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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
INTERNATIONAL VOLTAGE Afghanistan 220 V Australia 240 V Bahamas 120 V Brazil 120/220 V Canada 120 V China 220 V Finland 230 V Guam 110 V HongKong 220 V Japan 100 V Mexico 127 V Spain 230 V United Kingdom 230 V United States 120 V Vietnam 127/220 V
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
Curved Paths & Nonuniform Fields Staight path, uniform field: Curved path, nonuniform field:
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.
Conceptual Example 22.1. An Irregular Condutor Sketch some equipotentials & electric field lines for an isolated egg-shaped conductor. • 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.
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. Strongest field : Weakest field : 7 mm 12 mm
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.