝑒 −𝑖𝑥
+ − ⋯ = ∑(−1)𝑗
=
(2𝑗)!
2! 4!
2
𝑗=0
Sin[𝑥 + 𝑦] = Sin[𝑥]Cos[𝑦] + Cos[𝑥]Sin[𝑦]
Cos[𝑥 + 𝑦] = Cos[𝑥]Cos[𝑦] − Sin[𝑥]Sin[𝑦]
Sin[2𝑥] = 2Sin[𝑥]Cos[𝑥]
Cos[2𝑥] = Cos[𝑥]2 − Sin[𝑥]2
= 2Cos[𝑥]2 − 1 = 1 − 2Sin[𝑥]2
Gauss’s Law
Surface Integrals
∮ 𝑑𝑎
𝑄 in
𝐸ሬ⃗ = Electric field
∮ 𝐸ሬ⃗ [𝑟⃗] ∙ 𝑑𝑎⃗ =
1
𝜖0
𝜖0 =
surface
4𝜋𝑘
𝑑𝑎⃗ = Normal vector of infinitesimal area
Integrate over
infinitesimal
area elements
surface
∮ 𝑓[𝑟⃗]𝑑𝑎
surface
𝑄 in = Charge enclosed in surface
∮ 𝐹⃗ [𝑟⃗] ∙ 𝑑𝑎⃗ =
∮ ൫𝐹⃗ [𝑟⃗] ∙ 𝑑𝑎̂൯𝑑𝑎
surface
surface
𝑑𝑎̂
Choose Gaussian surface
𝑑𝑎̂ is the normal direction
(
(perpendicular to a surface)
∮ 𝑑𝑎 = ∫
surface
0
𝜋
∫ 𝑅 2 Sin[𝜃] 𝑑𝜃 𝑑𝜙
2𝜋
surface
2𝜋
0
)
𝑑𝑎⃗ = 𝑟̂ 𝑑𝑎
∮ 𝐸ሬ⃗ [𝑟⃗] ∙ 𝑑𝑎⃗ = ∮ 𝐸[𝑟]𝑑𝑎 = 𝐸[𝑟] ∮ 𝑑𝑎 = 𝐸[𝑟](4𝜋𝑟 2 ) =
0
∮ 𝐹⃗ [𝑟⃗] ∙ 𝑑𝑎⃗ = ∫
surface
Gaussian sphere ⇒
𝜋
∫ 𝑓[𝑟⃗]𝑅 2 Sin[𝜃] 𝑑𝜃 𝑑𝜙
0
or 𝐸ሬ⃗ ∙ 𝑑𝑎⃗ = 𝐸 𝑑𝑎
(𝐸ሬ⃗ points radially and only depends on distance 𝑟 from origin)
0
∮ 𝑓[𝑟⃗]𝑑𝑎 = ∫
that utilizes symmetry
on every surface 𝐸ሬ⃗ ∙ 𝑑𝑎⃗ = 0
Electric field from point charge 𝑞 at the origin
Radial symmetry ⇒
𝐸ሬ⃗ [𝑟⃗] = 𝐸[𝑟]𝑟̂
Sphere of radius 𝑅 centered at the origin
2𝜋
𝑑𝑎̂
𝐸ሬ⃗
𝜋
∫ ൫𝐹⃗ [𝑟⃗] ∙ 𝑟̂ ൯𝑅 2 Sin[𝜃] 𝑑𝜃 𝑑𝜙
𝐸ሬ⃗ [𝑟⃗] =
0
𝑞
4𝜋𝜖0 𝑟2
𝑟̂
Example: Infinite Sheet of Charge
Infinite plane at 𝑥 = 0 with uniform charge density 𝜎
𝐸ሬ⃗ [𝑟⃗] = 𝐸[𝑥]𝑥̂
Translational symmetry ⇒
(𝐸ሬ⃗ points away from plane and only depends on distance 𝑥 from plane)
Gaussian cube with side length 2𝑠 centered on the plane

𝐸[𝑥]𝑥̂ ∙ 𝑑𝑎⃗ = 0 for all sides that intersect the plane

𝐸[𝑥]𝑥̂ ∙ 𝑑𝑎⃗ = 𝐸[𝑥]𝑑𝑎 for both surfaces parallel to plane
∮ 𝐸ሬ⃗ [𝑟⃗] ∙ 𝑑𝑎⃗ = 2𝐸[𝑥](4𝑠 2 ) =
𝑑𝑎̂
𝐸ሬ⃗
𝜎
𝑥̂
2𝜖0
ሬ⃗
𝐸 [𝑟⃗] = ൞ 𝜎
−
𝑥̂
2𝜖0
𝜎(4𝑠 2 )
𝜖0
𝑥>0
𝑥<0
Exactly on the plane, electric field
must be zero by symmetry!
Discontinuity in 𝐸ሬ⃗ across a sheet
𝜎
of charge is always
𝜖0
Value of 𝐸ሬ⃗ on the sheet is the
average of its values on both sides
𝑞
𝜖0
Electrostatics
𝜙 = − ∫ 𝐸ሬ⃗ ⋅ 𝑑𝑠⃗
Fundamental quantities:
𝐸ሬ⃗
𝜙
𝜌 – Charge distribution
ሬ⃗𝜙
𝐸ሬ⃗ = −∇
𝐸ሬ⃗ – Force per unit charge on a test particle
𝜙 – Potential energy per unit charge
to place a test particle
Work required to assemble charge distribution:
𝑁
𝑘 𝑞𝑖 𝑞𝑗
1
𝑈 = ∑∑
2
𝑟𝑗𝑘
(Point charges)
𝑖=1 𝑗≠𝑖
𝜖0
1
𝑈=
∫ 𝐸 2 𝑑𝑣 = ∫ 𝜌 𝜙 𝑑𝑣 (Charge distribution)
2
2
𝜌
Conductors
Key properties

ሬ⃗ inside a conductor
𝐸ሬ⃗ = 0

𝜌 = 0 inside a conductor

Any net charge resides on the surface

A conductor surface is equipotential

𝐸ሬ⃗ perpendicular to surface just outside a conductor
Example: Parallel Plates
(Ignoring Edge Effects)

Uniform charge density 𝜎 =
𝑄
𝐴
spread over both
inner surfaces

𝐸ሬ⃗ =
𝜎
(−𝑧̂ )
𝜖0
ሬ⃗ outside plates
between plates, 𝐸ሬ⃗ = 0
𝑄
𝜖0 𝐴
=
𝜙1 − 𝜙2
𝑠
2
2
𝑄
𝜖0 𝐸
𝑈≡
=
𝐴𝑠
2𝐶
2
𝐶≡
(Capacitance)
Energy Stored
(
)
in Capacitor