superposition of forces

Superposition of Forces

0

1 212 2

e4 r

q qF

r

We find the total force by adding the vector sum of the individual forces.

rr

Work Problem 21-18

21-18. (III) Two charges, and are a distance apart. These two charges are free to move but do not because there is a third charge nearby. What must be the magnitude of the third charge and its placement in order for the first two to be in equilibrium?

Q 4Q

Electric Field of a Point Charge

0

21

4k

qE kr

r2

0

ˆ= eF kq

Eq r

Relation between F and E

1

1

1 1

If we put a charge in an electric field , then the charge feels a force of value

***

q Eq

F q E

This is the really useful part.

Don’t confuse this charge q1 with the test charge q0 or the original charges q that produced E. The test charge q0 was used to find the electric field. This is a real charge q1 placed in the electric field.

Electric Field Lines for a Point Charge

Electric Field Lines for Systems of Charges

We call this a dipole. It is a dipole field.

The Electric Field of a Charged Plate

02E

A Parallel-Plate Capacitor

The electric field near a conducting surface must be perpendicular to the surface when in equilibrium.

If we place conductor in electric field,the E lines must be to surface. If not,charges would move. must be zero inside.E

Conductor placed around a charge +Q

Conductor

21-86.21-86. An electron moves in a circle of An electron moves in a circle of radius radius rr around a very long uniformly around a very long uniformly charged wire in a vacuum chamber, as charged wire in a vacuum chamber, as shown in the figure. The charge density on shown in the figure. The charge density on the wire is the wire is λλ = 0.14 = 0.14 μμC/m.C/m. ( (aa) What is the ) What is the electric field at the electron (magnitude and electric field at the electron (magnitude and direction in terms of direction in terms of rr and and λλ? (? (bb) What is the ) What is the speed of the electron? speed of the electron?

F qE ma

qEa

m

= =

=

We can work all kinds of problems with charged particles moving in electric fields.

Electron entering charged parallel plates

Electric flux:

Electric flux through an area is proportional to the total number of field lines crossing the area.

cosE E A EA qF = × =

Flux through a closed surface:

0 for closed surface that contains no chargeE

negative

positive

The net number of field lines through the surface is proportional to the charge enclosed, and also to the flux, giving Gauss’s law:

This can be used to find the electric field in situations with a high degree of symmetry.

enclE

o

QdE A

Electric field of charged sheetElectric field of charged sheet

0 0

0

0

(2 )

where is charge/area on the sheet.

2

2

Q AE A

AE A

E

Electric Potential V

Electric potential, or potential, is one of the most useful concepts in electromagnetism. This is a biggie!!

0 0

*** Unit: J/C = volt, VU W

Vq q

definition!!

Copyright © 2009 Pearson Education, Inc.

The electrostatic force is conservative – potential energy can be defined.

Change in electric potential energy is negative of work done by electric force:

Electrostatic Potential Energy and Potential Difference

b aU U W qEd- =- =-

The Potentials of Charge Distributions

If the electric field is known:

For many point charges:

For one point charge:

b

a

r

rV E dsD =- ò

04

qV

rpe=

0

1

4i

i i

qV

rpe= å

The Potentials of Charge Distributions

If the electric field is known:

For a continuous charge distribution:

For many point charges:

For differential charge:

b

a

r

rV E dsD =- ò

04

dqdV

rpe=

0

1

4i

i i

qV

rpe= å

0

1

4

dqV

rpe= ò


An equipotential is a line or surface over which the potential is constant.

Electric field lines are perpendicular to equipotentials.

The surface of a conductor is an equipotential.

Equipotential Surfaces

04

qV

rpe=


Another case showing electric field lines are perpendicular to equipotentials.

The surface of a conductor is an equipotential.

Equipotential Surfaces

We can also see that equipotentials are perpendicular to electric fields from the equation

V E dD =- ò


Equipotential SurfacesEquipotential surfaces are always perpendicular to field lines; they are always closed surfaces (unlike field lines, which begin and end on charges).

