ch 23 notes - fcps...the electric force caused by any collection of charges at rest is a...

Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley

PowerPoint® Lectures forUniversity Physics, Twelfth Edition

– Hugh D. Young and Roger A. Freedman

Lectures by James Pazun

Chapter 23

Electric Potential


Goals for Chapter 23

• To study and calculate electrical potential energy

• To define and study examples of electric potential

• To trace regions of equal potential as equipotential surfaces

• To find the electric field from electrical potential


How to achieve the goals:This PowerPoint

Reading the Chapter

Attending the Review Sessions

Homework:

1, 7, 11, 13, 27, 31, 33, 37, 39, 41, 43 and 45


Introduction• Electrical potential is

sometimes modeled as a river. The width of the river defines how much water will be able to flow through its banks.

• The arc welder in the picture at right is taking advantage of a potential between the welding rod and material to be joined. The arc of electrical flow is so hot that the metals and the rod actually melt into one material.


Using energy concerns can simplify problemsWhen a charged particle moves in an electric field, the field exerts a force that can do work on the particle.

This work can always be expressed as the electric potential energy. Just like how gravitational potential relied on position, so will electric potential energy.

We will describe the energy as electric potential or just potential.

The difference in potential is called potential difference, or voltage.


Lets start with a “review” of Mechanics conceptsWhen a force acts on a particle that moves from point a to point b, the work Wa b done by the force is given by a line integral:

where dl is an infinitesimal displacement along the particle’s path and φ is the angle between the force and dl at each point on the path.If the force is conservative, as defined in section 7.3, the work done by F can always be expressed in terms of a potential energy U. When the particle moves from a point where the potential energy is Ua to where it is Ub the change in potential energy is ΔU = Ub – Ua and the work done by the force is expressed as:

cosb b

a ba a

W F d F dφ→ = =∫ ∫i

( )a b a b b aW U U U U U→ = − = − − = −Δ


Work and Energy

When the work is positive, Ua is greater than Ub and the change in potential is negative.This is what happens when Ty falls while running. He goes from a high point a, his running form, to a low point b, on the ground. The force of gravity does positive work and gravitational potential decreases.The work-energy theorem says that the change in kinetic energy is equal to the total work done, or W = Kb – KaSubstituting our previous expression for work yields:

This means that the total mechanical energy is conserved.

( )a b a b b aW U U U U U→ = − = − − = −Δ

a a b bK U K U+ = +


Electric Potential energy in a Uniform FieldCheck out figure 23.1 on page 870. A set of parallel plates sets up a uniform electric field with magnitude E. The field exerts a downward force on a test charge with magnitude F = qoE.The work done on the charge by this force as it moves a distance of d is independent of the path the charge takes: Wa b= Fd = qoEdThis is positive because the force and displacement are in the same direction.This is analogous to the gravitational force on a mass near the earth’s surface.


Work, energy, and the path from start to finishLike the gravitational force, the force on qo has only a y-component. We can also conclude that the force exerted on qo is conservative, just like the gravitational force.This means the work required to move the particle from a to b is independent of the path. We can express the work as potential energy function: U = qoEy where y is the displacement. The work can now be expressed as:

Check out 23.2 and 23.3 to see the consequences of this conservative force.

( )( ) ( )

a b b a

a b o b o a o a b

W U U U

W q Ey q Ey q E y y→

→

= −Δ = − −

= − − = −


Figure 23.2


Figure 23.3


General Potential Energy TrendsU increases if the test charge qo moves in the direction opposite to the electric force

U decreases if the test charge qo moves in the same direction as the electric force

The relationship between electric potential energy change and motion in an electric field is an important one that we’ll use often. It’s also a relationship that takes a little effort to truly understand. Take the time to review page 871 carefully. Doing so now will help you tremendously later!

EqF o=

EqF o=


Electric Potential Energy of Two Point ChargesThe force on qo is:

Integrate to find the work as qo moves from ra to rb:

Page 872 shows that the work is independent of the path. This independence and the fact that work with no displacement equals zero defines a conservative force.

