Review MTE 2P208

MTE 2: Ch 2103 5:30-7pm on Oct 31

Alternate Exams:

Wed Ch 2103 6:30pm-8:00pm

(people attending the alternate exam will not be allowed to go out of the room while others from

previous exam are still around)

Thu @ 9:00-10:30 am

Thu @ 6:30pm-8:00pm in the lab Ch 3254You have received an email saying that your request for an alternate exam was approved. If you did not, please notify me at the ed of this lecture!

2

Contents of MTE2

! Electric Potential and Field

! Capacitors and capacitance

! Current and resistance, Ohm’s law

! DC Circuits and Kirchoff’s laws

! RC circuits

! Lorentz force and motion of charge in a magnetic field

! Biot and Savart

! Ampere’s law (MTE3, so study it!)

! Read chapters 28 (no 28.2)-29-30-31-32

Electric potential energy

4

!

Whand

= "U since they repel! potential energy increases

Work is Force x distance (taking into account cos! between 2 vectors!)

>0

If opposte charges they attract => W <0 and potential energy decreases

Electric Potential

5

Q source of the electric potential, q ‘experiences’ it

6

Example E Potential

Calculate the electric potential at B

Calculate the work YOU must do to move a Q=+5 mC charge

from A to B.

Calculate the electric potential at A

x

+-

B

A

d1=3 m 3 m

d2=4 m

3 m

y

-12 µC +12 µC

d

!

VB = kq

d"q

d

#

$ %

&

' ( = 0

!

VA = kq

d1

"q

3d1

#

$ %

&

' ( = k

2q

3d1

!

WYou = "U =UB #UA =Q(VB #VA ) = #k2qQ

3d1

A. W = +19.8 mJ

B. W = -19.8 mJ

C. = 0

7

Work and electrostatic potential energy

!1µ

!3µ

!2µ

5 m

5 m5 m

Question: How much work would it take YOU to

assemble 3 negative charges?

Likes repel, so YOU will still do positive work!

q3

q2q1

!

W1

= 0

W2

= kq1q2

r12

= 9 "109 #1"10

#6"#2 "10

#6

5= 3.6mJ

W3

= kq1q3

r13

+ kq2q3

r23

=16.2mJ

Wtot = kq1q2

r12

+ kq1q3

r13

+ kq2q3

r23

= +19.8mJ

!

UE

=19.8mJ electric potential energy of the system increases

Electric Potential and FieldPoints A at 2m and B at 5m on the x axis are in a region where

the electric field is uniform and given by E = 4i+3j N/C. What

is the potential difference VA - VB?

8

B(5,0)

E = 4i N/Cx(m)

A(2,0)0

A) -12V

B) +12V

C) -24V

D) +24V

!

E = "#V $ Ex

= "dV

dx$ dV

A

B

% = "Ex

dxA

B

% $

VA"V

B= 4 & 3 =12V

Remember Capacitors!!

Capacitance of a capacitor:

Parallel plate capacitor:

Energy stored in a capacitor:!

C =Q

"V

!

U =Q2

2C=1

2C"V

2=1

2Q"V

Depends on geometry and

material between plates

The parallel plates of an isolated capacitor with vacuum between

them are given equal and opposite charge. If the plates are pulled

from a distance d to D > d, does the energy stored in the capacitor

1) Increase

2) Decrease

3) Remain equal

A)

B)

C)

10

Energy stored in a parallel plate capacitor

q is the same because C isolated

!

U =q2

2C

Cini ="0A

d#Cfin =

"0A

D$ Cfin < Cini $Ufin >Uini

q is the same

E is the same = q/(A#0)

"V increases = Ed

C decreases

U increases

Capacitors in parallel and series

11

"V1 = "V2 = "V Qtotal = Q1 + Q2

Ceq = C1 + C2

Q1=Q2 =Q "V = "V1+"V2 1/Ceq = 1/C1 + 1/C2

12

Calculate the equivalent CapacitanceC1 = 10 µF

C2 = 20 µF

C3 = 30 µF

C4 = 40 µF

V = 50 Volts

C2VC3

C1

C4

!

