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R-C circuits. So far we considered stationary situations only meaning:. Next step in generalization is to allow for time dependencies. For consistency we follow the textbook notation where time dependent quantities are labeled by lowercase symbols such as i =I(t), v=V(t), etc .

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Presentation Transcript
slide1

R-C circuits

So far we considered stationary situations only meaning:

Next step in generalization is to allow for time dependencies

For consistency we follow the textbook notation where time dependent quantities are labeled by lowercase symbols such as i=I(t), v=V(t), etc

Let’s charge a capacitor again

-q

+q

Vab=Va-Vb

Vb

Va

slide2

What happens during the charging process

The capacitor starts to accumulate charge

A voltage vab starts to build up which is given by

The transportation of charge to the capacitor means a current i is flowing

The current gives rise to a voltage drop across the resistor R

which reads:

Using Kirchhoff’s loop rule we conclude

With the definition of current as

Inhomogeneous first order linear differential equation

Since in the very first moment when the switch is closed there is not charge in the capacitor we have the initial condition q(t=0)=0

slide3

Lets consider common strategies to solve such a differential equation

We use intuition to get an idea of the solution.

We know that initially q=0 and after a long time when the charging process is finished i=0 and hence

q

t

Let’s guess

and check if it works and, if so, let’s determine the time constant 

Substitution into

slide4

Our guess works if we chose

Guessing is a common strategy to solve differential equations

In the case of a linear first order differential equation there is a systematic

integration approach which always works

with q(t=0)=0

with

slide5

q

t

After time

63% of Qf reached

How does the time dependence of the current look like

i

I0/e

t

slide6

Now let’s discharge a capacitor

Using Kirchhoff’s loop rule in the absence of an emf we conclude

homogeneous first order

linear differential equation

In the very first moment when the switch is closed the current is at maximum

and determined by the charge Q(t=0)=Q0 initially in the capacitor

Current is opposite to the

direction on charging

slide7

We close the chapter with an energy consideration for a charging capacitor

When multiplying

i

by the time dependent current i

Rate at which energy

is stored in the capacitor

Power delivered

by the emf

Rate at which energy

is dissipate by resistor

Total energy supplied by the battery (emf)

Half of the energy delivered by the battery is stored

in the capacitor no matter what the value of R or C is !

Total energy stored in capacitor is

slide8

Let’s check this surprising result

by calculating the energy dissipated in R which must be the remaining half of the energy delivered by the emf

with

with