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22. Electric Potential 1. 2. 3. 4. 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: 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. GOT IT? 22.1 What would happen to VAB in the figure if doubles doubles becomes 0 (a)E were doubled; (b) r were doubled; (c) the points were moved so the path lay at right angles to E; reverses sign (d) the positions of A & B are interchanged. The Volt & the Electronvolt [ V ] = J/C = Volt = V U q 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 heart’s electrical activity 1 mV 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 transmission line & ground 365 kV Between base of thunderstorm cloud & ground 100 MV GOT IT? 22.2 10 eV (a) A proton ( charge e ), 20 eV (b) an particle ( charge 2e ), and 10 eV (c) a singly ionized O atom each moves through a 10-V potential difference. What’s the work in eV done on each? 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. VAB E r 300 kN / C 0.10 m 30 kV U AB e VAB 30 keV K AB U AB 30 keV 4.8 1015 J 4.8 fJ 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 Curved Paths & Nonuniform Fields Staight path, uniform field: VAB E rAB Curved path, nonuniform field: Ei ri E dr r 0 B VAB lim i A 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 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 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 (a) greater, (b) less, or (c) 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 (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: qi rP ri V P k i Potential of a set of charge sources: V P Vi 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 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. 2 3 Hence, W = 0 for all 3 paths. P 1 q Work along path 2 is 0 since V = 0 on it. q Continuous Charge Distributions Superposition: V dV k dq 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 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 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 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. Conductor in the Presence of Another 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 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.