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for capacitance : i = Cdv
dt
for inductance : v = Ldi
dt
for resistance : i =v
R
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urface of
the plates
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E(t) = S (t) =
q(t)
S
Gauss
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d
0
v(t) = E(t)dl =0d E(t) d = q(t)
d
S
q(t) =S
d v(t)
C =S
d
i(t) =dq(t)
dt
q(t) = C v(t)
i(t) = Cdv(t)
dt
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i(t) = Cdv(t)
dt
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i(t) = Cdv(t)
dt
i(t) = Cdv(t)
dt
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i(t) = Cdv(t)
dt dv(t) =
1
Ci (t)dt
dv(t)v ( tO )
v ( t )
=1
Ci (t)dt
tO
t
v(t) v(tO ) =1
Ci (t)dt
tO
t
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v(t) = v(tO ) +1
Ci (t)dt
tO
t
q(tO ) =v(tO )
C
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i(t) = I 0 v(t) =I 0
Ct + v(0)
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dv
dt=I 0C
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i(t) =dq(t)
dt dq(t) = i (t)dt q(t) = i (t)dt
t
v(t) =q(t)
C v(t) =1
Ci(t)dt
t
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q(t) = i(t)dtt
= i (t)dttO
+ i (t)dttO
t
= q(tO ) + i (t)dttO
t
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v(t) =1
Ci(t)dt
t
=1
Ci (t)dt
tO
+ i (t)dttO
t
=q(tO )
C+
i (t)dttO
t
C= v(t0) +
1
Ci (t)dt
tO
t
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v(t) = v(t0) +1
Ci(t)dt
tO
t
+
t t
v
t
v
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pE (t) =dwE (t)
dt= v(t) i (t) = v(t) C
dv(t)
dt
dwE = v(t) Cdv(t) = v(t) dq(t)
dwE = d
F •
d =
= E dq ( d) =v
d
dq d = v dq
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dwE = C v dv dwE
wE ( t )
= C v dvv ( )
v ( t )
wE (t) =C v 2 (t)
2
wE (t) =C v 2 (t)
2=q2 (t)
2 C
v =q
C
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wE (t) =C v 2 (t)
2=q2 (t)
2 C
Mass : f = Mdv
dt Capacitor : i = C
dv
dt
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v(t) =q(t)
CS
= v1 (t) + v2 (t) =q(t)
C1
+q(t)
C2
1
CS
=1
C1
+1
C2
i (t) = i1 (t) = i2 (t)
q(t) = i(t)dtt
=
= q1 (t) = i1 (t)dt =t
q2 (t) = i2 (t)dtt
v(t) = v1 (t) + v2 (t)
i
C=dv
dt =
d
dt(v1 + v2) =
dv1
dt+dv2
dt=i
C1
+i
C2
1
C=
1
C1
+1
C2
1
C=d1
1A+d2
2A=1
C1+1
C2
i = CP
dv
dt= i1 + i2 = C1
dv
dt+C2
dv
dt CP = C1 +C2
q(t) = i(t)dt
t = i1 (t)dt
t + i2 (t)dt
t = q1 (t) + q2 (t)
v(t) =q(t)
C= v1 (t) =
q1 (t)
C1
= v2 (t) =q2 (t)
C2
Cp =1A1 + 2A2
d= 1A1
d+ 2A2
d= C1 + C2
vS (t) =1
CiS (t)dt
tO
t
iS (t) = CdvSdt
v(t) =1
CiS (t)dt
t
i(t) = iS (t)
v(t) = vS (t)
i(t) = CdvS (t)
