PHYS208 Exam 2
Formula/Information Sheet
Basic constants:
Gravitational acceleration
Electric permittivity in vacuum
𝑔
𝜖𝑜
Electric constant in vacuum
𝑘
Speed of light in vacuum
Elementary charge
Atomic mass unit
Rest mass of electron
Rest mass of proton
𝑐
𝑒
1𝑢
𝑚𝑒
𝑚𝑝
= 9.8 𝑚/𝑠 2
= 8.854 ∙ 10−12 𝐹/𝑚
1
=
= 8.99 ∙ 109 𝑁𝑚2 /𝐶 2
4𝜋𝜖𝑜
≅ 3.00 ∙ 108 𝑚/𝑠
= 1.60 ∙ 10−19 𝐶
= 1.66 ∙ 10−27 𝑘𝑔
= 9.11 ∙ 10−31 𝑘𝑔
= 1.67 ∙ 10−27 𝑘𝑔
Electron-volt
Kilowatt-hour
1 𝑒𝑉
1𝑘𝑊ℎ
= 1.60 ∙ 10−19 𝐽
= 3.6 ∙ 106 𝐽
Units of energy:
Some indefinite integrals:
𝑑𝑥
𝑥
𝑑𝑥
∫ 2
(𝑥 + 𝑎2 )3/2
𝑑𝑥
∫ 2
(𝑥 ± 𝑎2 )1/2
𝑑𝑥
∫
𝑎 + 𝑏𝑥
𝑥𝑑𝑥
∫ 2
(𝑥 + 𝑎2 )3/2
𝑥 𝑑𝑥
∫ 2
(𝑥 ± 𝑎2 )1/2
= 𝑙𝑛 𝑥 + 𝐶
∫
=
= 𝑙𝑛 (𝑥 + √𝑥 2 ± 𝑎2 ) + 𝐶
1
ln(𝑎 + 𝑏𝑥) + 𝐶
𝑏
−1
= 2
+𝐶
(𝑥 + 𝑎2 )1/2
=
= (𝑥 2 ± 𝑎2 )1/2 + 𝐶
⃗=
∇
Definition of the gradient operator:
Some basic geometry:
𝑥
+𝐶
𝑎2 (𝑥 2 + 𝑎2 )1/2
𝜕
𝜕
𝜕
𝑖̂ +
𝑗̂ + 𝑘̂
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝐴
= 4𝜋𝑅2
4𝜋 3
=
𝑅
3
= 2𝜋𝑅ℎ
V
= 𝜋𝑅2 ℎ
𝛺
= 4𝜋
𝐴
Sphere of radius R
V
Cylinder of radius R and height h
(Area here refers to the cylindrical barrel,
not including the ends)
Solid angle of a sphere
Coulomb’s Law
for point charges
𝐹
Unit vector directed from a source to an
observation point
Definition
𝑟̂
for a point charge
𝐸⃗ (𝑟)
Definitions and Laws:
=
1 𝑞1 𝑞2 𝑟
∙
4𝜋𝜖𝑜 𝑟 2 𝑟
|𝑟̂ | = 1
=
1 𝑞
𝑟̂
4𝜋𝜖𝑜 𝑟 2
𝑁
Electric field [1N/C=1V/m]
Electric force [1N]
𝑁
1 𝑞𝑖
𝑟̂
4𝜋𝜖𝑜 𝑟𝑖 2 𝑖
for a group of point charges
(discrete charge distribution)
𝐸⃗𝑛𝑒𝑡
for a continuous charge distribution
𝐸⃗𝑛𝑒𝑡
on q placed in 𝐸⃗
𝐹
≡ 𝑞𝐸⃗
through a small flat area ∆𝐴
∆Φ𝐸
= 𝐸⃗ ∙ ∆𝐴 =|𝐸⃗ ||∆𝐴| 𝑐𝑜𝑠𝜃
through an entire surface
Φ𝐸
= lim ∑ ∆Φ𝐸,𝑖 = ∯ 𝐸⃗ ∙ 𝑑𝐴
≡ ∑ 𝐸⃗𝑖 = ∑
𝑖=1
=∫
𝑖=1
1 𝑑𝑞
𝑟̂
