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22. Electric Potential. Electric Potential Difference Calculating Potential Difference Potential Difference & the Electric Field Charged Conductors. This parasailer landed on a 138,000-volt power line. Why didn’t he get electrocuted?.

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22 electric potential

22. Electric Potential

Electric Potential Difference

Calculating Potential Difference

Potential Difference & the Electric Field

Charged Conductors


This parasailer landed on a 138,000-volt power line. Why didn’t he get electrocuted?

He touches only 1 line – there’s no potential differences & hence no energy transfer involved.

22 1 electric potential difference
22.1. Electric Potential Difference

Conservative force:

( path independent )

Electric potential difference electric potential energy difference per unit charge

[ V ] = J/C = Volt = V

if reference potential VA = 0.

For a uniform field:



table 22 1 force field potential energy electric potential
Table 22.1. Force & Field, Potential Energy & Electric Potential




Electric field

E = F / q

N/C or V/m

Potential energy difference


J/C or V

Electric potential difference

potential difference is path independent
Potential Difference is Path Independent

Potential difference VAB depends only on positions of A & B.

Calculating along any paths (1, 2, or 3) gives VAB = E r.

got it 22 1
GOT IT? 22.1
  • What would happen to VAB in the figure if
  • E were doubled;
  • r were doubled;
  • the points were moved so the path lay at right angles to E;
  • the positions of A & B are interchanged.



becomes 0

reverses sign

the volt the electronvolt
The Volt & the Electronvolt

[ V ] = J/C = Volt = V

E.g., for a 12V battery, 12J of work is done on every 1C charge that moves from its negative to its positive terminals.

Voltage = potential difference when no B(t) is present.

Electronvolt (eV) = energy gained by a particle carrying 1 elementary charge when it moves through a potential difference of 1 volt.

1 elementary charge = 1.61019 C = e

1 eV = 1.61019 J

table 22 2 typical potential differences
Table 22.2. Typical Potential Differences

Between human arm & leq due to 1 mV

heart’s electrical activity

Across biological cell membrane 80 mV

Between terminals of flashlight battery 1.5 V

Car battery 12 V

Electric outlet (depends on country) 100-240 V

Between lon-distance electric 365 kV

transmission line & ground

Between base of thunderstorm cloud & ground 100 MV

got it 22 2
GOT IT? 22.2
  • A proton ( charge e ),
  • an  particle ( charge 2e ), and
  • a singly ionized O atom
  • each moves through a 10-V potential difference.
  • What’s the work in eV done on each?

10 eV

20 eV

10 eV

example 22 1 x rays
Example 22.1. X Rays

In an X-ray tube, a uniform electric field of 300 kN/C extends over a distance of 10 cm, from an electron source to a target; the field points from the target towards the source.

Find the potential difference between source & target and the energy gained by an electron as it accelerates from source to target ( where its abrupt deceleration produces X-rays ).

Express the energy in both electronvolts & joules.

example 22 2 charged sheet
Example 22.2. Charged Sheet

An isolated, infinite charged sheet carries a uniform surface charge density .

Find an expression for the potential difference from the sheet to a point a perpendicular distance x from the sheet.


curved paths nonuniform fields
Curved Paths & Nonuniform Fields

Staight path, uniform field:

Curved path, nonuniform field:

got it 22 3
GOT IT? 22.3

The figure shows three straight paths AB of the same length, each in a different electric field.

The field at A is the same in each.

Rank the potential differences ΔVAB.

Smallest ΔVAB .

Largest ΔVAB .

22 2 calculating potential difference
22.2. Calculating Potential Difference

Potential of a Point Charge

For A,B on the same radial

For A,B not on the same radial, break the path into 2 parts,

1st along the radial & then along the arc.

Since, V = 0 along the arc, the above equation holds.

the zero of potential
The Zero of Potential

Only potential differences have physical significance.

Simplified notation:

R = point of zero potential

VA = potential at A.

