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Lecture 7

Physics 1202: Lecture 7Today�s Agenda

• Announcements:– Lectures posted on: www.phys.uconn.edu/~rcote/

• Office hours: – Monday 2:30-3:30 – Thursday 3:00-4:00

• Homework #2: due this coming Friday/• Labs: Already begun last week• Policy on clicker questions

– 80 % of total points gives 100%– No make-up for missed clicker questions …

• Policy on Homework– Lowest homework will be dropped– No extension

Today�s Topic :• Chapter 21: Electric current & DC-circuits

– Review » Electric current, resistance, Ohm’s law & power » Resistance in series & parallel » Kirchhoff’s rules» Capacitances in series & parallel

– RC-circuits– Measuring devices

• Chapter 22: Magnetism– Magnetic field (B) & force– Motion of a charged particle in B-field

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Lecture 7

e R

I

e = R I

I = DQ / Dt 21-1: Electric current

21-2: Resistance & Ohm’s Law• Resistance

Resistance is defined to be the ratio of the applied voltage to the current passing through.

• What does it mean ?it is the a measure of the friction slowing the motion of charges

V

I IR

UNIT: OHM = W

• Analogy with fluids

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Lecture 7

21-3: Energy & PowerBatteries & Resistors Energy expended

What�s happening?

Assert:

chemical to electrical

to heat

Charges per time

Energy �drop� per charge

Units:

For Resistors:

Rate is:

e R

I

e = R I

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Lecture 7

V R1R2

V

R1

R2

Summary• Resistors in series

– the current is the same in both R1 and R2

– the voltage drops add

• Resistors in parallel– the voltage drop is the same

in both R1 and R2

– the currents add

e1

I1e2

e3

R

R

RI2

I3

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Lecture 7

Capacitors in Parallel

V

a

b

Q2Q1 º V

a

b

Q

Þ C = C1 + C2

Capacitors in Seriesa bº +Q -Q

a b+Q -Q

Þ

21-7: RC Circuits• Consider the circuit shown:

– What will happen when we close the switch ?

– Add the voltage drops going around the circuit, starting at point a.

IR + Q/C – V = 0

– In this case neither I nor Q are known or constant. But they are related,

V

a

b

c

R

C

•This is a simple, linear differential equation.

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Lecture 7

RC Circuits• Case 1: Charging

Q1 = 0, Q2 = Q and t1 = 0, t2 = t

•To get Current, I = dQ/dt

Q

t t

I

V

a

b

c

R

C

RC Circuits

c

• Case 2: Discharging: Q1 = Q0 , Q2 = Q and t1 = 0, t2 = t

• To discharge the capacitor we have to take the battery out of the circuit (V=0)

•To get Current, I = dQ/dt

tIQ

t

V

a

b

c

R

C

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Lecture 7

Lecture 7, ACT 1

c

• Consider the simple circuit shown here. Initially the switch is open and the capacitor is charged to a potential VO. Immediately after the switch is closed, what is the current ?

A) I = VO/R B) I = 0 C) I = RC D) I = VO/R exp(-1/RC)

V

a

b

c

R

C

21-8: Electrical InstrumentsThe Ammeter

The device that measures current is called an ammeter.

Ideally, an ammeter should have zero resistance so thatthe measured current is not altered.

A

eI

R1 R2

+

-

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Lecture 7

Electrical InstrumentsThe Voltmeter

The device that measures potential difference is called a voltmeter.

An ideal voltmeter should have infinite resistance so thatno current passes through it.

V

e

R1 R2

I Iv

I2

Problem Solution Method:Five Steps:

1) Focus on the Problem- draw a picture – what are we asking for?

2) Describe the physics- what physics ideas are applicable- what are the relevant variables known and unknown

3) Plan the solution- what are the relevant physics equations

4) Execute the plan- solve in terms of variables- solve in terms of numbers

5) Evaluate the answer- are the dimensions and units correct?- do the numbers make sense?

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Lecture 7

Example: Power in Resistive Electric Circuits

A circuit consists of a 12 V battery with internal resistance of 2 W connected to a resistance of 10 W. The current in the resistor is I, and the voltage across it is V. The voltmeter and the ammeter can be considered ideal; that is, their resistances are infinity and zero, respectively.

What is the current I and voltage V measured by those two instruments ? What is the power dissipated by the battery ? By the resistance ? What is the total power dissipated in the circuit ?Comment on these various powers.

