Short Version : 22. Electric Potential

22.1. Electric Potential Difference Conservative force: Electric potential difference  electric potential energy difference per unit charge if reference potential V A = 0. [ V ] = J/C = Volt = V For a uniform field: ( path independent ) r AB E

Table 22.1.Force & Field, Potential Energy & Electric Potential QuantitySymbol / EquationUnits Force Electric field Potential energy differenc e Electric potential difference FN E = F / qN/C or V/m J J/C or V

Potential Difference is Path Independent Potential difference  V AB depends only on positions of A & B. Calculating along any paths (1, 2, or 3) gives  V AB = E  r.

Example 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

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, 1 st 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 V A = potential at A. Some choices of zero potential Power systems / CircuitsEarth ( Ground ) Automobile electric systemsCar’s body Isolated charges Infinity

Example 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 (a) the potential at its surface, (b) the work needed to bring a proton from infinity to the sphere’s surface, (c) the potential difference between the sphere’s surface & a point 2R from its center. (a) (b) (c)

Example 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 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. r >> a  p = 2qa = dipole moment +q: hill  q: hole V = 0

Continuous Charge Distributions Superposition:

Example 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 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. sheet point charge disk

22.3. Potential Difference & the Electric Field W = 0 along a path  E   V = 0 between any 2 points on a surface  E. Equipotential  Field lines. Equipotential = surface on which V = const. V > 0 V < 0 V = 0 Steep hill Close contour Strong E

Calculating Field from Potential   =  ( Gradient of V ) E is strong where V changes rapidly ( equipotentials dense ).

Example 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 falling, E x > 0 V flat, E x = 0 V rising, E x < 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 If we connect them with a thin wire, there’ll be charge transfer until V 1 = V 2, i.e., In terms of the surface charge densities we have  Smaller sphere has higher field at surface.  Same V

Ans. 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. Conceptual Example An Irregular Condutor Sketch some equipotentials & electric field lines for an isolated egg-shaped conductor.

Conductor in the Presence of Another Charge

Making the Connection The potential difference between the conductor and the outermost equipotential shown in figure is 70 V. Determine approximate values for the strongest & weakest electric fields in the region, assuming it is drawn at the sizes shown. 7 mm 12 mm Strongest field : Weakest field :

Application: Corona Discharge, Pollution Control, and Xerography Air ionizes for E > MN/C. Recombination of e with ion  Corona discharge ( blue glow ) Corona discharge across power-line insulator. Electrostatic precipitators: Removes pollutant particles (up to 99%) using gas ions produced by Corona discharge. 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.