1/17/07

PHY 184

Spring 2007 Lecture 6

Title: Gauss’ Law

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Announcements

    Homework Set 2 is due Tuesday at 8:00 am.

We will have clicker quizzes for extra credit starting next week (the clicker registration closes January 19).

Homework Set 3 will open Thursday morning.

Honors option work in the SLC will start next week • Honors students sign up after class for time slots.

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Outline

   /1/ Review /2/ Electric Flux /3/ Gauss’ Law 1/17/07

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Review - Point Charge

► The electric field created by a point charge, as a function of position r,

E

 (

r

 )  1 4  0

q r

2

r

ˆ ► x The force exerted by an electric field on a point charge q located at position

F

 

q E

 (

x

) … direction tangent to the field line through x 1/17/07

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Review – Electric Dipole

 2 equal but opposite charges, -q and +q  Dipole moment (direction: - to +)  

q d

 On the axis, far from the dipole,

E

 2

p

 0

r

3 1/17/07

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Force and Torque on an Electric Dipole

 Assume the 2 charges (-q and +q) are connected together with a constant distance d, and put the dipole in a uniform electric field E.

Net force = 0 Torque about the center:

τ τ

torque 

qE



pE



d

F 1  moment sin 2 sin   

qE d

2 arm sin   F 2   moment arm

qdE

sin  Torque vector    

E

 1/17/07

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Gauss’ Law

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Objective

    So far, we have considered point charges. But how can we treat more complicated distributions, e.g., the field of a charged wire, a charged sphere or a charged ring?

Two methods Method #1: Divide the distribution into infinitesimal elements dE and

integrate

to get the full electric field.

Method #2: If there is some special symmetry of the distribution, use Gauss’ Law to derive the field.

1/17/07 Gauss’ Law The flux of electric field through a closed surface is proportional to the charge enclosed by the surface.

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1/17/07 Gauss’ Law The flux of electric field through a closed surface is proportional to the charge enclosed by the surface.

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Electric Flux

 Let’s imagine that we put a ring with area with velocity

v A

perpendicular to a stream of water flowing

v A

  The product of area times velocity, per unit time • The units are m 3 /s

Av

, gives the volume of water passing through the ring If we tilt the ring at an angle projected area is

A

cos  flowing through the ring is

Av

 , then the , and the volume of water per unit time cos 

.

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 10

Electric Flux (2)

 We call the amount of water flowing through the ring the “

flux

of water”

Flux

 

Av

cos

  We can make an analogy with electric field lines from a constant electric field and flowing water

E



A

 We call the density of electric field lines through an area

A

the electric flux given by

Elecric Flux

 

EA

cos

 1/17/07

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

Math Primer Surfaces and Normal Vectors

For a given surface, we define the normal unit vector

n,

which points normal to the surface and has length 1. Electric Flux

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  

Math Primer - Gaussian Surface

A closed surface, enclosing all or part of a charge distribution, is called a

Gaussian Surface.

Example: Consider the flux through the surface on the right. Divide surface into small squares of area  A.

Flux through surface: 1/17/07

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Electric Flux (3)

  constant everywhere

E

 (

r

 ) We define the electric flux through a closed surface in terms of an integral over the closed surface   

S E

 

d



A

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Gauss’ Law

 Gauss’ Law states (named for German mathematician and scientist Johann Carl Friedrich Gauss, 1777 - 1855)  0  

q

  • (q = net charge enclosed by S).

If we add the definition of the electric flux we get   0 

S E

 

d A



q

Gauss’ Law : the electric field flux through S is proportional to the net charge enclosed by S.

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Theorem: Gauss’ Law and Coulomb’s Law are equivalent.

   Let’s derive Coulomb’s Law from Gauss’ Law.

We start with a point charge

q.

We construct a spherical surface with radius

r

surrounding this charge.

• This is our “Gaussian surface” r q 

= E A

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Theorem (2)

 The electric field from a point charge is radial, and thus is perpendicular to the Gaussian surface everywhere.

/1/ The electric field direction is parallel to the normal vector for any point.

/2/ The magnitude of the electric field is the same at every point on the Gaussian surface.

 

E

 

d A

/1/ /2/  

E dA



E



dA



EA

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

= E A

17

Theorem (3)

1/17/07 Now apply Gauss’ Law

ε

0

Φ



q where Φ



EA Area A



4π r

2

E

 1 4  0

q r

2

Q. E. D.

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Shielding

   An interesting application of Gauss’ Law:

The electric field inside a charged conductor is zero.

Think about it physically… • • The conduction electrons will move in response to any electric field.

Thus the excess charge will move to the surface of the conductor.

• So for any Gaussian surface inside the conductor - encloses no charge! – the flux is 0. This implies that the electric field is zero inside the conductor.

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Shielding Illustration

    Start with a hollow conductor.

Add charge to the conductor.

The charge will move to the

outer

surface We can define a Gaussian surface that encloses zero charge • Flux is 0 • Ergo - No electric field!

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 

Cavities in Conductors

a) Isolated Copper block with net charge. The electric field inside is 0.

b) Charged copper block with cavity. Put a Gaussian surface around the cavity. The E field inside a conductor is 0. That means that there is no flux through the surface and consequently, the surface does not enclose a net charge. There is no net charge on the walls of the cavity 1/17/07

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Shielding Demonstration

   We will demonstrate shielding in two ways We will place Styrofoam peanuts in a container on a Van de Graaff generator • In a metal cup • In a plastic cup We will place a student in a wire cage and try to fry him with large sparks from a Van de Graaff generator • Note that the shielding effect does not require a solid conductor • A wire mesh will also work, as long as you don’t get too close to the open areas 1/17/07

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Lightning Strikes a Car

The crash-test dummy is safe, but the right front tire didn’t make it … High Voltage Laboratory, Technical University Berlin, Germany 1/17/07

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