4𝜋𝜖𝑜 𝑟 2
Electric flux [1V∙m]
∆𝐴→0
Gauss’s Law (for a closed surface)
in vacuum
Φ𝐸,𝑡𝑜𝑡
in dielectric
Φ𝐸,𝑡𝑜𝑡
𝑄𝑒𝑛𝑐𝑙
𝜖𝑜
𝑄𝑒𝑛𝑐𝑙
≡ ∯ 𝐸⃗ ∙ 𝑑𝐴 =
𝜖
≡ ∯ 𝐸⃗ ∙ 𝑑𝐴 =
𝑏
∆𝑉
≡ 𝑉𝑏 −𝑉𝑎 = − ∫ 𝐸⃗ ∙ 𝑑𝑙
for a point charge, with 𝑉(∞) = 0
𝑉(𝑟)
1 𝑞
=
4𝜋𝜖𝑜 𝑟
for a group of point charges, with 𝑉𝑖 (∞) =
0
𝑉(𝑟 )
Definition
𝑎
Electric Potential [1V=1J/C]
Electric potential energy[1J=1N∙m]
𝑁
𝑖=1
𝑖=1
for a continuous charge distribution, with
𝑉𝑖 (∞) = 0
𝑉(𝑟 )
1
𝑑𝑞
=
∫
4𝜋𝜖𝑜 |𝑟⃗⃗⃗𝑜 − 𝑟 |
for a test charge 𝑞𝑜 in 𝐸⃗
∆𝑈
≡ 𝑈𝑏 − 𝑈𝑎 = 𝑞𝑜 (𝑉𝑏 − 𝑉𝑎 )
for a system of two point charges
𝑈12
𝐸⃗
Electric field 𝐸⃗ from potential V
For a capacitor
=
1 𝑞1 𝑞2
4𝜋𝜖𝑜 𝑟12
⃗ 𝑉 (∇
⃗ = gradient operator)
≡ −∇
≡ 𝑄/𝑉
For an insulated conductor
Electric capacitance[1F]
𝑁
1
𝑞𝑖
≡ ∑ 𝑉𝑖 =
∑
4𝜋𝜖𝑜
|𝑟𝑖 − 𝑟|
C
≡ 𝑄/𝑉𝑎𝑏
𝜖𝑜 𝐴
𝑑
𝜖0 𝐾𝑒 𝐸 2
=
2
=𝜅
Formula for parallel–plates
Density of electric field energy [J/m3]
𝑢𝐸
Moment of an electric dipole [1C∙m]
𝑝
≡ 𝑞𝑙
Torque of the electric dipole [1N∙m]
𝜏
=𝑝
⃗⃗⃗ × 𝐸⃗
Potential energy of an electric dipole [1J]
𝑈𝐸
= −𝑝 ∙ 𝐸⃗
Energy stored in a capacitor [1J]
𝑈𝐸 (𝑡)
Intensity of the electric current [1A]
Current density [1A/m2]
Definition
I
for a current density 𝑗
I
for a steady current I through a surface of
area A
for charges in motion
j
j
Resistivity [1Ω∙m]
Definition
ρ
Resistance [1Ω]
Definition for a steady current
R
for a metallic wire of length l and crosssection A
R
Energy-rate dissipated on resistors [1W]
P
=
1 𝑄(𝑡)2
2 𝐶
𝑑𝑞
≡
𝑑𝑡
= ∯ 𝑗 ∙ 𝑑𝐴
𝐼
𝐴
= 𝑛𝑞v𝑑
1 |𝐸⃗ |
= =
𝜎 |𝑗|
𝑉
=
𝐼
𝑙
=𝜌
𝐴
=
= 𝐼𝑉 = 𝑅𝐼 2 =
𝑉2
𝑅
𝜏𝑅𝐶
= 𝑅𝐶
Charging a capacitor
𝑞(𝑡)
= 𝑄𝑚𝑎𝑥 ∙ (1 − exp(−𝑡/𝜏))
Discharging a capacitor
𝑞(𝑡)
= 𝑄𝑜 ∙ exp(−𝑡/𝜏)
Time constant for an RC circuit [1s]
Definition