2

14

or

o

qqFrπε

=

( )2

b

o o o

a b

a

rkqq kqq kqq

a b r rrr

W dr→ = = −∫


Electric Potential EnergyThe previous work equation is valid for a potential energy function U:

This is valid for any combination of charges.U must be zero when r = ∞. The potential energy U represents the work that must be done on qo if it is moved from initial distance r to infinity. See Fig 23.6 and consider Example 23.1 on 874.

14

o

o

qqUrπε

=


Electrical potential and multiple point charges

Be sure to read page 876. Some highlights:U = 0 is arbitrary, the potential at qo below is the algebraic sum of the potentials of q1, q2 and q3.Try example 23.2.


23.1 Summary and HomeworkThe electric force caused by any collection of charges at rest is a conservative force. The work W done by the electric force on a charged particle moving in an electric field can be represented by the change in a potential-energy function U.The electric potential energy for two point charges q and qo depends on their separation r. The electric potential energy for a charge qo in the presence of a collection of charges q1, q2and q3 depends on the distance from qo to each of these other charges. Read 878 to 890 and do #’s 1, 7 and 11on page 899


Electric PotentialWhen we need to determine an electric field, it is often easier to determine the potential first and then find the field from it.

Potential is potential energy per unit charge, or U per unit charge:

The unit for potential is the volt: 1V = 1J/C

The potential difference (voltage) is the work done by the electric force during a displacement from a to b divided by the magnitude of the moving charge:

( )a b b ab a a b

o o o o

W U UU V V V Vq q q q→ ⎛ ⎞Δ

= − = − − = − − = −⎜ ⎟⎝ ⎠

o

UVq

=


The electrical potential• Vab can be

measured between point a and point b(the positive and negative terminals).

• Moving with the electrical field decreases the electrical potential. Moving against the field increases it.


Voltage—Potential DifferenceVab, the potential of a with respect to b, equals the work done by the electric force when a UNIT charge moves from a to b.Vab, the potential of a with respect to b, equals the work that must be done to move a UNIT charge slowly from bto a against the electric force.The potential due to a single point charge:

A collection of point charges:

For a continuous distribution of charge:

14o o

U qVq rπε

= =

14

i

io o i

qUVq rπε

= = ∑

14 o

dqVrπε

= ∫


CAUTIONBefore getting too involved in the details of how to calculate electric potential, you should stop and remind yourself what potential is. The electric potential at a certain point is the potential energy that would be associated with a unit charge placed at that point. That’s why potential is measured in joules per coulomb, or volts. Keep in mind, too, that there doesn’t have to be a charge at a given point for a potential V to exist at that point. (In the same way, an electric field can exist at a given point even if there’s no charge there to respond to it.)


Potential Difference as an Integral of E In some cases it is easier to calculate the potential difference using the electric field.

The force on a test charge can be written as:

The work done by the electric force as the test charge moves from a to b is given by:

If we divide this by qo and compare the result with a previous equation:

b b

a b oa a

W F dl q E dl→ = =∫ ∫i i

oF q E=

cosb b

a ba a

V V E dl E dlφ− = =∫ ∫i


Other useful informations . . .The potential difference is independent of the path taken, like the work.

Moving with the field is moving in the direction of decreasing V, moving against the field is moving in the direction of increasing V.

One volt per meter is equal to one newton per coulomb: 1V/m = 1N/C

One electron volt is equal to 1.602 x 10-19 Joules: 1eV = 1.602 x 10-19 J


A particle accelerator imparts amazingly large energiesA particle accelerator can bring a charged particle to motion at velocities great enough to impart millions, even billions, of eV as kinetic energy.

Figure 23.12 at right shows a particle accelerator at the Fermi Lab in Illinois.

Refer to Example 23.3.


Finding the potential


22.2 Summary and HomeworkPotential, denoted by V, is potential energy per unit charge. The potential difference between two points equals the amount of work that would be required to move a unit positive test charge between those points. The potential V due to a quantity of charge can be calculated by summing (if the charge is a collection of point charges) or by integrating (if the charge is a distribution). (See Examples 23.3, 23.4, 23.5, 23.7, 23.11, and 23.12.)The potential difference between two points a and b, also called the potential of a with respect to b, is given by the line integral of E. The potential at a given point can be found by first finding E and then carrying out this integral. Read 890 to 897 and do 13, 27 and 31


Calculation of electrical potential• Consider Problem-Solving

Strategy 23.1.

