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Short Version : 22. Electric Potential 22.1. Electric Potential Difference Conservative force: U AB U B U A WAB B F dr A ( path independent ) Electric potential difference electric potential energy difference per unit charge VAB B U AB E dr A q VB [ V ] = J/C = Volt = V if reference potential VA = 0. For a uniform field: VAB E rAB E rB rA rAB E Table 22.1. Force & Field, Potential Energy & Electric Potential Quantity Force Electric field Potential energy difference Electric potential difference Symbol / Equation F Units N E=F/q N/C or V/m B U F dr A F U U V q B V E dr A E V J J/C or V 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. 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. V0 x E x 0 E x 2 0 22.2. Calculating Potential Difference Potential of a Point Charge B VAB VB VA E dr A B A kq rˆ d r 2 r For A,B on the same radial rˆ d r rˆ rˆ dr VAB rB rA dr 1 1 kq d r k q 2 r rB rA 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: VRA VA VR VA R = point of zero potential VA = potential at A. Some choices of zero potential Power systems / Circuits Automobile electric systems Isolated charges Earth ( Ground ) Car’s body Infinity 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 (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) Q V R k R 9.0 10 Vm / C 9 640 10 2.30 m (b) W e V R 2.50 MeV 1.6 1019 C (c) VR,2 R V 2R V R k k Q 1.25 MV 2R 6 Q Q k 2R R C 2.50 MV 2.50 MV 4.0 1013 J 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? rB r A rA VAB E d r B r 2 0 rB rA 1 dr r rˆ rˆ d r 2 0 r r ln B 2 0 rA 22 m 2 9.0 10 V m / C 2.6 10 C / m ln 0.01 m 9 360 kV 6 Finding Potential Differences Using Superposition Potential of a set of point charges: V P k i Potential of a set of charge sources: qi rP ri V P Vi P i 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. V P k q q k r1 r2 1 1 r r kq kq 2 1 r2 r1 r1 r2 r12 r 2 a 2 2r a cos +q: hill r22 r 2 a 2 2r a cos r22 r12 4 r a cos r2 r1 r1 r2 r >> a V P k q r2 r1 2 a cos r2 r1 2 qa cos p cos k k r2 r2 r2 V=0 q: hole p = 2qa = dipole moment Continuous Charge Distributions Superposition: V dV V r k k dq r r d 3r r r k dV r 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. dq V x k r k Same r for all dq k x a 2 Q x k 2 x a 2 Q x a 2 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. V x dV a 0 disk sheet 2k Q a2 k x r 2 2 dq Q 2 r dr 2 2 a2 x r 2Q k 2 a point charge k a 0 r x2 r 2 2Q k 2 a dr x2 a2 x x r 2 a 0 2kQ 2k Q a x a 2 x a2 0 kQ x 2 x a x a 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 Calculating Field from Potential rB VAB E d r rA dV E d r Ei dxi i Ei V xi i V d xi xi E V = ( Gradient of V ) V V V E i j k x y z 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: V x 2k Q a2 2k Q V 2 Ex a x E y Ez 0 x2 a2 x x x2 a2 1 dangerous conclusion x>0 x<0 Tip: Field & Potential Values of E and V aren’t directly related. V flat, Ex = 0 V falling, Ex > 0 V rising, Ex < 0 Ex V x 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 V1 k Same V Q1 R1 V2 k Q2 R2 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 j 1 R1 2 R2 Qj 4 R 2j Smaller sphere has higher field at surface. E1 R1 E2 R2 Q1 Q2 R1 R 2 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. 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. Strongest field : E 70 V 7 103 m 10 kV / m Weakest field : E 7 mm 12 mm 70 V 12 103 m 5.8 kV / m 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 across power-line insulator. Corona discharge. Laser printer / Xerox machines: Ink consists of plastic toner particles that adhere to charged regions on lightsensitive drum, which is initially charged uniformy by corona discharge.