A) Ohm’s law is obeyed since the current still increases when V increases

B) Ohm’s law is not obeyed

C) This has nothing to do with Ohm’s law

You double the You double the voltagevoltage

across a certain conductor across a certain conductor

and you observe the and you observe the currentcurrent

increases three times. increases three times.

What can you conclude?What can you conclude?

B) Ohm’s law is not obeyed

Ohm’s law says I is prop. to V, here V=RI at first, then 2V=3R I, so the “proportionality constant” changes

You double the You double the voltagevoltage

across a certain conductor across a certain conductor

and you observe the and you observe the currentcurrent

increases three times. increases three times.

What can you conclude?What can you conclude?

Consider two identical resistors wired in Consider two identical resistors wired in series. If there is an electric current through series. If there is an electric current through the combination, the current in the second the combination, the current in the second resistor is resistor is

A.A. equal to equal to

B.B. half half

C.C. smaller than, but not necessarily half the smaller than, but not necessarily half the current through the first resistor.current through the first resistor.

Two wires of identical length and identical Two wires of identical length and identical metal but with different diameters are metal but with different diameters are connected to identical batteries. At which connected to identical batteries. At which point is the point is the currentcurrent greater? greater?

A.A. Point 1Point 1

B.B. Point 2Point 2

C.C. They are the sameThey are the same

1 2

Skinny wire Fat wire

A wire connected to a battery has sections A wire connected to a battery has sections of different diameters. How do the of different diameters. How do the currentscurrents at points 1, 2 and 3 compare?at points 1, 2 and 3 compare?

A.A. 1>2>31>2>3

B.B. 1=3>21=3>2

C.C. 1=3<21=3<2

D.D. 1=2=31=2=3

1

3

2

A wire connected to a battery has sections A wire connected to a battery has sections of different diameters. How do the of different diameters. How do the current current densitiesdensities at points 1, 2 and 3 compare? at points 1, 2 and 3 compare?

A.A. 1>2>31>2>3

B.B. 1=3>21=3>2

C.C. 1=3<21=3<2

D.D. 1=2=31=2=3

1

3

2

A wire connected to a battery has sections A wire connected to a battery has sections of different diameters. How do the of different diameters. How do the potential differencespotential differences VVabab, , VVbcbc and and VVcdcd

compare?compare?A.A. VVab ab > > VVbcbc > > VVcdcd

B.B. VVab ab = = VVcdcd > > VVbc bc

C.C. VVab ab = = VVcdcd < < VVbcbc

D.D. VVab ab = = VVcdcd = = VVbcbc

a

b

c

d

In the circuit below, what is the voltage In the circuit below, what is the voltage across across RR11??

12 V

R1= 4 R2= 2

A. 12 V

B. 6 V

C. 8 V

D. 4 V

In the circuit below, what is the current In the circuit below, what is the current through through RR11??

A. 10 AB. 5 AC. 2 AD. 7 A

10 V

R1= 5

R2= 2

Points P and Q are connected to a battery Points P and Q are connected to a battery of fixed voltage. As more resistors of fixed voltage. As more resistors RR are are added to the parallel circuit, what happens added to the parallel circuit, what happens to the to the total currenttotal current in the circuit? in the circuit?

A. It increases

B. It remains the same

C. It decreases but not all the way to zero

D. It drops to zero

Current flows through a light bulb. Suppose Current flows through a light bulb. Suppose a wire is connected across the bulb as a wire is connected across the bulb as shown. When the wire is connected,shown. When the wire is connected,

A.A. all the current continues to flow through the bulb. all the current continues to flow through the bulb.

B.B. half the current flows through the wire, the other half half the current flows through the wire, the other half continues through the bulb. continues through the bulb.

C.C. all the current flows through the wire. all the current flows through the wire.

D.D. none of the abovenone of the above

The circuit below consists of two identical The circuit below consists of two identical light bulbs burning with equal brightness light bulbs burning with equal brightness and a single 12 V battery. When the switch and a single 12 V battery. When the switch is closed, the brightness of bulb is closed, the brightness of bulb AA

A.A. increases. increases.

B.B. remains unchanged.remains unchanged.

C.C. decreases. decreases.

The circuit shown has three identical light The circuit shown has three identical light bulbs, each of them with resistance bulbs, each of them with resistance RR. . Which if the following expresses the relative Which if the following expresses the relative brightnesses of the bulbs:brightnesses of the bulbs:

A.A. A > B = CA > B = C

B.B. A < B = CA < B = C

C.C. A < B < CA < B < C

D.D. Cannot determineCannot determine

A

B

C

If the four light bulbs in the figure are If the four light bulbs in the figure are identical, which circuit puts out more (total) identical, which circuit puts out more (total) light?light?

