Thursday, Feb. 10, 2011 1PHYS 1444-002 Dr. Andrew BrandtPHYS 1444 – Section 02Lecture #6• Chapter 23:Tuesday Feb 10, 2011Dr. Mark Sosebee for Dr. Andrew Brandt• Electric Potential due to Point Charges• Shape of the Electric Potential• V due to Charge Distributions• Equi-potential Lines and Surfaces• Electric Potential Due to Electric Dipole• E determined from V• Electrostatic Potential Energy of a System of ChargesThursday, Feb. 10, 2011 2PHYS 1444-002 Dr. Andrew BrandtElectric Potential and Electric Field• The potential energy is an (independent of path) function expressed in terms of a (conservative) force.• The potential difference is the potential energy difference per unit charge– This formula can be used to determine Vbawhen the electric field is given. • When the field is uniformbaUUbaVbaVVbaV EdorUnit of the electric field in terms of potential?V/m Can you derive this from N/C?baVVbaUUqbaFdlqbaE dlbaE dlbaE dlEdbaF dlThursday, Feb. 10, 2011 3PHYS 1444-002 Dr. Andrew Brandt50V5cmExample 23 – 3 Uniform electric field obtained from voltage: Two parallel plates are charged to a voltage of 50 V. If the separation between the plates is 5.0 cm, calculate the magnitude of the electric field between them, ignoring any fringe effects.EVd505.0VcmWhat is the relationship between electric field and the potential for a uniform field? V EdSolving for E2505 10Vm1000 /VmThursday, Feb. 10, 2011 4PHYS 1444-002 Dr. Andrew BrandtElectric Potential due to Point Charges• What is the electric field due to a point charge Q at a distance r? • Electric potential due to the field E for moving from point rato rbaway from the charge Q isbaVVbarrE dlE2014Qr2Qkr20ˆˆ4barrQrrdrr2014barrQdrr0114baQrrNotice how the integral is carried out in the radial direction.Thursday, Feb. 10, 2011 5PHYS 1444-002 Dr. Andrew BrandtElectric Potential due to Point Charges• Since only the differences in potential have physical meaning, we can choose at .• The electrical potential V at a distance r from a single point charge is• So the absolute potential from a single point charge depends only on the magnitude of the point charge and the distance from itV014Qr0bVbrThursday, Feb. 10, 2011 6PHYS 1444-002 Dr. Andrew Brandt• What are the differences between the electric potential and the electric field?– Electric potential• Electric potential energy per unit charge• Inversely proportional to the distance• Simply add the potential from each of the charges to obtain the total potential from multiple charges, since potential is a scalar quantity– Electric field• Electric force per unit charge• Inversely proportional to the square of the distance• Need vector sums to obtain the total field from multiple charges• Potential for a positive charge is large near the charge and decreases to 0 at large distances.• Potential for the negative charge is small (large magnitude but negative) near the charge and increases with distance to 0 Properties of the Electric Potential2014QEr014QVrThursday, Feb. 10, 2011 7PHYS 1444-002 Dr. Andrew BrandtShape of the Electric Potential• So, how does the electric potential look like as a function of distance?– What is the formula for the potential by a single charge?V014QrPositive ChargeNegative ChargeA uniformly charged sphere would have the same potential as a single point charge.What does this mean?Uniformly charged sphere behaves like all the charge is on the single point in the center.Thursday, Feb. 10, 2011 8PHYS 1444-002 Dr. Andrew BrandtSince we obtainExample 23 – 6Work to bring two positive charges close together: What minimum work is required by an external force to bring a charge q=3.00 μC from a great distance away ( ) to a point 0.500 m from a charge Q=20.0 μC?What is the work done by the electric field in terms of potential energy and potential? W0.500,bar mrWIn other words, the external force must input work of +1.08J to bring the charge 3.00 C from infinity to 0.500m from the 20.0 C charge.baqV04baq Q Qrr004bqQr04bqQr922 6 68.9910 3.001020.00101.080.500NmC CCJmrThursday, Feb. 10, 2011 9PHYS 1444-002 Dr. Andrew BrandtElectric Potential from Charge Distributions• Let’s consider that there are n individual point charges in a given space and V=0 at• Then the potential due to the charge Qiat a point a, distance riafrom Qiis• Thus the total potential Vaby all n point charges isiaV014iiaQr1niaiV0114niiaiQraV• For a continuous charge distribution, we obtain014dqrVrThursday, Feb. 10, 2011 10PHYS 1444-002 Dr. Andrew BrandtExample 23 – 8 • Potential due to a ring of charge: A thin circular ring of radius R carries a uniformly distributed charge Q. Determine the electric potential at a point P on the axis of the ring a distance x from its center.• Each point on the ring is at the same distance from the point P. What is the distance?22r R x• So the potential at P is014dqrV014dqr22014dqxR2204QxRWhat’s this?Thursday, Feb. 10, 2011 11PHYS 1444-002 Dr. Andrew BrandtEqui-potential Surfaces• Electric potential can be visualized using equipotential lines in 2-D or equipotential surfaces in 3-D• Any two points on equipotential surfaces (lines) are on the same potential• What does this mean in terms of the potential difference?– The potential difference between the two points on an equipotential surface is 0.• How about the potential energy difference?– Also 0.• What does this mean in terms of the work to move a charge along the surface between these two points?– No work is necessary to move a charge between these two points.Thursday, Feb. 10, 2011 12PHYS 1444-002 Dr. Andrew BrandtEqui-potential Surfaces• An equipotential surface (line) must be perpendicular to the electric field. Why?– If there are any parallel components to the electric field, it would require work to move a charge along the surface.• Since the equipotential surface (line) is perpendicular to the electric field, we can draw these surfaces or lines easily.• There can be no electric field inside a conductor in static case, thus the entire volume of a conductor must be at the same potential. • So the electric field must be perpendicular to the conductor surface.Point chargesParallel PlateJust like a topographic map13Electric Potential due to Electric Dipoles• What is an electric dipole?– Two equal point charge Q of opposite sign separated by a distance l
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