ch17-electgricpotential

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Chapter 17 Electric Potential

17.1 Electric Potential Energy

1. Newton’s law for the gravitational force and Coulomb’s law

for the electrostatic force are mathematically identical. Thus

the general feature we have discussed for the gravitational

force should apply to the electrostatic force.

2. In particular, we can infer that the electrostatic force is a

conservative force. Thus when that force acts between two or

more charged particles within a system of particles, we can

assign an electric potential energy U to the system.

3. Moreover, If the system changes its configuration from an

initial state i to a different final state f, the electrostatic force

does work W on the particles. The resulting change U ∆ in the

potential energy of the system is W U U U i f −=−=∆ . As with

other conservative forces, the work done by the electrostatic

force is path independent.

4. For convenience, we usually take the reference configuration

of a system of charged particles to be that in which the

particles are all infinitely separated from each other. And we

usually set the corresponding reference potential energy to be

zero.

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17.2 Electric Potential

1. The potential energy of a charged particle in an electric field

depends on the magnitude of the charge. However, the

potential energy per unit charge has a unique value at any

point in the electric field. Thus the potential energy per unit

charge, which can be symbolized as U/q, is independent of the

charge q of the particle and is characteristic only as the electric

field we are investigating. The potential energy per unit charge

at a point in an electric field is called the electric potential V

(or simply the potential) at that point. Thusq

U V = . Electric

potential is a scalar, not a vector.

2. The electric potential difference V ∆ between any two points i

and f in an electric field is equal to the difference in potential

energy per unit charge between the two point:

q

W

q

U V V V i f

−=

∆=−=∆ . The potential difference between two

point is thus the negative of the work done by the electrostatic

force per unit charge that move from one point to the other .

3. The SI unit for electric potential is the joule per coulomb. This

combination occurs so often that a special unit, the volt

(abbreviated V) is used to represent it.

4. One electron-volt (eV) is the energy equal to the work

required to move a single elementary charge e through a

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potential difference of exactly one volt, so J eV 19

1060.11−×= .

17.3 Equi-potential Surfaces

1. Adjacent points that have the same electric potential form an

equipotential surface, which can be either an imaginary

surface or a real, physical surface. No net work W is done on a

charged particle by an

electric field when the

particle moves between

two points i and f on the

same equipotential surface. See the Figure.

2. Figure shows the electric field lines and cross sections of

equipotential surface for several cases. We can find that

equipotential surfaces are always perpendicular to electric

field lines and thus to E which is always tangent to these

lines.

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17.4 Calculating the Potential from the Field

1. We can calculate the potential difference between any two

points i and f in an electric field if we know the field vector

E at all positions along any path connecting those points.

2. Consider an arbitrary

electric field,

represented by the field

lines in the right figure,

and a positive test charge 0q that moves along the path shown

from point i to point f. The differential work done on the

particle by the electrostatic force during a displacement is

sd E q sd F dW ⋅=⋅= 0 . Thus the total work done on the particle by

the fields is the integration of the differential work done on

the charge for all the differential displacement along the path.

∫ ⋅=f

i sd E qW

0 . Therefore, ∫ ⋅−=−=−f

ii f sd E

q

W V V

0

.

3. If we choose the potential iV at point i to be zero, then

∫ ⋅−=

f

i sd E V

, in which we have drooped the subscript f. It

gives us the potential V at any point f in the electric field

relative to the zero potential at point i.

4. Potential due to a point charge:r

qV

04

1

πε = .

5. Potential due to a group of point charges: We can find the net

potential at a point due to a group of point charges to sum up

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the potential resulting from each charge at the given point.

∑∑==

==n

i i

in

i

ir

qV V

101 4

1

π ε . Here iq is the value of the ith charge, and

ir is the radial distance of the given point from the ith charge.

6. Potential due to an electric dipole: (1) 2

0

cos

4

1

r

pV

θ

π ε = . (2)

Induced dipole moment: Many molecules such as water

(Seeing right figure) have

permanent electric dipole moments.

In other molecules (nonpolar

molecules) and in every atom, the

centers of the positive and negative

charges coincide and thus no dipole moment is set up.

However, if we place an atom or nonpolar molecule in an

external electric field, the field distorts the electron orbits and

separates the centers of positive and negative charge and sets

up a dipole moment that points in the direction of the field.

This dipole moment p said to be induced by the field, and the

atom or molecule is said to be polarized by the field. When

the field is moved, the

induced dipole moment and

the polarization disappear.

See the figure.

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7. Potential due to a continuous charge distribution:

∫ ∫ ==r

dqdV V

04

1

πε . Here the integral is to be taken over the

entire charge distribution. (1) lines of charge; (2)Charged

disk.

17.5 Calculating Electric Field from Electric Potential

1. The component of E in any direction is the negative of the

rate of change of the electric potential with distance in that

direction. z

V E

y

V E

x

V E

z y x∂

∂−=

∂

∂−=

∂

∂−= ;; .

17.6 Electric Potential Energy of a System of Point Charges

1. The electric potential energy of a system of fixed point-

charges is equal to the work that must be done by external

agent to assemble the system, bringing each charge in from an

infinite distance.

17. 7 Potential of a Charged Isolated Conductor

1. An excess charge placed on an

isolated conductor will distribute

itself on the surface of that

conductor so that all points of the

conductor-whether on the surface

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or inside-come to the same potential. This is true regardless of

whether the conductor has an internal cavity.

2. If an isolated conductor is placed in an external electric field,

as in the right figure, all points of the conductor still come to

a single potential regardless of whether the conductor has an

excess charge.

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