dt
iS (t) =0 for t 0
IO for t > 0
vS (t) =0 for t 0
VO for t > 0
iS (t) =0 for t 0
IO for t > 0
v(t) =1
CiS (t)dt =
t
0 for t 0IOt
C for t > 0
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iS (t) =0 for t 0
IO for t > 0
v(t) =
0 for t 0IOt
C for t > 0
vS (t) =0 for t 0
VO for t > 0
i(t) = Cdv(t)
dt =
0 for t < 0
? for t = 0
0 for t > 0
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vS (t) =0 for t 0
VO for t > 0
u(t;T ) T 0 u(t)
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du(t;T )
dt= (t;T )
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for
T 0
1) (t) = 0 for t 0
2) (t)dt = u(t) du(t)
dt= (t)
t
3) (t)dt = 1+
vS (t) =0 for t 0
VO for t > 0
= VO limT 0
u(t;T ) = VOu(t)
i(t) = Cdv(t)
dt = C
d
dtVO lim
T 0u(t;T )[ ] = CVO lim
T 0(t;T ) = CVO (t)
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v1 (t) = v2 (t) v(t) for t tO +
v1 (t) =q1 (t)
C1
= v2 (t) =q2 (t)
C2
for t tO +
i1 (t) + i2 (t) = 0 for t tO +
dq1 (t)
dt+dq2 (t)
dt=d
dtq1 (t) + q2 (t)[ ] = 0 for t tO +
q1 (t0 ) + q2 (t0 ) = q1 (t) + q2 (t) with t t0 +
Q1 +Q2 = C1v1 (t) +C2v2 (t) with t t0 +
Q1 +Q2 = C1v(t) +C2v(t) with t t0 +
Q1 +Q2 = (C1 +C2) v(t) with t t0 +
v(t) =Q1 +Q2
C1 +C2
=C1V1 +C2V2
C1 +C2
with t t0 +
v(t) =Q1 +Q2
C1 +C2
=C1V1 +C2V2
C1 +C2
V with t t0 +
q1 (t) = C1v(t) = C1
Q1 +Q2
C1 +C2
= C1
C1V1 +C2V2
C1 +C2
with t t0 +
q2 (t) = C2v(t) = C2
Q1 +Q2
C1 +C2
= C2
C1V1 +C2V2
C1 +C2
with t t0 +
v(t) =Q1 +Q2
C1 +C2
=C1V1 +C2V2
C1 +C2
V with t t0 +
v =m1v1 +m2v2m1 +m2
wE (t < t0) =Q1
2
2C1
+Q2
2
2C2
wE (t > t0) =C1 (Q1 +Q2)
C1 +C2
2
1
2C1
+C2 (Q1 +Q2)
C1 +C2
2
1
2C2
=(Q1 +Q2)
C1 +C2
2
C1
2
2C1
+C2
2
2C2
=
=(Q1 +Q2)
C1 +C2
2
C1
2+C2
2
=(Q1 +Q2)
2
2(C1 +C2)
wE (t < t0) wE (t > t0) =Q1
2
2C1
+Q2
2
2C2
(Q1 +Q2)2
2(C1 +C2)=
=Q1
2 (C1 +C2)C2 +Q2
2 (C1 +C2)C1 Q1
2C1C2 Q2
2C1C2 2Q1Q2C1C2
2C1C2 (C1 +C2)=
=Q1
2C2
2+Q2
2C1
2 2Q1Q2C1C2
2C1C2 (C1 +C2)=
(Q1C2 Q2C1)2
2C1C2 (C1 +C2) 0
wE (t < t0) wE (t > t0) =(Q1C2 Q2C1)
2
2C1C2 (C1 +C2) 0
Q1C2 Q2C1 = 0 Q1C2 =Q2C1 Q1
C1
=Q2
C2
V1 = V2
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Ampere' s Law :
B • d
l C
= μI enclosed
B • d
l C
= 0 B = 0
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B cos 0 dla
b
+ B cos 90O
b
c
dl + 0 cos 0O dlc
d
+ B cos 90O
d
a
dl = μNi
Bl = μNi B =μNi
l
Ampere' s Law :
B • d
l C
= μI enclosed
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B(t) =μNi(t)
l
(t) = B(t)S
L (t) = N (t) = N B(t) S =μN 2S
l i (t)
L =μN 2S
l
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L (t) =μ N 2 S
l i (t)
v(t) =d L (t)
dt= L
di (t)
dt
L =μN 2S
l
L (t) = Li(t)
v(t) =d L (t)
dt
v(t) = Ldi (t)
dt
L