A.A. Circuit I Circuit I

B.B. The two emit the same amount of light. The two emit the same amount of light.

C.C. Circuit IICircuit II

A.It increases

B.It decreases

C.It stays the same

What happens to the voltage across the resistor What happens to the voltage across the resistor RR11

when the switch is closed? (All resistances are when the switch is closed? (All resistances are

equal.)equal.)

V

R1

R3R2

S

V

R1

R3 R4R2

S

What happens to the voltage across the resistor What happens to the voltage across the resistor RR44

when the switch is closed? (All resistances are equal.)when the switch is closed? (All resistances are equal.)

A.It increases

B.It decreases

C.It stays the same

Kirchhoff’s RulesKirchhoff’s Rules

At any junction point, the sum of all At any junction point, the sum of all currents entering the junction is the same currents entering the junction is the same as the sum of all currents leaving the as the sum of all currents leaving the junction.junction.

The sum of all changes in potential around The sum of all changes in potential around any closed loop of a circuit must be zero.any closed loop of a circuit must be zero.

Solving Problems with Kirchhoff’s Solving Problems with Kirchhoff’s RulesRules

1.1. Label all currents: ILabel all currents: I11, I, I22, …and choose directions!, …and choose directions!

2.2. Identify the unknownsIdentify the unknowns

3.3. Apply junction rule: write on current as the sum of Apply junction rule: write on current as the sum of othersothers

4.4. Apply loop rule: Apply loop rule: (choose direction of loop)(choose direction of loop)

- For resistors: Ohm’s law, negative sign if loop direction For resistors: Ohm’s law, negative sign if loop direction is the same as chosen current directionis the same as chosen current direction

- For batteries: sign positive if loop direction chosen For batteries: sign positive if loop direction chosen points from negative to positive terminal points from negative to positive terminal

5.5. Solve equations algebraicallySolve equations algebraically

RC CircuitsRC Circuits

There is a change of charge on the capacitor, or There is a change of charge on the capacitor, or voltage across C and across R and of the voltage across C and across R and of the current, so I is not constant!current, so I is not constant!How fast does it happen? Determined by the How fast does it happen? Determined by the circuit, not how the circuit is operated! circuit, not how the circuit is operated! Typical time (time constant) t=RCTypical time (time constant) t=RCNote: t is not a function of V,I,Q! Note: t is not a function of V,I,Q! After a long time (ca 5t), nothing happens After a long time (ca 5t), nothing happens anymore!anymore!– No current, charge maximal, voltage across C No current, charge maximal, voltage across C

maximal, voltage across R zero.maximal, voltage across R zero.

RC CircuitsRC Circuits

There is a change of charge on the capacitor, or There is a change of charge on the capacitor, or voltage across C and across R and of the voltage across C and across R and of the current, so I is not constantcurrent, so I is not constantHow fast does it happen? Determined by the How fast does it happen? Determined by the circuit, not how the circuit is operated.circuit, not how the circuit is operated.Typical time (time constant) t=RCTypical time (time constant) t=RCNote: t is not a function of V,I,Q Note: t is not a function of V,I,Q After a long time (ca 5t), nothing happens After a long time (ca 5t), nothing happens anymore!anymore!– No current, charge maximal, voltage across C No current, charge maximal, voltage across C

maximal, voltage across R zero.maximal, voltage across R zero.

RC Switch OnRC Switch On

Capacitor gets chargedCapacitor gets chargedCharges flow from battery Charges flow from battery through resistor to capacitorthrough resistor to capacitorCharges flow fast when Charges flow fast when capacitor holds little chargecapacitor holds little chargeCurrent (charge flowing by a Current (charge flowing by a point per time) is strong, point per time) is strong, becomes weakerbecomes weakerInitially, no charge on Initially, no charge on capacitor, i.e. no voltage capacitor, i.e. no voltage V=Q/C across capacitor, i.e. V=Q/C across capacitor, i.e. all voltage drop has to all voltage drop has to happen at resistor (100%)happen at resistor (100%)

Voltage across Capacitor

At the same time:Voltage across Resistor

At the same time:Current in circuit

RC Switch OffRC Switch Off

Capacitor gets Capacitor gets dischargeddischargedCharges flow from one Charges flow from one plate of the capacitor to plate of the capacitor to the other (total charge of the other (total charge of circuit: zero!)circuit: zero!)Charges flow fast when Charges flow fast when capacitor has a lot of capacitor has a lot of chargechargeCurrent (charge flowing Current (charge flowing by a point per time) is by a point per time) is strong, becomes weakerstrong, becomes weakerSimulationSimulation

