21. gauss’s law “the prince of mathematics” carl friedrich gauss (1777 – 1855) wikemedia...

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21. Gauss’s Law The Prince of MathematicsCarl Friedrich Gauss (1777 – 1855) Wikemedia Commons

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Page 1: 21. Gauss’s Law “The Prince of Mathematics” Carl Friedrich Gauss (1777 – 1855) Wikemedia Commons

21. Gauss’s Law

“The Prince of Mathematics”Carl Friedrich Gauss(1777 – 1855)

Wikemedia Commons

Page 2: 21. Gauss’s Law “The Prince of Mathematics” Carl Friedrich Gauss (1777 – 1855) Wikemedia Commons

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Topics

Electric Field Lines Electric Flux Gauss’s Law Using Gauss’s Law Gauss’s Law and Conductors

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Electric Field Lines

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Electric Field Lines

A field lineshows the direction ofthe electricforce on apositivepoint charge

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Electric Field Lines

By using a convention for thenumber of linesper unit charge, one can use field lines to indicate the strength of an electric field.

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Electric Flux

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Electric Flux

Flux is the “flow” of any quantity through a surface.

For example, it could be sunlight through a window, or water through a hole.

In particular, it can be electric field through a surface

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The electric flux through a small surfaceelement A is defined by

E A

where

ˆA An

The unit vector is normal to the surface element

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Surface

i ii

E A

E dA

This is an example of a surface integral

The total electric flux through a surface is the sum of the individual fluxes

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Electric Flux – A Closed Surface

Let’s compute the flux through a spherical surface about a point charge:

+

2

2 22

ˆ

4

4

ˆ ndAq

k rr

q qk

E dA

kr rk

r

q

dA

n̂E

We see that the flux is proportionalto the enclosed charge

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Gauss’s Law

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Gauss’s Law

The electric flux through any closed surface is proportional to the net charge enclosed by the surface:

which is usually written as

4E dA kq

+ -

- -+

+ +

+

enclosed

0

qE dA

+ where 0 = (4k)-1 = 8.85 x 10-12 C2/Nm2

is called the permittivity constant

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Gauss’s Law

Gauss’s law is always true for any closed surface. However, it is most useful when the

charge distribution and the enclosing surface have a high degree of

symmetry.+ -

- -+

+ +

+

enclosed

0

qE dA

+

a

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Using Gauss’s Law

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A Uniformly Charged Sphere

The enclosed charge is Q. Gauss’s law is

0

QE dA

Because of the spherical symmetry of the chargedistribution, we can infer that the magnitude of the electric field is constant on any spherical surface enclosing the charge

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A Uniformly Charged Sphere

The symmetry makes is easy to evaluate thesurface integral

2

0

2 20

4

4

QE dA E dA E R

Q QE k

R R

The electric field of a spherically symmetric charge distribution is like thatof a point charge

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A Uniformly Charged Sphere

3r

q QR

We can apply Gauss’s law within the sphereby drawing a Gaussian surface of radius r.

The charge enclosed within this surface is

32

0 0

3 30

4

4

q Q rE dA E r

R

Q QE r k r

R R

therefore,

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A Uniformly Charged Sphere

Within the sphere the field varieslinearly with radius

Outside, the fieldlooks like that ofa point charge

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A Hollow Spherical Shell

The shell contains a net charge Q distributed uniformlyover its surface. Because of the sphericalsymmetry, the field outside the shell is like that of a point charge.

But the field inside is zero!Why? Because the field from Acancels that from B.

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Recap

By convention, the electric field points away from a positive charge and towards a negative charge.

Electric field lines can be used to visualize an electric field. By convention, the number of field lines is proportional to the charge.

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Recap

Electric field is additive: the field at any point is the vector sum of the electric fields of all charges.

Gauss’s law: the net electric flux through any closed surface is proportional to the net enclosed charge:

enclosed

0Closed Surface

qE dA

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An Infinite Line of Charge

By symmetry, the electric field is radial.

Therefore, a suitable Gaussian surface is a

cylinder of length L, radius r

placed symmetrically

about the line charge.