Flux Linkage : L
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v(t) = Ldi (t)
dt
v(t) = Ldi (t)
dt
v(t) = Ldi(t)
dt v(t)dt = Ldi(t)
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v(t)dtt0
t
= L di(t)i( t0 )
i( t )
i(t) = i(t0) +1
Lv(t)dt
t0
t
(t0) = Li(t0)
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v(t) =V0 i(t) =V0Lt + i(0)
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v(t) =d (t)
dtv(t)dt = d (t) (t) = v(t)dt
t
(t) = L i(t)
i(t) =1
Lv(t)dt
t
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(t) = v(t)dtt0
+ v(t)dt = (t0) +t0
t
v(t)dtt0
t
(t)
L=
(t0)
L+
v(t)dtt0
t
L i (t) = i0 (t) +
1
Lv(t)dt
t0
t
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i(t) = i0 (t) +1
Lv(t)dt
t0
t
+
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wM (t) = pM (t)dt = i (t)v(t)dt = i (t)Ldi (t)
dt
ttt
= L i (t)di (t) =1
2
i ( t )
Li 2 (t)
pM (t) =dwM (t)
dt= i (t)v(t) = i (t)L
di (t)
dt
dwM (t) = i (t)Ldi (t) = i (t)d (t)
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wM (t) =Li 2 (t)
2=
2 (t)
2L
(t) = Li(t)
Spring : f (t) = f (t0) + K v(t)dtt0
t
Inductor : i(t) = i (t0) +1
Lv(t)dt
t0
t
• i(t) = i1 (t) + i2 (t) + i3 (t)
di
dt=d (i1 + i2 + i3)
dt
di
dt=di1dt+di2
dt+di3
dt
v(t)
Leq
=v(t)
L1
+v(t)
L2
+v(t)
L3
v(t) =d (t)
dt
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(t) = 1 (t) = 2 (t)
L i(t) = L1 i1 (t) = L2 i2 (t)
L =μN 2A
l(t) = L i (t)
i(t0) = ik (t0) with k = 1,2,...,m
v(t) = v1 (t) + v2 (t) + v3 (t)
Leq
di
dt= L1
di
dt+ L2
di
dt+ L3
di
dt
v(t) =d (t)
dt
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L =μN 2A
l(t) = L i (t)
i(t) = i1 (t) = i2 (t) (t)
L= 1 (t)
L1
= 2 (t)
L2
L i(t) = L1 i (t) + L2 i (t) (t) = 1 (t) + 2 (t)
iS (t) =1
LvS (t)dt
t
vS (t) = LdiS (t)
dt
v(t) = vS (t)
i(t) =1
LvS (t)dt
t
i (t)
i(t)
i(t) = iS (t)
v(t) = LdiS (t)
dt
vS (t) =0 for t 0
VO for t > 0
i(t) =1
LvS (t)dt =
t
0 for t 0VOt
L for t > 0
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i(t) =
0 for t 0VOt
L for t > 0
iS (t) =
0 for t t0
I 0 for t > t0
= I 0u(t)
v(t) = Ldi (t)
dt= L
d
dtI 0u(t)[ ] = LI 0 (t)
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1 (t0 ) = 1
2 (t0 ) = 2
i1 (t0 ) = I1 =1
L1
i2 (t0 ) = I 2 =2
L2
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i(t) = 1 + 2
L1 + L2
=L1I1 + L2I 2
L1 + L2
with t t0 +
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1 (t) = L1i (t) = L11 + 2
L1 + L2
= L1
L1I1 + L2I 2
L1 + L2
with t t0 +
2 (t) = L2i (t) = L21 + 2
L1 + L2
= L2
L1I1 + L2I 2
L1 + L2
with t t0 +
i(t) = 1 + 2
L1 + L2
=L1I1 + L2I 2
L1 + L2
I with t t0 +
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