Step 1: Focus on the problem• Drawing with relevant parameters

– Voltmeter can be put a two places

e

RI I

rA

• What is the question ?– What is I ?– What is V ?– What is Pbattery ?– What is PR ?– What is Ptotal ?– Comment on the various

P�s

V

V10 W2 W

12 V

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Lecture 7

Step 2: describe the physics

• What concepts are relevant ?– Potential difference in a loop is zero– Energy is dissipated by resistance

• What are the known and unknown quantities ?– Known: R = 10 W ,r = 2 W, e = 12 V– Unknown: I, V, P�s

Step 3: plan the solution

• What are the relevant physics equations ?

• Kirchoff�s first law:

• Power dissipated:

For a resistance

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Lecture 7

Step 4: solve with symbols• Find I: e - Ir - IR = 0

e

RI I

rA

• Find V:

• Find the P�s:

Step 4: solve numerically• Putting in the numbers

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Lecture 7

Step 5: Evaluate the answers • Are units OK ?

– [ I ] = Amperes– [ V ] = Volts– [ P ] = Watts

• Do they make sense ?– the values are not too big, not too small …– total power is larger than power dissipated in R

» Normal: battery is not ideal: it dissipates energy

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Lecture 7

Magnetism• Magnetic effects from natural magnets have been known

for a long time. Recorded observations from the Greeks more than 2500 years ago.

• The word magnetism comes from the Greek word for a certain type of stone (lodestone) containing iron oxide found in Magnesia, a district in northern Greece – or maybe it comes from a shepherd named Magnes who got the stuff stuck to the nails in his shoes

• Properties of lodestones: could exert forces on similar stones and could impart this property (magnetize) to a piece of iron it touched.

• Small sliver of lodestone suspended with a string will always align itself in a north-south direction. ie can detect the magnetic field produced by the earth itself. This is a compass.

Bar Magnet• Bar magnet ... two poles: N and S

Like poles repel; Unlike poles attract.

• Magnetic Field lines: (defined in same way as electric field lines, direction and density)

• Does this remind you of a similar case in electrostatics?

You can see this field by bringing a magnet near a sheet covered

with iron filings

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Lecture 7

Magnetic Field Lines of a bar magnet

Electric Field Linesof an Electric Dipole

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Lecture 7

Magnetic Monopoles• One explanation: there exists magnetic charge, just like

electric charge. An entity which carried this magnetic charge would be called a magnetic monopole (having + or -magnetic charge).

• How can you isolate this magnetic charge?Try cutting a bar magnet in half:

NS N NS S

• In fact no attempt yet has been successful in finding magnetic monopoles in nature.

• Many searches have been made• The existence of a magnetic monopole could give an

explanation (within framework of QM) for the quantization of electric charge (argument of P.A.M.Dirac)

Source of Magnetic Fields?• What is the source of magnetic fields, if not magnetic

charge?• Answer: electric charge in motion!

– eg current in wire surrounding cylinder (solenoid) produces very similar field to that of bar magnet.

• Therefore, understanding source of field generated by bar magnet lies in understanding currents at atomic level within bulk matter.

Orbits of electrons about nuclei

Intrinsic �spin� of electrons (more important effect)

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Lecture 7

22-2:Forces due to Magnetic Fields• Electrically charged particles come under various

sorts of forces.

• As we have already seen, an electric field provides a force to a charged particle, F = qE.

• Magnets exert forces on other magnets.

• Also, a magnetic field provides a force to a charged particle, but this force is in a direction perpendicular to the direction of the magnetic field.

Definition of Magnetic FieldMagnetic field B is defined operationally by the magnetic force on a test charge. (We did this to talk about the electric field too)

• What is "magnetic force"? How is it distinguished from "electric" force?

Start with some observations:

• Empirical facts: a) magnitude: µ to velocity of q

b) direction: ^ to direction of qq

F

v

mag

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Lecture 7

Magnetic Force on a Moving Charge• When moving through a magnetic field, a

charged particle experiences a magnetic force

where B is called Magnetic Field:• It is a vector quantityo The SI unit of magnetic

field is the Tesla (T)o The cgs unit is a Gauss (G)

o 1 T = 104 G

Earth B-field: 0.5 G or 5 x 10-5 T

• Given by the right-hand rule– direction of Fmag on a positive charge– Fmag on a negative charge:opposite direction

© 2017 Pearson Education, Inc.

Direction of Magnetic Force

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Lecture 7

Direction of Magnetic Force• This relationship between the three vectors—

magnetic field, velocity, and force—can also be written as a vector cross product:

Right Hand Rule:

Your thumb points in the direction of the force, F , for a positive charge

max if v & B perpendicular0 if v & B parallel

• Consider a positive charge• in an electric field – force in the direction of field E

• in a magnetic field – force is perpendicular to field B

• This leads to very different motions

© 2017 Pearson Education, Inc.