Power systems / Circuits

Earth ( Ground )

Automobile electric systems

Car’s body

Isolated charges


got it 22 4
GOT IT? 22.4
  • You measure a potential difference of 50 V between two points a distance 10 cm apart in the field of a point charge.
  • If you move closer to the charge and measure the potential difference over another 10-cm interval, will it be
  • greater,
  • less, or
  • the same?
example 22 3 science museum
Example 22.3. Science Museum
  • The Hall of Electricity at the Boston Museum of Science contains a large Van de Graaff generator, a device that builds up charge on a metal sphere.
  • The sphere has radius R = 2.30 m and develops a charge Q = 640 C.
  • Considering this to be a single isolate sphere, find
  • the potential at its surface,
  • the work needed to bring a proton from infinity to the sphere’s surface,
  • the potential difference between the sphere’s surface & a point 2R from its center.




example 22 4 high voltage power line
Example 22.4. High Voltage Power Line

A long, straight power-line wire has radius 1.0 cm

& carries line charge density  = 2.6 C/m.

Assuming no other charges are present,

what’s the potential difference between the wire & the ground, 22 m below?

finding potential differences using superposition
Finding Potential Differences Using Superposition

Potential of a set of point charges:

Potential of a set of charge sources:

example 22 5 dipole potential
Example 22.5. Dipole Potential

An electric dipole consists of point charges q a distance 2a apart.

Find the potential at an arbitrary point P, and approximate for the casewhere the distance to P is large compared with the charge separation.

+q: hill

V = 0

r >> a

q: hole

p = 2qa

= dipole moment

got it 22 5
GOT IT? 22.5

The figure show 3 paths from infinity to a point P on a dipole’s perpendicular bisector.

Compare the work done in moving a charge to P on each of the paths.

  • V is path independent
  • work on all 3 paths are the same.
  • Work along path 2 is 0 since V = 0 on it.
  • Hence, W = 0 for all 3 paths.







example 22 6 charged ring
Example 22.6. Charged Ring

A total charge Q is distributed uniformly around a thin ring of radius a.

Find the potential on the ring’s axis.

Same r for all dq

example 22 7 charged disk
Example 22.7. Charged Disk

A charged disk of radius a carries a charge Q distributed uniformly over its surface.

Find the potential at a point P on the disk axis, a distance x from the disk.

point charge



22 3 potential difference the electric field
22.3. Potential Difference & the Electric Field

Equipotential = surface on which V = const.

  • W = 0 along a path  E
  • V = 0 between any 2 points on a surface  E.
    • Equipotential  Field lines.

Steep hill

Close contour

Strong E

V > 0

V < 0

V = 0

got it 22 6
GOT IT? 22.6

The figure show cross sections through 2 equipotential surfaces.

In both diagrams, the potential difference between adjacent equipotentials is the same.

Which could represent the field of a point charge? Explain.

(a). Potential decreases as r1 , so the spacings between equipotentials should increase with r.

calculating field from potential
Calculating Field from Potential

=  ( Gradient of V )

E is strong where V changes rapidly ( equipotentials dense ).

example 22 8 charged disk
Example 22.8. Charged Disk

Use the result of Example 22.7 to find E on the axis of a charged disk.

Example 22.7:

x > 0

x < 0

dangerous conclusion

tip field potential
Tip: Field & Potential

Values of E and V aren’t directly related.

V flat, Ex = 0

V falling, Ex > 0

V rising, Ex < 0

22 4 charged conductors
22.4. Charged Conductors

In electrostatic equilibrium,

E = 0 inside a conductor.

E// = 0 on surface of conductor.

 W = 0 for moving charges on / inside conductor.

 The entire conductor is an equipotential.

  • Consider an isolated, spherical conductor of radius R and charge Q.
  • Q is uniformly distributed on the surface
  • E outside is that of a point charge Q.
  • V(r) = k Q / R. for r  R.

Consider 2 widely separated, charged conducting spheres.

Their potentials are

Same V

If we connect them with a thin wire,

there’ll be charge transfer until V1 = V2 , i.e.,

In terms of the surface charge densities

we have

 Smaller sphere has higher field at surface.


Isolated conductor with irregular shape.

  • Surface is equipotential  | E | is larger where curvature of surface is large.
  • More field lines emerging from sharply curved regions.
  • From afar, conductor is like a point charge.
application corona discharge pollution control and xerography
Application: Corona Discharge, Pollution Control, and Xerography

Air ionizes for E > MN/C.

Recombination of e with ion

 Corona discharge ( blue glow )

Electrostatic precipitators:

Removes pollutant particles (up to 99%) using gas ions produced by Corona discharge.

Corona discharge across power-line insulator.

Laser printer / Xerox machines:

Ink consists of plastic toner particles that adhere to charged regions on light-sensitive drum, which is initially charged uniformy by corona discharge.