Electric field and equipotentials for electric dipole.

23-74. Four point charges are located at the corners of a square that is 8.0 cm on a side. The charges, going in rotation around the square, are Q, 2Q, -3Q and 2Q, where Q = 3.1 μC. What is the total electric potential energy stored in the system, relative to U = 0 at infinite separation?


When a capacitor is connected to a battery, the charge on its plates is proportional to the voltage:

The quantity C is called the capacitance.

Q CV=

QC

V=

Parallel plate capacitor

0 0

0

0

So and

Q dV Ed d

AAQ

V d

AQC

V d

The capacitance value depends only on geometry!


Capacitors in parallel have the same voltage across each one. The equivalent capacitor is one that stores the same charge when connected to the same battery:

Capacitors in Parallel

eq 1 3 (parallel)C C C C= + +


Capacitors in series have the same charge. In this case, the equivalent capacitor has the same charge across the total voltage drop. Note that the formula is for the inverse of the capacitance and not the capacitance itself!

Capacitors in Series

1 2 3eq 1 2 3

eq 1 2 3

eq 1 2 3

1 1 1

1 1 1 1

Q Q Q QV V V VC C C C

Q QC C C C

C C C C

Effect of a Dielectric on the Electric Field of a Capacitor

0E E 00 and /

where is the dielectric constant

/E E V V


A dielectric is an insulator, and is characterized by a dielectric constant .

Capacitance of a parallel-plate capacitor filled with dielectric:

Using the dielectric constant, we define the permittivity:

0

0

0

for parallel-plate capacitorACd

C C

k

Energy in electric fieldThe energy U in a capacitor is

The volume is Ad, and the energy density u is

22 0

20

1 12 2

12

AU CV Ed

d

U E Ad

20 2

0

1energy 12

2volume

E Adu E

Ad

Defibrillator

Potential energy of a charged capacitor:

All three expressions are equivalent!

221 1

2 2 2

QU QV CV

C


A complete circuit is one where current can flow all the way around. Note that the schematic drawing doesn’t look much like the physical circuit!

Open circuit

Direction of Current and Electron Flow

Ohm's law in this case is the emf of battery is the resistance of bulb

V IRVR

Resistors are color coded to indicate the value of their resistance.

622 10 10%


The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area:

The constant ρ, the resistivity, is characteristic of the material.

Resistivity

RA

r=

Energy and Power

The unit of electrical power is watt W (J/s).

If we use Ohm’s law with this equation, we have

( )Power

U dU dQ VP IV

t dt dt

2

2

( )P IV I IR I R

V VP IV V

R R

AC Voltage and Current for a Resistor Circuit

0 0

00

0 max

sin 2 sin

sin sin

peak current(maximum current)

V V ft V t

VVI t I t

R RI I

Note that I and V are in phase!!

25-36. (II) A 120-V hair dryer has two settings: 850 W and 1250 W. (a) At which setting do you expect the resistance to be higher? After making a guess, determine the resistance at (b) the lower setting; and (c) the higher setting.

25-43. (II) How many 75-W lightbulbs, connected to 120 V as in Fig. 25–20, can be used without blowing a 15-A fuse?

Resistors in Series

1 2 3

1 2 3

1 2 3

( ) eq

eq

IR IR IR

I R R R IR

R R R R

Current is not used up in each resistor. Same current I passes through each resistor in series.

1 2 3V V V eqIR


A parallel connection splits the current; the voltage across each resistor is the same:

1 2 3 1 2 3

1 2 3

1 2 3

1 1 1

1 1 1 1

eq

eq

V V V VV

R R R R R R R

R R R R

I I I I

Analyzing a Complex Circuit of Resistors

'eq

'eq

1 1 1 2

/ 2

R R R R

R R

eq

eq

/ 2

2.5

R R R R

R R

Kirchhoff’s Junction Rule

The sum of currents meeting at a junction must be zero. I1 – I2 – I3 = 0 or I1 = I2 + I3

In +

Out -

Kirchhoff’s Loop Rule

The sum of potential differences around any closed circuit loop is zero.

