Electricity and Magnetism Review
Notes by L. Qian 1
Electric
==============================
Charge:
Symbol: q
Unit: C [Coulomb]
Magnetic
=============================
Current Element:
r
Symbol: i ds
Unit: A⋅m [Ampere-meter]
--------------------------------------------------Electric Field:
r
Symbol: E
Unit: V/m = N/C
Current:
Symbol: i
Unit: A (Ampere)
------------------------------------------------Magnetic Field:
r
Symbol: B
Unit: T = N⋅s /(C⋅m) [T: Tesla]
- Electric field due to a static charge q
r
1 q
E=
rˆ
4πε 0 rr 2
r
- Magnetic field due to a current element i ds
r μ 0 idsr × rˆ μ 0 idsr × rr
B=
=
4π rr 2
4π rr 3
(from Coulomb’s Law)
(Biot-Savart Law)
y
y
r̂
q
r
ids
r
r
P
r̂
r
r
P
x
x
- Magnetic field due to an infinitely long
current-carrying straight wire i
μi
B= 0
2πR
R
1
Notes are for your reference only. Please check with the textbook for accuracy. Notations may be different from
that of the textbook.
- Electric field in a parallel-plate capacitor:
V
q
E= =
d εo A
(where A is the area of the plate, and
d is the plate separation)
--------------------------------------------------Electric Field Lines:
- Start from positive charges or ∞
- End at negative charges or ∞
- The denser the lines, the higher the
E field magnitude
- The tangent of the field line at any point
indicates the E field direction
- Magnetic field in an ideal solenoid:
(where n is the number of turns per unit
length)
------------------------------------------------Magnetic Field Lines:
- External field lines start from the N-pole
- External field lines end at the S-pole
- The denser the lines, the higher the
B field magnitude
- The tangent of the field line at any point
indicates the B field direction
Sample field lines:
Sample field lines:
B = μ 0 ni
note the right-hand rule
- Uniform E field inside a parallel-plate
capacitor
- Uniform B field inside an ideal solenoid
(inductor)
V+
r
E
V−
V = Ed
---------------------------------------------------
Φ B = BA
-----------------------------------------------------Magnetic Flux:
Symbol: ΦΒ
Unit: Wb = T⋅m [Wb: Weber]
Definition:
r r
Φ B = ∫ B ⋅ dA
S
Magnetic Flux Linkage (of a solenoid):
NΦB (N= number of turns)
--------------------------------------------------Electric Force:
r
Symbol: F
Unit: N [Newton]
- Force on a charge in an external E field
r
r
F = qE
- Force between two charges q1 and q2
F =
q1q 2
4πε 0 rr 2
1
-----------------------------------------------------Magnetic Force:
r
Symbol: F
Unit: N [Newton]
- Force on a moving charge in an external B
field
r
r r
F = qv × B
- Force on a current element in an external B
field
r
r r
F = idL × B
- Force between two current-carrying wires
ia and ib
μii
Fba = 0 a b L
2π d
(note, above formula is the force exerted on
wire b over length L)
ib
ib
d
ia
ia
--------------------------------------------------Electric Potential:
Symbol: V
Unit: V = J/C [V: Volt]
- Electric potential due to a single charge q:
1 q
V=
4πε 0 r
- Potential created by multiple point charges:
1
qi
V=
∑
4πε 0 i ri
- Potential difference between capacitor plates:
V = q/C
------------------------------------------------------
- Equal potential lines are perpendicular
to field lines
--------------------------------------------------Electric Potential Energy:
Symbol: UE
Unit: J [Joule]
U E = qV or ΔU E = q ΔV
---------------------------------------------------
------------------------------------------------------
-----------------------------------------------------Induction (Faraday’s Law):
- Induced emf by one turn of coil:
dΦ B
unit: V
E=−
dt
- Induced emf by a solenoid of N turns:
dΦ B
unit: V
E = −N
dt
--------------------------------------------------Capacitance:
Symbol: C
Unit: F = C/V [F: Farad]
Definition:
q
C=
V
- Parallel-plate capacitor
ε A
C= 0
d
(where A is the area of the plate, and
d is the plate separation)
-----------------------------------------------------Inductance:
Symbol: L
Unit: H = Wb/A [H: Henry]
Definition:
NΦ B
i
1
or L =
=
L NΦ B
i
- Ideal solenoid of length l
- Capacitance is a geometry- and materialrelated parameter
- Inductance is a geometry- and materialrelated parameter
--------------------------------------------------Capacitors in parallel
-----------------------------------------------------Inductors in parallel
1
1
1
= +
+ ...
Leq L1 L2
(Flux linkage is the same for all inductors)
Ceq = C1 + C2 + ...
(Voltage is the same for all capacitors)
L = μ 0 n 2 Al
(where n is the number of turns per unit
length, A is cross-sectional area)
Capacitors in series
1
1
1
=
+
+ ...
Ceq C1 C 2
(Charge is the same for all capacitors)
Inductors in series
--------------------------------------------------Energy stored in a capacitor
q2 1
UE =
= CV 2
2C 2
-----------------------------------------------------Energy stored in an inductor
1
U B = Li 2
2
Electric energy density
Magnetic energy density
UE
1
= εoE2
volume 2
==============================
UB
1 2
B
=
volume 2 μ o
=============================
uE =
Leq = L1 + L2 + ...
(Current is the same for all inductors)
uB =