Energy Stored in a Capacitor
z
z
z
What is the potential energy, U, of a
charged capacitor?
Think of U as being stored in E field
between plates
Calculate W required to charge plates to
potential V
ΔU = W = ΔVq
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Recover energy by discharging capacitor
Energy Stored in a Capacitor
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Charge capacitor by transferring electrons
with a battery
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More charge moved, E field between
plates gets bigger, harder to move
charges so takes positive work to charge
capacitor
Energy Stored in a Capacitor
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At given instance potential across plates is
q′
V′ =
C
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Transfer increment of charge dq´, work
required is
W =V q
q′
dW = V ′ dq′ = dq′
C
Energy Stored in a Capacitor
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Work required from 0 to total charge q is
1 q
q2
W = ∫ q ′ dq ′ =
C 0
2C
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Potential energy = work
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Or, use
q = CV
2
q
U =
2C
1
2
U = CV
2
Energy Stored in a Capacitor
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Advantage of capacitor
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Get more power than from just a battery
1
2
U = CV
2
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Slowly charge capacitor with battery and then
discharge quickly
Examples – photo flash, medical defibrillator
Capacitance
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What if we have more than one
capacitor in a circuit?
z Replace
combination with an
equivalent capacitance
C eq
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Two basic combinations
z Parallel
z Series
Capacitors in Parallel
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z
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Capacitors in parallel
Capacitors are directly
wired together at each
plate and V applied
across the group of plates
V is same across all
capacitors
V1 = V2 = V3 = V
Capacitors in Parallel
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z
Capacitors in parallel
Total q stored on
capacitors is sum of the
charges of all capacitors
q = q1 + q2 + q3
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Ceq has total charge q
and same V as original
capacitors
C eq
q
=
V
Capacitors in Parallel
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Capacitors in parallel
q = q1 + q2 + q3
q1 = C1V
q2 = C2V q3 = C3V
q = (C1 + C2 + C3 ) V
Ceq = C1 + C2 + C3
n
Ceq = ∑ Ci
i
Capacitors in Series
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z
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Capacitors in series
Capacitors are wired one
after the other and V is
applied across the two ends
of the series
Capacitors have identical q
q1 = q 2 = q 3 = q
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Battery produces q only on
top and bottom plates,
induced q on other plates
Capacitors in Series
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Capacitors in series
Sum of V across all capacitors
is equal to applied V
V = V1 + V 2 + V 3
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Ceq has same q and total V
as original capacitors
C eq
q
=
V
Capacitors in Series
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Capacitors in series
V = V1 + V 2 + V 3
q
V1 =
C1
q
V2 =
C2
V3 =
q
C3
⎛ 1
1
1 ⎞
+ ⎟⎟
V = q ⎜⎜ +
⎝ C1 C2 C3 ⎠
⎛ 1
1
1
1 ⎞
⎟⎟
= ⎜⎜
+
+
C eq
C3 ⎠
⎝ C1 C 2
Capacitors in Series
z
z
Capacitors in series
Charge can only be shifted
from one capacitor to
another.
1
=
C eq
z
n
∑
i
1
Ci
Ceq is always less than
smallest capacitance
Capacitors in Parallel & Series
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Capacitors in parallel
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z
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V across each is equal
Total q is sum
C eq =
n
∑C
i
i
Capacitors in series
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z
q is equal on each
Total V is sum
1
=
C eq
n
∑
i
1
Ci
Capacitors in Parallel and Series
Capacitance (Exercise)
These are all
the same
Capacitance (Exercise)
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z
•
•
A battery with V stores total charge q on
two identical capacitors
a) What is V across and q on either
capacitor if they are in parallel?
V is same for each and equal to V of
battery.
q1 = q2
Total charge conserved and
q = q1 + q2 = 2q1
qcap
q
=
2
Capacitance (Exercise)
z
z
•
•
A battery with V stores charge q on two
identical capacitors
b) What is V across and q on either
capacitor if they are in series?
q is same for each
V1
V is sum of V across capacitors
V = V1 + V2 = 2V1
Vcap
V
=
2
= V2