Our rules:

1) When going from – to + across an emf the

V is +. (+ to -, it is -).

2) When going across resistor in direction of

assumed I, the V is -. (Opposite, it is +).

Measuring the Current in a Circuit

We want ammeter to have very low resistance so it will not affect circuit. Ammeters go in series.

Ammeter

Measuring the Voltage in a Circuit

We want voltmeter to have very large resistance so it will not affect circuit. Voltmeters go in parallel across what is being measured.

Voltmeter

eq voltmeter

1 1 1

R R R


An ohmmeter measures resistance; it requires a battery to provide a current. These circuits are much more complicated. Rsh is a shunt resistor to change scales. Rser is a resistor to adjust galvanometer scale zero.

Magnetic Field Lines for a Bar Magnet

Imagine using a test pole N; place it at any point and see where the force is. Just like we do for electric fields. We actually use small compasses to do this.

B

S N

The Magnetic Force Right-Hand Rule

BF qv B

Units of magnetic field: teslas

The Lorentz force is the sum of the electric and magnetic forces acting on the same object:

kg1 T 1 C s

=×

F q E v Bæ ö÷ç ÷ç ÷ç ÷ç ÷ç ÷ç ÷÷çè ø

= + ´


The Earth’s magnetic field is similar to that of a bar magnet.

Note that the Earth’s “North Pole” is really a south magnetic pole, as the north ends of magnets are attracted to it.

Operating Principle of a Mass Spectrometer

mvr

q B

Several applications

Magnetic Force on a Current-Carrying Wire

sin

F q v B

dF I d B

F I d B

F I L B

F ILB

ddq

The Magnetic-Field Right-Hand Rule

discuss~ I

Br

Put thumb along direction of current, and fingers curl in direction of B.

r

Magnetic Forces on a Current Loop

0F

0F

Forces cause a torque

F I L B

I I

I

Ftotal = 0

F IhB=


An electric motor uses the torque on a current loop in a magnetic field to turn magnetic energy into kinetic energy.

27-23. (II) A 6.0-MeV (kinetic energy) proton enters a 0.20-T field, in a plane perpendicular to the field. What is the radius of its path? See Section 23–8.

We have to look closely at fields and forces to see how the forces occur.

2 2 1F I B

Ampère’s Law

We have seen that

02

IB

r

00

(2 ) 22

IB ds B ds B r r I

r

enclosed0B ds I

To find the field inside, we use Ampère’s law along the path indicated in the figure.

0

enclosed

0

0

0

0

#turns, turns/length

The dot product of and is zero everywhere except from cdd

cB ds B NI N n

NB I nI

B ds

B ds I

B ds

B nI

Induced Current Produced by a Moving Magnet

v

v

We conclude that it is the change in magnetic flux that causes induced current.

ind

B B AF =

This is called Faraday’s Law of Induction after Michael Faraday.

is number of turns

Induced emf BdN

dt

N

Lenz’s Law

The induced current will always be in the direction to oppose the change that produced it.

Induced emf Induced currentÛ

Applying Lenz’s Law to a Magnet Moving Toward and Away From a Current Loop

v v

Induced current

An Electrical Generator

Falling water,steam

Produces AC power

Magnetic flux changes!

Current is induced

A Simple Electric Motor/Generator

Inductance magnetic flux depends on current

is called inductance (actually self inductance here).

B

B

LI

d dIL

dt dt

L

The inductance L is a proportionality constant that depends on the geometry of the circuit

There will be a magnetic flux in Loop 1 due to current I1 flowing in Loop 1 and due to current I2 flowing in Loop 2.