22-3: motion of a charges particle

• Because Fmag is perpendicular to the direction of motion, the path of a particle is circular

• Also, while E can do work on a particle, B cannot—the particle�s speed remains constant

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Lecture 7

Lorentz Force• The force F on a charge q moving with velocity v

through a region of space with electric field E and magnetic field B is given by:

F

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

v

B

q

® ® ® ® ®

® ® ® ® ®v

B

qF = 0

v

B

qF�

Units: 1 T (tesla) = 1 N / Am1G (gauss) = 10-4 T

Lecture 7, ACT 2• Two protons each move at speed v (as

shown in the diagram) toward a region of space which contains a constant B field in the -y-direction. – What is the relation between the

magnitudes of the forces on the two protons?

(a) F1 < F2 (b) F1 = F2 (c) F1 > F2

1A B

x

y

z

1

2

v

v

1B

(a) F2x < 0 (b) F2x = 0 (c) F2x > 0

– What is F2x, the x-component of the force on the second proton?

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Lecture 7

Circular motion• Force is perp. to v

• q = 90o so sinq = 1 or F=qvB

• W=0 Þ DK=0– Kinetic energy not changed– Velocity constant: UCM ! R

• Work proportional to cos f (recall 1201)! f :angle between F and Dx– cos f =0 (perpendicular)

Lecture 7, ACT 3• Cosmic rays (atomic nuclei stripped bare

of their electrons) would continuously bombard Earth�s surface if most of them were not deflected by Earth�s magnetic field. Given that Earth is, to an excellent approximation, a magnetic dipole, the intensity of cosmic rays bombarding its surface is greatest at the (The rays approach the earth radially from all directions).

A) Poles B) Equator C) Mid-lattitudes

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Lecture 7

Motion of Charged Particles• If a v makes an angle with B

– the component of v along B will not change

– a particle with initial v at an angle to B will move in a helical path.

Trajectory in Constant B Field

FFvR

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

v B

q

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

• Suppose charge q enters B field with velocity v as shown below. (v^B) What will be the path q follows?

• Force is always ^ to velocity and B. What is path? – Path will be circle. F will be the centripetal force needed to keep

the charge in its circular orbit. Calculate R:

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Lecture 7

Radius of Circular Orbit• Lorentz force:

• centripetal acc:

• Newton's 2nd Law:

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

v

F

B

qFvR

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

This is an important result, with useful experimental consequences !

Þ

Þ

Ratio of charge to mass for an electron e-

3) Calculate B … next week; for now consider it a measurement4) Rearrange in terms of measured values, V, R and B

1) Turn on electron �gun�

DV

�gun�

2) Turn on magnetic field B

R

&

Þ

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Lecture 7

Lawrence's Insight"R cancels R"

• We just derived the radius of curvature of the trajectory of a charged particle in a constant magnetic field.

• E.O. Lawrence realized in 1929 an important feature of this equation which became the basis for his invention of the cyclotron.

• R does indeed cancel R in above eqn. So What??– The angular velocity is independent of R!! – Therefore the time for one revolution is independent of the

particle's energy! – We can write for the period, T=2p/w or T = 2pm/qB– This is the basis for building a cyclotron.

• Rewrite in terms of angular velocity w !

Þ ÞÞ

The Hall Effect

cd

l

a

c

B

BI

I-

vd F

Hall voltage generatedacross the conductor

qEH

Force balance

Using the relation between drift velocity and current we can write:

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Lecture 7

22-4: Magnetic Force on a Current• Consider a current-carrying wire in the

presence of a magnetic field B. • There will be a force on each of the charges

moving in the wire. What will be the total force DF on a length Dl of the wire?

• Suppose current is made up of ncharges/volume each carrying charge q and moving with velocity v through a wire of cross-section A.

N S

Simpler: For a straight length of wire L carrying a current I, the force on it is:

• Force on each charge =

• Total force =

Þ• Current =

or

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Lecture 7

Lecture 7, ACT 4• A current I flows in a wire which is formed in the

shape of an isosceles triangle as shown. A constant magnetic field exists in the -z direction. – What is Fy, net force on the wire in the y-

direction?

(a) Fy < 0 (b) Fy = 0 (c) Fy > 0x

y

Recap of Today�s Topic :• Chapter 21: Electric current & DC-circuits

– Review » Electric current, resistance, Ohm’s law & power » Resistance in series & parallel » Kirchhoff’s rules» Capacitances in series & parallel

– RC-circuits– Measuring devices

• Chapter 22: Magnetism– Magnetic field (B) & force– Motion of a charged particle in B-field