1

Ceq

=1

C1

+1

C2

+ C3

+1

C4

" Ceq = 6.9µF

!

V =V1

+V23

+V4 V

23=V

2=V

3

Q =Q1

=Q23

=Q4 Q

23=Q

2+Q

3

V =Q

Ceq

=Q

C1

+Q

C2

+ C3

+Q

C4 parallel

C1, C23, C4 in series

Electric current

!Current density J= I/A = nqvd

(direction of + charge carriers)

L

! SI unit: ampere 1 A = 1 C / s

! Average current:

! Instantaneous value:

n = number of electrons/volume

n x AL electrons travel distance L = vd "t

Iav = %Q/ %t = neAL vd /L

! Ohm’s Law: %V = R I (J = & E or E = $ J)

! %V = EL and E = ' J => ' I/A = %V/L

! R = $L/A Resistance in ohms (")

Equivalent resistance

R1= R = 2 k"

R2=R

R3=R

#=12

V

Req=3/2 R = 3 k"

#=12V

Which is the correct equivalent

circuit?

Req=3R = 6 k"

#=12V

A) B)

Quiz on a simple DC circuit

R1=R =2 k"

R2=R

R3=R

#=12

V

R’eq = Req + R1 = 3/2R=3 k"

R’eq= 3 k"#=12V

R1=R=2 k"

Req=R/2

#=12V

!

1

Req

=1

R+1

R=2

R" Req =

R

2

2 Identical Light Bulbs in parallel and series

Light Bulbs in series: which is more

luminous?

A)

B)

A) and B) have the same luminosity

A) B)A)

B)

Light Bulbs in parallel: which is more

luminous?

A)

B)

A) and B) have the same luminosity

In which of the 2 cases the bulbs will be more luminous?

- parallel - series

I

I

I/2

I/2!

PR

= RI2

4="2

R

!

PR

= RI2

= R"2

4R2

="2

4R

I=2!/R

Again power dissipated in resistors

17

The branch with A and B has larger resistance

than the one with C so less current flows in it!

Kirchhoff’s Rules

! Junction Rule: $ Iin = $ Iout! A statement of Conservation of Charge

! Loop Rule:

! A statement of Conservation of Energy

I1 = I2 + I3

!

"Vloop = "Vk = 0k

#

Kirchoff’s laws application

19

I1

I2

2 loops

Assume 1 current verse

per loop

I3

(given this you can consider 1

current per loop I1 and I2 and

remember that in the branch

in common between the 2

loops te current is I1-I2)

!

I1 = I2 + I3 " I3 = I1 # I2

8V + 4V # 4V #1$I1 # 2$I1 # 2$(I1 # I2) = 0

4V # 2$(I2 # I1) # 6$I2 = 0

Energy balance in a circuit

20

I1

I2

Chemical energy provided by

batteries is dissipated in

resistors (Joule heating)

BUT take into account the

verse of batteries and currents

flowing in resistors!

Power = rate of energy loss

!

P ="U

"t="Q

"t"V

P = I"V = RI2

="V

2

R

!

PR

=1"I12

+ 2"I12

+ 2"(I1 # I2)2

+ 6"I22

P$ = 8V % I1 + 4V % I1 # 4V % (I1 # I2)

Light bulbs and capacitor

! The circuit below contains 3 identical light bulbs and a fully

charged capacitor. Which light bulb(s) is (are) brightest?

21

equivalent

circuit

RC Circuits

22

What is the value of the time constant of this circuit?

A) 6 ms

B) 12 ms

C) 25 ms

23

RC Circuits: CHARGE/discharge OF C

R=10", C=30 mF

and #=20 Volts

R

C#

S1

!

" = Rdq

dt+q

C

!

q(t) = C"(1# e# t /RC )

I(t) ="

Re# t /RC

R

C

S1

!

RI = "Rdq

dt=q

C

!

q(t) = q0e" t /RC

I(t) =q0

RCe"t /RC

!

