PHYS 012A Lecture Notes - Lecture 8: Electric Flux, Dot Product, Surface Charge

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12 Jun 2018

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Chapter 24: Gauss’s Law 9/25/17

Gauss’s La

- Can be used as an alternative procedure for calculating electric fields

- Based on the inverse square behavior of the electric force between point charges.

- Convenient for calculating the electric field of highly symmetric charge distributions

Electric Flux (units = N*m2/C)

- The amount of electric field that penetrates a surface is the electric flux

- If uniform electric field is directed straight through a surface then:

- The electric flux is proportional to the number of electric field lines penetrating the surface.

- So, if the surface is not perpendicular to the field, it is the component of the area perpendicular

to the field that contributes to the flux.

Uniform Electric Field

- Making angle theta with normal at a surface.

- Flux is a maximum when the surface is perpendicular to the field and theta = 0 degrees.

- Flux is zero when the surface is parallel to the field and theta = 90 degrees.

Dot Product

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Open and Closed Surfaces

- A rectangle is an open surface – does’t otai a olue

o Surface normal can be up or down

- A sphere is a closed surface – it does contain a volume

o Surface normal is always outward

Closed Surface

- The vectors point in different directions at different points on the surface

o At each point, they are perpendicular to the surface, and they point outward

Flux through Closed Surface

- The net flux through a closed surface is proportional to the net number of electric field lines

leaving the surface.

o This net number of lines is the number of lines leaving the surface minus the number

entering the surface

Gauss’s La – General

- A positive point charge, q, is located at the center of a sphere of radius r.

- The magnitude of the electric field everywhere on the surface of the sphere is E = keq/r2.

- To find electric flux through the whole sphere:

- The field lines are directed radially outward and are perpendicular to the surface at every point.

- Closed surfaces of various shapes can surround the charge. The net flux through any closed

surface surrounding a point charge q is given by q/epsilono and is independent of the shape of the

surface. The net electric flux through a closed surface that surrounds no charge is zero.\

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Document Summary

Can be used as an alternative procedure for calculating electric fields. Based on the inverse square behavior of the electric force between point charges. Convenient for calculating the electric field of highly symmetric charge distributions. The amount of electric field that penetrates a surface is the electric flux. If uniform electric field is directed straight through a surface then: The electric flux is proportional to the number of electric field lines penetrating the surface. So, if the surface is not perpendicular to the field, it is the component of the area perpendicular to the field that contributes to the flux. Making angle theta with normal at a surface. Flux is a maximum when the surface is perpendicular to the field and theta = 0 degrees. Flux is zero when the surface is parallel to the field and theta = 90 degrees. A rectangle is an open surface does(cid:374)"t (cid:272)o(cid:374)tai(cid:374) a (cid:448)olu(cid:373)e: surface normal can be up or down.

Related Questions

Learning Goal: To work through astraightforward application of Faraday's law to find the EMF andthe electric field surrounding a region of increasing flux

Faraday's law describes how electric fields and electromotiveforces are generated from changing magnetic fields. This problem isa prototypical example in which an increasing magnetic fluxgenerates a finite line integral of the electric field around aclosed loop that surrounds the changing magnetic flux through asurface bounded by that loop. A cylindrical iron rod withcross-sectional area is oriented with its symmetry axis coincidentwith the z axis of a cylindrical coordinate system as shown. It hasa uniform magnetic field inside that varies according to . In otherwords, the magentic field is always in the positive z direction,and it has no other components.

For your convenience, we restate Faraday's law here: , where is theline integral of the electric field, and the magnetic flux is givenby , where is the angle between the magnetic field and the localnormal to the surface bounded by the closed loop.
Direction: The line integral and surface integral reverse theirsigns if the reference direction of or is reversed. The right-handrule applies here: If the thumb of your right hand is taken along ,then the fingers point along . You are free to take the loopanywhere you choose, although usually it makes sense to choose itto lie along the path of the circuit you are considering.

Find , the electromotive force (EMF) around a loop that is atdistance from the z axis, where is restricted to the region outsidethe iron rod as shown. Take the direction shown in the figure aspositive.
Express in terms of , , , , and any needed constants such as , ,and .
Due to the cylindrical symmetry of this problem, the inducedelectric field can depend only on the distance from the z axis,where is restricted to the region outside the iron rod. Find thisfield.
Express in terms of quantities given in the introduction (andconstants), using the unit vectors in the cylindrical coordinatesystem,r^ ,theta^ , and z^ .

1a) Faraday discovered that a current is induced in a coil

A) only when the N pole of a magnet is inside the coil
B) only when the S pole of a magnet is inside the coil
C) only when there is a magnetic field inside the coil
D) only when the magnetic field inside the coil is changing

1b). A conductor moves through a uniform magnetic field at speed v and has a motional emf V created across it. If the conductor were to move twice as fast through the magnetic field, the motional emf would be:

A) V
B) 2V
C) 4V
D) V/2
E) V/4

1c) In the conducting rail discussed in section 25.2 (where the rail is moving at constant speed), what can you say about the power input due to the person pulling on the rail and the power dissipated in the resistance?

A) Power input from pull = power dissipated in R
B) Power input from pull > power dissipated in R
C) Power input from pull < power dissipated in R

1d) The electric generator as shown in the reading generates electricity by:

A) rotating a loop inside a magnetic field
B) moving a magnet in and out of a loop
C) sliding a loop over a magnet
D) running current through a loop in a magnetic field

1e) The magnetic flux through a loop:

A) is proportional to the number of magnetic field lines going through it
B) depends just on the B field going through the loop
C) is maximized when the B field lines in the plane of the loop
D) is zero when the B field is perpendicular to the plane of the loop
E) doesn't depend on the direction of the B field

1f) According to Lenz's Law, the induced B field (due to the induced current) will:

A) be in the opposite direction of the existing B field
B) be in the same direction as the existing B field
C) be in the same direction as the change in the existing B field
D) be in the opposite direction of the change in the existing B field

1g) A stationary coil experiences a doubling of its magnetic field (no change in direction of the field) in a time T and has a given induced EMF. If the same change were to have happened in half the time (T/2), the induced EMF would have been:

A) half as large
B) the same
C) twice as large
D) one-fourth as large
E) four times as large

1h) Eddy currents:

A) make it hard to pull a nonmagnetic material through a magnetic field
B) make it hard to pull a magnetic material through a magnetic field
C) speed up a magnetic material moving through a magnetic field
D) speed up a nonmagnetic material moving through a magnetic field

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PHYS 012A Lecture Notes - Lecture 8: Electric Flux, Dot Product, Surface Charge

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