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Capacitors in Circuits -Q A V E d +Q Capacitance Two parallel plates charged Q and –Q respectively constitute a capacitor C = Q / V The relationship C = Q / V is valid for any charge configuration (Indeed this is the definition of capacitance or electric capacity)

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capacitance
-Q

A

V

E

d

+Q

Capacitance

Two parallel plates charged

Q and –Q respectively

constitute a capacitor

C = Q / V

The relationship C = Q / V is valid for any charge configuration

(Indeed this is the definition of capacitance or electric capacity)

In the particular case of a parallel plate capacitor

C = 0 A / d [vacuum] or C =  0 A / d [dielectric]

The capacitance is directly proportional to the area of the plates

and inversely proportional to the separation between the plates

slide3
Capacitors in Circuits

(Symbol for

a capacitor)

+Q

-Q

C

V

A piece of metal in equilibrium has a constant value of potential.

Thus, the potential of a plate and attached wire is the same.

The potential difference between the ends of the wires is V,

the same as the potential difference between the plates.

slide4
Parallel and Series

Series

Parallel

slide5
Capacitors in Parallel

C1 - q1

  • Suppose there is a potential
  • difference V between a and b.
  • Then q1 V = C1 & q2 V = C2

a

b

C2 - q2

  • We want to replace C1 and C2 with an
  • equivalent capacitance C = q V
  • The charge on C is q = q1 + q2
  • Then C = q V = (q1 + q2 ) V = q1 V + q2 V = C1 + C2

V

b

a

C - q

C = C1 + C2

  • This is the equation for capacitors inparallel.
  • Increasing the number of capacitors increases the capacitance.
slide6
Capacitors in Series

C1

C2

C

a

-q

+q

-q

+q

b

a

-q

+q

b

V1

V2

V

  • Here the total potential difference between a and b is V = V1 + V2
  • Also V1 = (1/C1) q and V2 = (1/C2) q
  • The charge on every plate (C1 and C2) must be the same (in magnitude)
  • Then: V = V1 + V2 = q / C1 +q / C2 = [(1/C1) + (1/C2)] q
  • or, V = (1/C) q 

1 / C = 1 / C1 + 1 / C2

  • This is the equation for capacitors in series.
  • Increasing the number of capacitors decreases the capacitance.
slide8
Series

Parallel

Ceq = C1 + C2 + C3

1/Ceq = 1/C1 + 1/C2 + 1/C3

slide10
Real circuit

Ideal circuit

What happens when

the switch is closed ?

How does the capacitor acquire the charge ?

rc circuits charging
open

closed

I

R

R

VR=IR

+

+

+++

V

V

- - -

VC=q/C

-

C

-

C

RC Circuits: Charging

V = I(t)R + q(t)/C

When the switch closes, at first a high current flows:

VR is big and VC is small. As q is stored in C, VC increases.

This fights against the battery, so I gradually decreases.

Finally, I stops (I = 0), C is fully charged (VC = Q/C = V),

and Q=C V

discharging an rc circuit
R

R

VR=IR

+q

+q

I

C

VC=V0

C

-q

VC=q/C

-q

Open circuit

After closing switch

Discharging an RC Circuit

Current will flow through the resistor for a while.

Eventually, the capacitor will lose all its charge,

and the current will go to zero.

During the transient: q(t) / C – I(t) R = 0

slide16
Charging and Discharging a Capacitor

Charging and discharging of a capacitor occurs gradually

with a characteristic time  = RC  time constant

At t = 0, (switch closed or open) a large current flows,

the capacitor behaves like a short circuit.

At t  , the current is essentially zero,

the capacitor behaves like an open switch.

The current decreases exponentially.

slide17
Measuring

Current and Voltage

in a circuit

The ammeter measures current,

and is connected in series.

The voltmeter measures voltage,

and is connected in parallel.

slide18
A modern digital multimeter combines the functions

of ammeter, voltmeter, and ohmmeter.

(i.e. can measure current, voltage, and resistance)

In addition, modern multimeters can measure

capacitance, temperature, and more,

and can be connected to computers too…

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