Udissipated = R I(t)2dt = RI

0

2e"2t /RC

dt0

#

$0

#

$ =

= Rq0

2

R2C2

RC

2=q0

2

2C

This is the energy stored in C!12/09/2002 U. Wisconsin, Physics 208, Fall 2006 24

Lorentz force and RH rule

An electron moves in the plane of this paper toward the top of

the page. A magnetic field is also in the plane of the page and

directed toward the right. The direction of the magnetic force

on the electron is

(A) toward the top of the page

(B) into the page

(C) toward the right edge of the page

(D) out of the pagev

B

F

12/09/2002 U. Wisconsin, Physics 208, Fall 2006 25

A charged particle traveling in the plane of this screen enters the region

of a uniform magnetic field pointing into the screen, as shown. The

subsequent path of the particle in the region of the magnetic field could

be represented by:

Quiz on Motion of a charged particle in B

A) Path 1

B) Path 2

C) Path 3

D) All of them

Centripetal force defines the

circular trajectory and points to

the center of the circle.

Is it a positive or a negative charge?

! Force on each charge

! Force on length ds of wire

26

Magnetic Force on wire with a Current and between wires

!

FB

= IL "B!

qv "B

!

qv" Ids

!

dF = Ids"B

! Attractive for parallel currents.

! Repulsive for antiparallel currents

! force/length

!

F /L =µ0

4"

2I1I2

a

27

Force on a wire and torque on a loop with current

vector direction: ( to plane of the loop in the

direction the thumb of right hand points if fingers curl in direction of current.

magnetic dipole moment:

µ = A I

) = A I B sin*

• Torque on loop

B

x

.F

F*

w

*

µ Force

couple

µ

)

I

BStable equilibrium when

and or plane of loop perpendicular

to direction of B

!

U = "r µ #

r B = "µB

!

r µ ||

r B

12/09/2002 U. Wisconsin, Physics 208, Fall 2006 28

In which of these cases the magnitude of the torque is largest?

(A) a (B) b (C) c

Another on Torque

Answer: (c). all loops enclose same area and carry same current +

magnitude of µ is the same for all. (c) µ upwards, µ ( B and " = µB. (a),

) = 0 (b) ) = µBsin*

ab

c

µµ "

"

29

Biot-Sarvart Law

! Each element of current produces a contribution to the magnetic field.

r *I

ds

!

dB =µo

4"

Ids# ˆ r

r2

B out of page

For a single charge in motion

!

B =µ0

4"

qv #ur

r2

For a wire with current

!

qv" Ids

12/09/2002 U. Wisconsin, Physics 208, Fall 2006 30

Question on Biot-Savart’s Law

How is the B-field directed for a wire with

a current flowing out of the page at a

distance r from the wire?i

r

dB

!

B =µ0

4"i

dL

r2

=#µ0

4"i2"r

r2

=µ0i

2r

Field lines are tangent to the circle

of radius r

What if there are 2 wires? B-field superimposes!

i B

r

i

r

B-B=0

!

B =µ0i

3r+

µ0i

r=4µ

0i

3r

12/09/2002 U. Wisconsin, Physics 208, Fall 2006

More Questions

31

What if the problem is bi-dimensional?

• Points A at (2,3)m and B at (5,7)m are in a region where the

electric field is uniform and given by E = 4i+3j N/C. What is

the potential difference VA - VB?

32

y

(m)

x(m)

A(2,3)

B(5,7)

E = 4i+3j N/C

Text

!

E = "#V $ dVA

B

% = " E & dsA

B

% =

" 4dx "2

5

% 3dy = "12 "12 = "24V $VA "VB = 24V3

7

%

33

Electrostatic Potential of 2 like charges

!

UA = kqQ

r,UB = k

2qQ

2r

!

Electrostatic potential

V (r) = kQ

r (point charge Q)

!

VA = kQ

r,VB = k

Q

2r

Potential depends on source charge and on distance, and it is independent on the test charge.

A)

B)

C)

34

Torque on a loop with current

! A loop of current-carrying wire is initially aligned with its plane in the direction of a uniform magnetic field. What happens to it?

Torque aligns loop ( ,

A. Stays parallel to magnetic field

B. Rotates till it is perpendicular to field

C. Rotates continuously