The enclosed charge

is q = L, where is the

charge per unit length

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An Infinite Line of Charge

From Gauss’s law, we deduce that the electric field of a long (strictly infinite) line charge is

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A Sheet of Charge

E+

E-

For an infinite sheet of charge the field is perpendicular to the sheet. The flux through a cylindrical Gaussian surface is

EA + EA.

The enclosed charge is

q = A,

where is the charge per unit area. Therefore, the field is

02E

A

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A Charged Disk

The electric field at a point P along the axis of a disk is closely related to the field of a sheet

of charge:

2 2

2 2

3/ 22 2

2 2

cosdq

kx r

dq

dE

x

x rk

x r

xdqk

x r

P

dq

r

x E

2 2k 0dis

12

xE dE

x R

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Gauss’s Law & Conductors

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Conductors

An applied electric field causes

the free positive and negative

charges to separate until the field

they create exactly cancels the

applied field, at which point the

charge migration stops. The

conductor is then in electrostatic

equilibrium.

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Charged Conductors

Since like charges repel, all

excess charge must reside on the

surface of a conductor. This

is also consistent with the fact

that, in equilibrium, the

electric field within a conductor

is zero.

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A Hollow Conductor

A conductor carries a net

charge of 1 C and has

a 2 C charge in the

internal cavity.

The charges must distribute

themselves as shown in order to be

consistent with Gauss’s law.

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A Hollow Spherical Conductor

Consider a neutral spherical

conductor in equilibrium with a

cavity containing a net charge +q.

1. The charge on the inner surface of the cavity is –q. Why?

2. The charge on the outer surface of the conductor must therefore be +q. Why?

3. And this charge is uniformly distributed. Why?

+qE = 0

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Field at a Conductor Surface

E+

E- = 0 (inside conductor)

The flux through a cylindrical Gaussian surface is just EA since the field inside the conductor is zero, in equilibrium. The enclosed charge

is q = A, therefore, the field at the surface of a charged conductor is

0

E

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Applications

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Electric Shielding

The tendency of conductors to exclude electric fields from their bulk has many applications. For example:

1. Co-axial cables

2. Lightning Safety

3. Sensitive Compartmented Information Facility (SCIF)

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Co-axial Cables

Co-axial cables connect, for example, iPods to ear-phones.

If the electric

fields are too strong,

the dielectric

can suffer

dielectric breakdownWikemedia Commons

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Co-axial Cables and Dielectrics

Some molecules, like H2O, have permanent dipole moments. Others can be distorted by an electric field, and become dipolar; that is, acquire induced dipole moments. These materials are called dielectrics

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Lightning Safety

http://www.lightningsafety.noaa.gov/lightning_map.htm

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SCIFs

Wright-Patterson Air Force Base in Dayton, Ohio, is one of the major command posts of the U.S. Air Force (USAF).

It contains a giant Faraday cage that houses a

Sensitive

Compartmented

Information

Facility (SCIF)

+ + + ++ + + +

+

+

+ + + ++ + + + +- - - -- - - - -

- - - -- - - - -

-

+-

+-

+-

+-

+-

+-

+-

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SCIFs

Any externally generated electric field causes electrons in the Faraday cage to migrate in the direction opposite the field.

+ + + ++ + + +

+

+

+ + + ++ + + + +- - -- -- - --

- - - -

-

-- - -

-

+-

+-

+-

+

-

+

- +

-

+-

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SCIFs

The induced field exactly cancels the externally generated fields. Consequently, any electronic equipment inside is immune from an electromagnetic attack.

+ + + ++ + + +

+

+

+ + + ++ + + + +- - -- -- - --

- - - -

-

-- - -

-

+-

+-

+-

+

-

+

- +

-

+-

0E

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Summary

Electric Flux

Gauss’s Law

The electric flux through a closed surface is determined by the enclosed charge

E dA

enclosed 0/E dA q

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Summary

Conductors

The electric field within a conductor, in electrostatic equilibrium, is zero because the charge rushes to, and distributes itself on, the surface of the conductor