Department of Physics and Applied Physics95.144 Danylov Lecture 13

Lecture 13

Chapter 31

RC Circuits

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsII

Lecture Capture: http://echo360.uml.edu/danylov201415/physics2spring.html

Physics II

Department of Physics and Applied Physics95.144 Danylov Lecture 13

Steady

current

In the preceding sections we dealt with circuits in which the circuits elements were resistors and in which the currents did not vary with time. Here we

introduce the capacitor as a circuit element, which will lead us to the study of time-varying currents.

Time-varying current

Department of Physics and Applied Physics95.144 Danylov Lecture 13

RC circuit (Charging a Capacitor)

Now, we know Kirchhoff’s rules and let’s apply them

to study an RC circuit

Department of Physics and Applied Physics95.144 Danylov Lecture 13

Charging a Capacitor

We need to analyze it:

denote RC (the time constant) and (full charge of the capacitor)

The capacitor charge at time t is:

, , ∆ ?

∆ + ∆ + ∆ 0

Let’s look at the circuit at some arbitrary moment of time t and apply Kirchhoff’s loop rule:

The figure shows an RC circuit, some time after the switch was closed.

(The resistor current is the rate at which charge is added to the capacitor)

There are two variables I(t),Q(t), which are dependent:

/

0

0

It is not hard to solve, but we just present the solution (see the solution at the end of this presentation)

0

∆1 ⁄

1 ⁄

Department of Physics and Applied Physics95.144 Danylov Lecture 13

Resistor Current and Capacitor VoltageLet’s calculate the resistor current:

⁄

RC ⁄ ⁄

⁄

(https://www.youtube.com/watch?v=jFrVoG-edFc)

No current. Electrons waiting for a switch to be closed.

The first photo of a traveling electron

1 ⁄

This current looks like “The Land Run” of 1893 (the Oklahoma Territory) shown in the movie “Far and Away

Race begins. Electrons are on the way to their lands.

A) No light.B) First, it is bright, then dim.C) First, it is dim, then bright.D) Steady bright.

In the circuit shown, the capacitor is originally uncharged. Describe the behavior of the lightbulb from the instant switch S is closed until a long time later.

ConcepTest RC circuit 1

When the switch is first closed, the current is highand the bulb burns brightly. As the capacitor charges,The voltage across the capacitor increases causingthe current to be reduced, and the bulb dims.

Department of Physics and Applied Physics95.144 Danylov Lecture 13

RC circuit (discharging)

At t = 0, the switch closes and the charged capacitor begins to discharge through the resistor.

We want to analyze the RC circuit:, , ∆ ?

Department of Physics and Applied Physics95.144 Danylov Lecture 13

RC circuit (discharging)

Kirchhoff’s loop law applied to this circuit clockwise is:

The figure shows an RC circuit, some time after the switch was closed.

The resistor current is the rate at which charge is removed from the capacitor:

Q and I in this equation are the instantaneous values of the capacitor charge and the resistor current.

denote time constant as:

The resistor current 0 ,

where Q0 is the charge at t = 0

The charge on the capacitor of an RC circuit

0

ln

Department of Physics and Applied Physics95.144 Danylov Lecture 13

RC circuit (discharging)Let’s plot it:

Department of Physics and Applied Physics95.144 Danylov Lecture 13

RC circuit (discharging)Let’s calculate the resistor current:

I0 is the initial current

The current undergoes the same exponential decay

∆

∆ ∆

Let’s calculate the voltage of the capacitor:

∆ / ∆ ∆ / the voltage across the capacitor

Now we know everything about the circuit [Q(t), I(t), and ΔV(t)]

2.7 0.37

A) Capacitor A.B) Capacitor B.C) They discharge at the

same rate.D) Can’t say without

knowing the initial amount of charge.

Which capacitor discharges more quickly after the switch is closed?

ConcepTest RC circuit 1

time constant = RC

= 12 µs = 15 µs

So the capacitor A discharges faster than B

A) 5 sB) 4 sC) 2 sD) 1 sE) The capacitor does not discharge because the resistors cancel each other

What is the time constant for the discharge of the capacitor shown in the figure?

ConcepTest RC circuit 3

time constant by definition = ReqC

+ =

= ReqC =4Ωx1F=4 seconds

How about this?

= ReqCeq

A) R1< R2< R3

B) R1< R3< R2.C) R2< R3< R1.D) Not enough information.

Figure shows the voltage as a function of time of a capacitor as it is discharged (separately) through three different resistors. Rank in order, from largest to smallest, the values of the resistances R1, R2, and R3.

ConcepTest RC circuit 2

time constant by definition = RC

< . ∆. ∆. ∆

From the figure we can see that:

∆

Department of Physics and Applied Physics95.144 Danylov Lecture 13

My application I used an RC circuit in my paper.

Charging

a Capacito

r

Discharging a Capacitor

Department of Physics and Applied Physics95.144 Danylov Lecture 13

What you should read

Chapter 31 (Knight)

Sections 31.9

Department of Physics and Applied Physics95.144 Danylov Lecture 13

Derivation(charging a capacitor)

Department of Physics and Applied Physics95.144 Danylov Lecture 13

Thank youSee you on Tuesday

Since all resistors are identical, the voltage drops are the sameacross the upper branch and the lower branch.

Thus, the potentials at points aand b are also the same. Therefore, no current flows.

ConcepTest Wheatstone BridgeA) lB) l/2C) l/3D) l/4E) zero

An ammeter A is connected between points a and b in the circuit below, in which the four resistors are identical. The current through the ammeter is:

I

V