1 1 12 2

2 2 21 1

(1)

Similarly,(2)

B

B

L I M I

L I M I

Now it is clearer why we call L self inductance and M mutual inductance.

For example, two nearby coils

Solenoid Self-Induction

Area A

Only depends on geometry.

0

20 0

20, so

B

B

B nI

NBA nNIA An I

LI L An

General energy density

2

0

20

22

00

1 general result

2

1

2

1

2

B

E

B E

Bu

u E

Bu u u E

30-34. (II) A 425-pF capacitor is charged to 135 V and then quickly connected to a 175-mH inductor. Determine (a) the frequency of oscillation, (b) the peak value of the current, and (c) the maximum energy stored in the magnetic field of the inductor.

RL Circuit

Think about what will happen when the switch is closed.

0 0dIV L IRdt

0V

I

Current as a Function of Time in an RL Circuit

/ /0 0

/

1 1t tR LV V

L R

I e eR R

Oscillations in LC Circuits

Start with charged capacitor. Close switch.

It will discharge through inductor, and then recharge in opposite sense.

If no resistance, will continue indefinitely.


The charge and current are both sinusoidal, but with different phases.

0

0cos( )

sin( )

Q Q t

I tI


LC Oscillations with Resistance (LRC Circuit)

Any real (nonsuperconducting) circuit will have resistance.

Damped Oscillations in RLC Circuits

Charge equation:

Solution:

where

and

2 4' 0 when LRC

w= =


This is a step-up transformer – the emf in the secondary coil is larger than the emf in the primary:

P P

S S

V N

V N

Lots of applications for transformers,the bug zapper.

Power distribution

Transformers work only if the current is changing; this is one reason why electricity is transmitted as ac.

Resistors, capacitors, and inductors have different phase relationships between current and voltage when placed in an ac circuit.

The current through a resistor is in phase with the voltage.

Single elements with AC Source

Resistive element0cosI I tw=


Therefore, the current through an inductor lags the voltage by 90°.

Inductive ElementThe voltage across the inductor is determined by

or0

0

00

cos 90

cos 90

V LI t

V V t

0

0

cos

sin

I I t

dIV L LI tdt

w

w w

=

= =- 0cosI I tw=

1t2t


The voltage across the inductor is related to the current through it:

The quantity XL is called the inductive reactance, and has units of ohms:

Inductive Circuit

0max 0 0 LX IV V LI

For very low frequencies the inductive reactance is small. That is because for direct currents (zero frequency), an inductor has little or no effect. Direct current passes right through an inductor.

2LX L f Lw pº =


Therefore, in a capacitor, the current leads the voltage by 90°.

The voltage across the capacitor is given by

Capacitive Circuit

00

00

1 cos 90

cos 90

QV I tC C

V t

1t

2t


Capacitive Circuit

The voltage across the capacitor is related to the current through it:

The quantity XC is called the capacitive reactance, and (just like the inductive reactance) has units of ohms:

0 00 0

0 0 0

1 cos 90 cos 90

1C

V I t V tC

V I I XC

1 12CX

C f Cw pº =

Effects of frequency on capacitive reactance

Note that when the frequency increases to large values that XC becomes very small. The current then becomes very large.

Why?

The frequency is so high that the capacitor doesn’t have time to fully charge. It almost acts as a short circuit. At low frequencies, it acts as an open circuit.

1CX

C

Either capacitors or inductors can be used to make either AC or DC filters:

AC & DC input


Analyzing the LRC series AC circuit is complicated, as the voltages are not in phase – this means we cannot simply add them. Furthermore, the reactances depend on the frequency.

LRC Series AC Circuit

Some Applications

Diodes and Rectifiers

• A diode conducts electricity in one direction only

• Can use diodes and combinations of diodes to make half- and full-wave rectifiers

Half-wave Rectifier

Some Applications

Full-wave Rectifier

superposition of forces

Documents

electric force

dipole field

electric fields

electric potential energy

total number of field

differential charge

charge q21

charge density