• Refer to Example 23.8 with Figure 23.16.


Example—oppositely charged parallel plates• Read about Ionization and Corona discharge on own.

• Refer to Example 23.9 using Figure 23.18.


Example—a charged, conducting cylinder• Refer to Example 23.10 using Figure 23.19.


Example—a ring of charge• Refer to Example 23.11 using Figure 23.20.


Example—a line of charge• Refer to Example 23.12 using

Figure 23.21.


23.3 HomeworkOn page 901: 33, 37, 39

Be sure to read 23.4 and 23.5, focus on page 892 and the example problems.


Equipotential Surfaces are like Topographical Maps


Equipotential surfaces and field linesAn equipotential surface is a 3-D surface on which the electric potential V is the same at every point, like elevation on a topographical map. If a test charge qo is moved from point to point on the surface, the electric potential energy qoV remains constant. Because the potential energy does not change the work done by the electric field is zero (work is the change in potential). If the work is zero then the electric field must be perpendicular to the surface at all points (the force on the charge by the field is perpendicular to the displacement of the charge).Field lines and equipotential surfaces are always mutually perpendicular.See Figure 23.23 for evidence of this result.There are equal potential differences between the lines representing the equipotential surfaces.


Refer to Figure 23.23On a given equipotential surface, the potential V has the same value at every point. In general, however, the electric-field magnitude Eis not the same at all points on an equipotentialsurface. For instance, on the equipotentialsurface labeled “V = -30 Volts” in 23b, the magnitude E is less to the left of the negative charge than it is between the two charges. On the figure-8-shaped equipotential surface in 23.c, E = 0 at the middle point halfway between the two charges; at any other point on this surface, E is nonzero.


Field lines and a conducting surfaceWhen all charges are at rest, the surface of a conductor is always an equipotentialsurface.When all charges are at rest, the electric field just outside a conductor must be perpendicular to the surface at every point.Read page 892 about charges on the inside of conductors.


23.4 SummaryAn equipotential surface is a surface on which the potential has the same value at every point. At a point where a field line crosses an equipotentialsurface, the two are perpendicular. When all charges are at rest, the surface of a conductor is always an equipotential surface and all points in the interior of a conductor are at the same potential. When a cavity within a conductor contains no charge, the entire cavity is an equipotential region and there is no surface charge anywhere on the surface of the cavity.


Potential GradientElectric fields and potential are closely related:

If we know E, we can calculate the potential difference. We can switch this around to get E if we know the potential difference. If we consider V to be a function of the coordinates (x,y,z) of a point in space, we will show that the components of V are related to the partial derivatives of V with respect to x, y and z. Lets start with a definition of V:Relating this to the previous equation yeilds:

For the integrals to be equal, the integrands must be equal too:

Rewritten in terms of the components of E and dl:

∫ ⋅=−b

aba ldEVV

∫∫ ⋅=−b

a

b

a

ldEdV

∫∫ −==−b

a

a

bba dVdVVV

dzEdyEdxEdV zyx ++=−

ldEdV ⋅=−


Partial DerivativesIf the only displacement is in the x-direction (y and z are constants) then –dV = Exdx or Ex = -(dV/dx). The y- and z-components of E are related to V in the same way:

In vector notation the following operation is called the gradient of the function f:So In the radial direction:

xVEx ∂∂

=yVEy ∂∂

=zVEz ∂∂

= ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

−= kzVj

yVi

xVE ˆˆˆ

fkz

jy

ix

f ⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

=∇ ˆˆˆ

rrVEr ˆ∂∂

=

VE ∇−=


Potential and fields of charge

• Follow Example 23.13 to calculate the potential and field of a point charge.

• Follow Example 23.14 to calculate the potential and field of a ring of charge.


23.5 Summary and HomeworkIf the potential V is known as a function of the coordinates x, y, and z, the components of electric field E at any point are given by partial derivatives of V.

On page 901: 41, 43 and 45

On page 902: to be assigned later.

ch 23 notes - fcps...the electric force caused by any collection of charges at rest is a...

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