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## 22. Electric Potential

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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?

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

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:

rAB

E

Table 22.1. Force & Field, Potential Energy & Electric Potential

Force

F

N

Electric field

E = F / q

N/C or V/m

Potential energy difference

J

J/C or V

Electric potential difference

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

- 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.

doubles

doubles

becomes 0

reverses sign

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 = 1.61019 C = e

1 eV = 1.61019 J

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

- 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

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

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.

E

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

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

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

Infinity

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

- 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.

(a)

(b)

(c)

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

Potential of a set of point charges:

Potential of a set of charge sources:

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

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.

2

3

P

q

q

1

Continuous Charge Distributions

Superposition:

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

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

disk

sheet

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

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 r1 , so the spacings between equipotentials should increase with r.

Calculating Field from Potential

= ( Gradient of V )

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

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

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

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

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.

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