Voltage across Capacitor

At the same time:Current in circuit

At the same time:Voltage across Resistor

Before the switch S is closed, the charges on the Before the switch S is closed, the charges on the capacitors arecapacitors are

A.A. the samethe same

B.B. differentdifferent

C.C. Impossible to tell without knowing moreImpossible to tell without knowing more

After the switch S is closed, the charges on the After the switch S is closed, the charges on the capacitors arecapacitors are

A.A. the samethe same

B.B. differentdifferent

C.C. Impossible to tell without knowing more Impossible to tell without knowing more

A short time after the switch S is closed, the A short time after the switch S is closed, the current in the resistors will havecurrent in the resistors will have

A.A. increasedincreased

B.B. decreaseddecreased

C.C. stayed the same stayed the same

A A long long time after the switch S is closed, the time after the switch S is closed, the current in the resistors will becurrent in the resistors will be

A.A. zerozero

B.B. constantconstant

C.C. maximal maximal

A A long long time after the switch S is closed, the time after the switch S is closed, the current through point b will becurrent through point b will be

A.A. zerozero

B.B. constantconstant

C.C. maximal maximal

Group WorkGroup WorkSwitch open:Switch open:

– What is the potential What is the potential at point a? (V=0 at at point a? (V=0 at negative terminal)negative terminal)

– What is the potential What is the potential at point b?at point b?

– What is the charge of What is the charge of the capacitors? the capacitors?

0.36

Switch is closed:Switch is closed:- What is the final potential at point b?- What is the final potential at point b?- What are the charges of the capacitors?- What are the charges of the capacitors?-How much charge flows through the switch How much charge flows through the switch after after it is closed?it is closed?

MagnetismMagnetism

Magnetic PhenomenaMagnetic Phenomena

Magnets always have two poles: N & SMagnets always have two poles: N & S

Opposite poles attractOpposite poles attract

Some metals (ferromagnetic) are attracted Some metals (ferromagnetic) are attracted by magnets even though they are not by magnets even though they are not “magnetic”“magnetic”

Earth has a (weak) magnetic field not Earth has a (weak) magnetic field not totally in line with the rotational axistotally in line with the rotational axis

Magnetic field lines can be visualizedMagnetic field lines can be visualized

Magnetic Phenomena IIMagnetic Phenomena II

Electric currents do produce magnetic Electric currents do produce magnetic fields, electric charges NOTfields, electric charges NOTMagnetic fields exert forces on currents, Magnetic fields exert forces on currents, moving chargesmoving chargesForce is perpendicular to field, current, Force is perpendicular to field, current, velocity of particlevelocity of particleA current loop in a magnetic field will A current loop in a magnetic field will rotate! (since there is a torque on it)rotate! (since there is a torque on it)The latter has many applications: motors, The latter has many applications: motors, generators, loudspeakers, galvanometersgenerators, loudspeakers, galvanometers

Compare to Electric Dipole FieldCompare to Electric Dipole Field

Group WorkGroup Work

Find the direction to the magnetic north Find the direction to the magnetic north polepole

Which pole of your little magnetic is the Which pole of your little magnetic is the North pole?North pole?

Categorize everyday objects: Categorize everyday objects: ferromagnetic or not? (keys, coins, paper ferromagnetic or not? (keys, coins, paper clips, rings, staples, stay clear of watches, clips, rings, staples, stay clear of watches, credit cards!)credit cards!)

If the Earth’s magnetic field would If the Earth’s magnetic field would be caused be a giant permanent be caused be a giant permanent magnet, where would the North magnet, where would the North

pole of this magnet be pointed to?pole of this magnet be pointed to?

Towards the geographic north poleTowards the geographic north pole

Towards the geographic south pole Towards the geographic south pole

Towards the magnetic north pole Towards the magnetic north pole

Towards the magnetic south pole Towards the magnetic south pole

What happens when a charged object is What happens when a charged object is brought near a magnet?brought near a magnet?

A.A. The south pole goes toward the positiveThe south pole goes toward the positive

B.B. The north pole rotates toward the positiveThe north pole rotates toward the positive

C.C. Neither pole is attracted. The magnet Neither pole is attracted. The magnet won’t rotatewon’t rotate

Right Hand RuleRight Hand Rule

What is the difference between your left What is the difference between your left and right hand? After all, index finger and right hand? After all, index finger always is between thumb and middle always is between thumb and middle finger.finger.Weak rule: magnetic field around a current Weak rule: magnetic field around a current in thumb direction is in direction of fingersin thumb direction is in direction of fingersStrong rule: force on a current in thumb Strong rule: force on a current in thumb direction in a field in index direction is in direction in a field in index direction is in middle finder direction middle finder direction

Where does the vector Where does the vector j x ij x i point to?point to?

in the direction of the positive x axisin the direction of the positive x axis

in the direction of the positive z axisin the direction of the positive z axis

in the direction of the negative z axis in the direction of the negative z axis

The resulting object is not a vector The resulting object is not a vector

In what direction is the magnetic field at point P?

