Ch. 21:
Coulomb’s Law:
=
,
=
F kq1q2r2 k 14πϵ0
= force
between two point charges (in a vacuum).
Electric field
:
=
→
=
=
E F0q0 F0 Eq0 ma
Electric field of a point charge
:
=
E qr4πϵ0r2
REMEMBER:
=
r rr
Electric field of a ring of charge
:
=
=
+
E Exi Qx4πϵ0x2 a232i
(for a ring with symmetry on x
axis, centered in yz plane).
Field of a finite line of charge:
=
+
E Q4πϵ0xx2 a2i
With charge density λ, λ=Q/2a:
=
+
E λ2πϵ0xx2a2 1i
thus
=
E λ2πϵ0r
for infinite line of charge
Field for finite, uniformly charged disk
:
=

Ex σx2ϵ01
+
1R2x2 1
=
E σ2ϵ0
for infinite charged disk
For two oppositely charged, infinite sheets, E inside is 2E,
outside either sheet is 0.
Dipole = pair of pt. charges with equal magnitude, opposite
sign
p=qd >
p
=q
d
(p = dipole moment)
τ=pEsin >
τ
=
p
x
E,
U=
p*E
Potential energy U at its lowest when dipole is in stable
equilibrium. U = pEcos
Ch 22 – Gauss’s Law, Eflux:
Φe = ∫ E cosφ dA = E
┴
dA =
E
.
d
A
=
Q
encl
/ε
o
=
Gauss’s Law
Single pt charge, E = (1/4πε
o
)(q/r
2
)
Charge on surface of conducting sphere,
E
= (1/4πε
o
)(q/r
2
)
Inside sphere = 0
Infinite wire, E = (1/2πε
o
)( λ /r)
Infinite conducting cylinder, E = (1/2πε
o
)( λ /r)
Inside = 0
Solid insulating sphere, Q distributed uniformly throughout, E
= (1/4πε
o
)(Q/r
2
)
Inside sphere E = (1/4πε
o
)(Qr/R
3
)
Infinite sheet with uniform charge, E = σ/2ε
o
2 oppositely charged conducting plates, E = σ/ε
o
Ch 23 – Potential energy, electric potential:
W
a
b
= U
a
 U
b
=(U
b
U
a
)
U=Fd=q
0
Ed
K
b
+U
b
=K
a
+U
a
U = (1/4πε
o
)(qq
0
/r) = (q
0
/4πε
o
)(q
1
/r
1
+ q
2
/r
2
+ …)
Equi triangle: U = (1/4πε
0
)(1/r)(q
1
q
2
+q
2
q
3
+q
1
q
3
)
V = U/q
0
= (1/4πε
o
)(q/r) or (q
0
/4πε
o
)(q
1
/r
1
+ q
2
/r
2
+ …)
V = (1/4πε
o
) ∫ (dq/r)
Va – Vb = ∫
E
d
l
= ∫E cosφ dl
E
= (dV/dx
i
+ dV/dy
j
+ dV/dz
k
)
V for a
point outside a sphere
(r away):
=
V q4πϵ0r
V for
inf. line charge or charged conducting cyl
.:
=
(
)
V λ2πϵ0ln Rr
V in
oppositely charged plates
(yaxis up is positive): V=Ey,

=
→ =
Va Vb Ed E Vabd
(a is top, +plate, b is bottom, plate)
Charge density
for above:
=
σ ϵ0Vabd
V for
rod of length 2a
:
=
(
+
+
+

V Q4πϵ02aln a2 x2 aa2 x2
)
a
V for
point x from center of ring of charge
:
=
+
V Q4πϵ0x2 a2
Ch 24 – Capacitance, Dielectrics: