E & M  Basic Physical Concepts
Electric force and electric field
Electric force between 2 point charges:

F

=
k

q
1
 
q
2

r
2
k
= 8
.
987551787
×
10
9
N m
2
/C
2
²
0
=
1
4
π k
= 8
.
854187817
×
10

12
C
2
/N m
2
q
p
=

q
e
= 1
.
60217733 (49)
×
10

19
C
m
p
= 1
.
672623 (10)
×
10

27
kg
m
e
= 9
.
1093897 (54)
×
10

31
kg
Electric field:
~
E
=
~
F
q
Point charge:

E

=
k

Q

r
2
,
~
E
=
~
E
1
+
~
E
2
+
· · ·
Field patterns:
point charge, dipole,
k
plates, rod,
spheres, cylinders,
. . .
Charge distributions:
Linear charge density:
λ
=
Δ
Q
Δ
x
Area charge density:
σ
A
=
Δ
Q
Δ
A
Surface charge density:
σ
surf
=
Δ
Q
surf
Δ
A
Volume charge density:
ρ
=
Δ
Q
Δ
V
Electric flux and Gauss’ law
Flux:
ΔΦ =
E
Δ
A
⊥
=
~
E
·
ˆ
n
Δ
A
Gauss law:
Outgoing Flux from S,
Φ
S
=
Q
enclosed
²
0
Steps:
to obtain electric field
–Inspect
~
E
pattern and construct
S
–Find Φ
s
=
H
surface
~
E
·
d
~
A
=
Q
encl
²
0
, solve for
~
E
Spherical:
Φ
s
= 4
π r
2
E
Cylindrical:
Φ
s
= 2
π r ‘ E
Pill box:
Φ
s
=
E
Δ
A
, 1 side;
= 2
E
Δ
A
, 2 sides
Conductor:
~
E
in
= 0,
E
k
surf
= 0,
E
⊥
surf
=
σ
surf
²
0
Potential
Potential energy:
Δ
U
=
q
Δ
V
1 eV
≈
1
.
6
×
10

19
J
Positive charge moves from high
V
to low
V
Point charge:
V
=
k Q
r
V
=
V
1
+
V
2
=
. . .
Energy of a chargepair:
U
=
k q
1
q
2
r
12
Potential difference:

Δ
V

=

E
Δ
s
k

,
Δ
V
=

~
E
·
Δ
~s
,
V
B

V
A
=

R
B
A
~
E
·
d~s
E
=

d V
dr
,
E
x
=

Δ
V
Δ
x
fl
fl
fl
fix y,z
=

∂V
∂x
, etc.
Capacitances
Q
=
C V
Series:
V
=
Q
C
eq
=
Q
C
1
+
Q
C
2
+
Q
C
3
+
· · ·
,
Q
=
Q
i
Parallel:
Q
=
C
eq
V
=
C
1
V
+
C
2
V
+
· · ·
,
V
=
V
i
Parallel platecapacitor:
C
=
Q
V
=
Q
E d
=
²
0
A
d
Energy:
U
=
R
Q
0
V dq
=
1
2
Q
2
C
,
u
=
1
2
²
0
E
2
Dielectrics:
C
=
κC
0
,
U
κ
=
1
2
κ
Q
2
C
0
,
u
κ
=
1
2
²
0
κ E
2
κ
Spherical capacitor:
V
=
Q
4
π ²
0
r
1

Q
4
π ²
0
r
2
Potential energy:
U
=

~
p
·
~
E
Current and resistance
Current:
I
=
d Q
dt
=
n q v
d
A
Ohm’s law:
V
=
I R
,
E
=
ρJ
E
=
V
‘
,
J
=
I
A
,
R
=
ρ‘
A
Power:
P
=
I V
=
V
2
R
=
I
2
R
Thermal coefficient of
ρ
:
α
=
Δ
ρ
ρ
0
Δ
T
Motion of free electrons in an ideal conductor:
a τ
=
v
d
→
q E
m
τ
=
J
n q
→
ρ
=
m
n q
2
τ
Direct current circuits
V
=
I R
Series:
V
=
I R
eq
=
I R
1
+
I R
2
+
I R
3
+
· · ·
,
I
=
I
i
Parallel:
I
=
V
R
eq
=
V
R
1
+
V
R
2
+
V
R
3
+
· · ·
,
V
=
V
i
Steps:
in application of Kirchhoff’s Rules
–Label currents:
i
1
, i
2
, i
3
, . . .
–Node equations:
∑
i
in
=
∑
i
out
–Loop equations:
“
∑
(
±E
) +
∑
(
∓
iR
)=0”
–Natural:
“+” for looparrow entering

terminal
“

” for looparrowparallel to current flow
RC circuit:
if
d y
dt
+
1
R C
y
= 0,
y
=
y
0
exp(

t
R C
)
Charging:
E 
V
c

R i
= 0,
1
c
d q
dt
+
R
d i
dt
=
i
c
+
R
d i
dt
= 0
Discharge:
0 =
V
c

R i
=
q
c
+
R
d q
dt
,
i
c
+
R
d i
dt
= 0
Magnetic field and magnetic force
μ
0
= 4
π
×
10

7
T m
/
A
Wire:
B
=
μ
0
i
2
π r
Axis of loop:
B
=
μ
0
a
2
i
2 (
a
2
+
x
2
)
3
/
2
Magnetic force:
~
F
M
=
i
~
‘
×
~
B
→
q ~v
×
~
B
Loopmagnet ID:
~
τ
=
i
~
A
×
~
B
,
~μ
=
i A
ˆ
n
Circular motion:
F
=
m v
2
r
=
q v B
,
T
=
1
f
=
2
π r
v
Lorentz force:
~
F
=
q
~
E
+
q ~v
×
~
B
Hall effect:
V
H
=
F
M
d
q
,
U
=

~μ
·
~
B
Sources of
~
B
and magnetism of matter
BiotSavart Law:
Δ
~
B
=
μ
0
4
π
i
Δ
~
‘
×
ˆ
r
r
2
,
B
=
μ
0
4
π
q~v
×
ˆ
r
r
2
Δ
B
=
μ
0
4
π
i
Δ
y
r
2
sin
θ
,
sin
θ
=
a
r
,
Δ
y
=
r
2
Δ
θ
a
Ampere’s law:
M
=
H
L
~
B
·
d~s
=
μ
0
I
encircled
Steps:
to obtain magnetic field
–Inspect
~
B
pattern and construct loop
L
–Find
M
and
I
encl
, and solve for
~
B
.
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 Spring '10
 ERSKINE
 Physics, Charge, Force, Magnetic Field, Correct Answer, Electric charge, shie

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