A. Into the screen

B.Out of the screen

C.Towards the wire

D.Away from the wire

I

P B

W

At B? At W?

A rectangular loop with counter-A rectangular loop with counter-clockwise current is hanging on a clockwise current is hanging on a spring into a magnetic field in +x spring into a magnetic field in +x

direction. What happens to the wire direction. What happens to the wire loop?loop?

NothingNothing

Moves downMoves down

Moves left Moves left

Moves forwardMoves forward

vv

A positive charge A positive charge

enters a uniform enters a uniform

magnetic field as magnetic field as

shown. What is shown. What is

the direction of the the direction of the

magnetic force?magnetic force?

A. out of the page

B. into the page

C. downwards

D. upwards

What is the direction of the force What is the direction of the force on an electron moving in the on an electron moving in the

negative x-direction in a magnetic negative x-direction in a magnetic field in the positive z-direction?field in the positive z-direction?

direction of the positive x axisdirection of the positive x axis

direction of the positive z axisdirection of the positive z axis

direction of the negative z axis direction of the negative z axis

direction of the negative y axisdirection of the negative y axis

xx

yy

A proton beam travels A proton beam travels

through a region of through a region of

magnetic field as shown. magnetic field as shown.

What is the direction of the What is the direction of the

magnetic field? magnetic field?

A. + y

B. – y

C. + z (out of page)

D. – z (into page)

x x x x x x x x x x x x

x x x x x x x x x x x x

x x x x x x x x x x x x

x x x x x x x x x x x x

x x x x x x x x x x x x

x x x x x x x x x x x x

A

B

C

D

A beam of charged A beam of charged

particles enters a region of particles enters a region of

magnetic field. What path magnetic field. What path

will the atoms follow?will the atoms follow?

MagnetosphereMagnetosphere

Magnetic Magnetic north pole north pole about 7about 7° ° west of west of geographic geographic north pole north pole

Motion of Charged ParticlesMotion of Charged Particles

Van Allen Radiation BeltsVan Allen Radiation Belts

Mainly Mainly heavier heavier protons in protons in the inner the inner beltbelt

electrons in electrons in outer beltouter belt

Aurora BorealisAurora Borealis

Aurora Borealis from SpaceAurora Borealis from Space

Particles of same mass are moving in a Particles of same mass are moving in a uniform field. Rank their magnitude of initial uniform field. Rank their magnitude of initial

acceleration. Magnetic field in positive x acceleration. Magnetic field in positive x directiondirection

# Charge/mC Speed/m/s dir. of velocity# Charge/mC Speed/m/s dir. of velocityA 5 3 +xA 5 3 +xB 5 3 -xB 5 3 -xC 5 3 +yC 5 3 +yD 5 3 -yD 5 3 -yE -10 3 +yE -10 3 +yF 10 3 -yF 10 3 -yG -10 5 +yG -10 5 +yH 10 5 -yH 10 5 -y

What is the magnetic field around a What is the magnetic field around a coaxial cable? coaxial cable?

Zero B=?Zero B=?

Perpendicular to cablePerpendicular to cable

Parallel to cable Parallel to cable

Some other Some other

directiondirection

II1 1 = = II22

The line integral of The line integral of BB around the loop around the loop shownshown

A.A. is positive.is positive.

B.B. is negative.is negative.

C.C. is zero.is zero.

D.D. cannot be determined without more information.cannot be determined without more information.

I

Eight wires cut the Eight wires cut the page at the points page at the points shown. Wire number shown. Wire number kk has current has current kIkI00. For . For

even numbered even numbered wires the current wires the current flows into the page, flows into the page, and vice versa for and vice versa for odd numbered wires.odd numbered wires.

What is forWhat is for

the path shown?the path shown?

1

2

34

5

6

78

ldB

Group Work

Eight wires cut the Eight wires cut the page at the points page at the points shown. Wire number shown. Wire number kk has current has current kIkI00. For . For

even numbered even numbered wires the current wires the current flows into the page, flows into the page, and vice versa for and vice versa for odd numbered wires.odd numbered wires.

What is for What is for the path shown?the path shown?

