Welcome to Physics 7C!Welcome to Physics 7C!

Lecture 5 -- Winter Quarter -- 2005

Professor Robin Erbacher

343 Phy/Geo

erbacher@physics.ucdavis.edu

AnnouncementsAnnouncements

• Course policy and regrade forms on the web: http://physics7.ucdavis.edu

• All lectures are posted on the web.

• Quiz today! ~20 minutes long on Block 12.

• Block 13 begins: DLMs 9, 10, and 11 this week.

• Turn off cell phones and pagers during lecture.

• If I’m speaking too loudly or softly, tell me!

Force ModelsForce ModelsIn Physics 7B you learned about contact forces: normal and friction, gravitational, electric. We will call this the Direct Model of Forces

It’s straightforward to think about a ball bouncing off the ground due to direct contact with the ground. But:How does Earth exert its gravitational force on the ball while in mid-air?

This is an example of action-at-a-distance, and leads toField Model of Forces

€

Object A Object B

exerts force

field Object Bexerts forceObject A field

creates

Recall: What is a Field?Recall: What is a Field?

€

What is a Field? => a “map” of a measurable quantity

Side out in

MAP? = > spatial variation (x,y,z)

* TemperatureScalars: * Elevation

* Atmospheric PressureQuantities? Scalar = magnitude only

* Wind VelocityVectors: * Gravity Field g = GM/r2

* Electric Field E = kQ/r2

Vector = magnitude and directionex: velocity v {3 components… (vx,vy,vz) }

Bold means vector or overstrike arrow: v

Gravity Field MapsGravity Field Maps

€

What is a Field? => a “map” of a measurable quantity

large

M

small

M

g field

vectors

g field

vectors

Notice that the magnitude of the vectors increasefor larger mass M: strength of field is greater!

Vector AdditionVector Addition

€

Side out in

Recall your vector addition rules:Whether we are discussing force vectors or field vectors, the rules of vector addition are simple.

• Always add vectors head-to-toe. Then it doesn’t matter what order you add them in.• The length of a vector is in general proportional to the magnitude.• The magnitude of a vector is a scalar: a simple numerical quantity.• You can break it down into x and y components to add the vectors.

ˆˆ

Gravitational Fields and Forces

Gravitational Fields and Forces

For gravity, we can think about an Object with mass M exerting a force on another object with mass m.

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Fdirect = -GMm

r 2

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Fgravity field = m• -GM

r 2

⎛

⎝ ⎜

⎞

⎠ ⎟ = mg

M

g

Alternatively, we can think about an Object with mass M creating a gravitational field. This field would then act on any other object nearby, such as one with mass m.

What does g depend on? What units does it have?

In which direction does it point?

– Q

+ Q

E

E

Alternatively, we can think about a charge Q creating an electric field. This field would then act on any other charges nearby, such as one with charge q.

Electric Fields and Forces Electric Fields and ForcesFor electricity, we can use the direct force model similarly to gravity. Consider a charge Q exerting a force on a new charge q:

What does E depend on?

€

Felectric field = q •kQ

r 2

⎛

⎝ ⎜

⎞

⎠ ⎟ = qE

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Fdirect = kQq

r 2

What units?

In which direction does it point?

– Q

+ Q

E

E

For an electric field E :

Magnitude: Direction: out from +Q

in toward -Q

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Edue to Q = kQ

r2

Electric Field/Force Directions

Electric Field/Force Directions

For the coulomb force on a test charge q in a field E :

Magnitude:

Direction: along the E field vector for +q opposite the E field vector for -q

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Ffield on q = qE

qQr

Electric Field LinesElectric Field Lines

This is an Electric Dipole!

Like charges (++) Opposite charges (+ -)

Electric Field StrengthsElectric Field StrengthsTypical electric field strengths:

•1 cm away from 1 nC of negative charge E = kq /r2 = 1010 *10-9/ 10-4 =105 N /C

Note: (N*m2/C2) C / m2 = N/C

• Fair weather atmospheric electricity

= 100 N/C downward

at 100 km high in the ionosphere

• Field due to a proton at the location of the electron in the

H atom. (The radius of the electron orbit is 0.5*10-10 m)

E = kq /r2 = 1010 *1.6*10-19/ (0.5 *10-10 )2 = 4*1011 N /C

qr

E

- - - - - - - - -

E+++++

+Hydrogen atom 1 N / C = Volt / meter

Example: Calculating E Fields

Example: Calculating E Fields

Finding an electric field from two charges:

We have q1

= +10 nC at the origin, q2

= +15 nC at x=4 m.

What is E at y=3 m and x=0? (point P)

x

y

q1=10 nC q2 =15 nC4

3

P

Use principle of superposition. (Find x and y components of electric field due to both charges and add them up.)

Example: Calculating E Fields

Example: Calculating E Fields

Recall: E =kq/r2

xq1=10 nC q2 =15 nC

4

5

y

3

E

Field due to q1: E = 1010 N.m2/C2 10 X10-9 C/(3m)2

= 11 N/C in the y direction. Ey= 11 N/C Ex= 0

Field due to q2: 1010 N.m2/C2 15 x10-9 C/(5m)2 = 6 N/C

at some angle Resolve into x and y components.

Ey= E sin C Ex= E cos C

Now add all components: Ey= 11 + 3.6 = 14.6 N/C Ex= -4.8 N/C

Magnitude:

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E = Ex2 + Ey

2

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E = 14.6( )2

+ −4.8( )2

=15.4N /C

atan Ey/Ex= atan (14.6/-4.8)= 72.8 deg

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rE = −4.8 i

∧

+14.6 j∧

Charge InductionCharge InductionInducing Charge on a Net Neutral Object:

How can a neutral object create an Electric field? (Where would the charges come from to produce such a field?)

Static Electricity: Charge can be transferred from one object to another by rubbing. Static is the imbalance of positive and negative charges.

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

Gradient Relations: PotentialGradient Relations: PotentialRecall: What is the potential energy of a mass m in a the Earth’s gravitational field, a height h above the surface of the Earth?

PE = mgh !

• Force on a mass m in gravity field g is F = mg.• Magnitude of force is the spatial derivative, or gradient, of the potential energy of the mass:

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Fon m = -d

drPE gravity( )

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PE gravity = - GMm

r

The direction of the force on the mass m is toward decreasing PEgrav (hence the negative sign!)

Gradients for E Fields: Potential

Gradients for E Fields: Potential

• Force on a charge q in an Electric field E is F = qE.• Magnitude of force is the spatial derivative, or gradient, of the potential energy of the mass:

€

Fon q = -d

drPE electric( )

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PE electric = - kQq

r

The direction of the force on the charge +/- q is toward decreasing PEgrav (hence the negative sign again!)