1

2

34

5

6

78

ldB

An amperian loop is in a An amperian loop is in a uniformuniform magnetic magnetic field as shown. The line integral of field as shown. The line integral of BB around this looparound this loop

A.A. is positive.is positive.

B.B. is negative.is negative.

C.C. is zero.is zero.

D.D. cannot be determined without more information.cannot be determined without more information.

B

An amperian loop is in a An amperian loop is in a non-uniformnon-uniform magnetic field as shown. The line integral of magnetic field as shown. The line integral of BB around this loop around this loop

A.A. is positive.is positive.

B.B. is negative.is negative.

C.C. is zero.is zero.

D.D. cannot be determined without more information.cannot be determined without more information.

B

If the magnetic field in some region is as If the magnetic field in some region is as shown we can concludeshown we can conclude

A.A. that current must be flowing into the page in this that current must be flowing into the page in this region of space.region of space.

B.B. that current must be flowing out of the page in that current must be flowing out of the page in this region of space.this region of space.

C.C. that no current is flowing in this region of space.that no current is flowing in this region of space.

D.D. nothing at all about the current flow here.nothing at all about the current flow here.

B

Directionality of Ampere’s lawDirectionality of Ampere’s law

The direction of the amperian loop can be The direction of the amperian loop can be chosen at willchosen at will

The result of the calculation will depend on the The result of the calculation will depend on the choice, but NOT the physics!choice, but NOT the physics!– Counterclockwise and clockwise integration will differ Counterclockwise and clockwise integration will differ

by a sign, and hence the enclosed current will differ by a sign, and hence the enclosed current will differ by a signby a sign

– The meaning is the same: if the loop is traversed in The meaning is the same: if the loop is traversed in the opposite direction, the current through its area has the opposite direction, the current through its area has to be assigned the opposite sign, see example to be assigned the opposite sign, see example

Example: Example:

Wire carrying Wire carrying upward currentupward current

Magnetic field Magnetic field circular around circular around wire, wire, counterclockwise counterclockwise when viewed from when viewed from above above

I

B

Loop 1: Loop 1:

Choose Choose amperian loopamperian loop counterclockwisecounterclockwiseThen the area vector Then the area vector points up (weak right points up (weak right hand rule)hand rule)That means that the That means that the current I has to be current I has to be counted positive, as in counted positive, as in “parallel to A”“parallel to A”Since dl is parallel with Since dl is parallel with B, integral will be B, integral will be positivepositive

Ampere’s law ok (+=+)Ampere’s law ok (+=+)

I

B

AA

Loop 2: Loop 2:

Choose Choose amperian loopamperian loop clockwiseclockwiseThen the area vector Then the area vector points down (weak right points down (weak right hand rule)hand rule)That means that the That means that the current I has to be current I has to be counted negative, as in counted negative, as in “anti-parallel to A”“anti-parallel to A”Since dl is anti-parallel to Since dl is anti-parallel to B, integral will be B, integral will be negativenegative

Ampere’s law ok (- = -)Ampere’s law ok (- = -)

I

B

AA

Two types of variablesTwo types of variables

Anything connected to the amperian loop Anything connected to the amperian loop is “hypothical” and can be chosen (size, is “hypothical” and can be chosen (size, direction of loop)direction of loop)Wires (shape) and currents (direction), Wires (shape) and currents (direction), magnetic fields are physical, cannot be magnetic fields are physical, cannot be chosenchosenEnclosed currentEnclosed current is the is the current as defined current as defined by the amperian loop, and not necessarily by the amperian loop, and not necessarily the the physical currentphysical current! !

Magnetic field due to a current loop

This is not a straight wire cannot use

straight-wire formula

Use Biot-Savart’s law

Check Biot-Savart law first on straight wire!

Magnetic field of straight wire

rr

? dB

dl

θ

RR

II

In which direction does dB point?

UpDownInto screenOut of screen

Solenoids (“lots of loops”)Solenoids (“lots of loops”)

L

What direction does the magnetic What direction does the magnetic field created by the current in the field created by the current in the

wire have at point P?wire have at point P?Little bit below +xLittle bit below +x

Away from wire above PAway from wire above P

Towards wire left of PTowards wire left of P

Hard to sayHard to say

º P

x

y

What direction does the magnetic What direction does the magnetic field created by the bit of current in field created by the bit of current in the wire above P have at point P?the wire above P have at point P?

Little bit below +xLittle bit below +x

Out of the pageOut of the page

Into the pageInto the page

Hard to sayHard to say

º P

x

y

Ferromagnetism

When the material is unmagnetized, the domains are randomly oriented. They can be partially or fully aligned by placing the material in an external magnetic field.

Hysteresis