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W03D2 Work, Potential Energy and Electric Potential 1 Announcements Math Review Tuesday Tues Feb 28 from 9-11 pm in 32-082 PS 3 due Tuesday Tues Feb 28 at 9 pm in boxes outside 32-082 or 26-152 W03D3 Reading Assignment Course Notes: Sections 3.5, 3.7-3.8, 4.8.4 Exam One Thursday March 1 7:30-9:30 pm Room Assignments (See Stellar webpage announcements) 2 Outline Electrical Work Electric Potential Energy Electric Potential Difference Calculating Electric Potential Difference 3 Electrical Work Electrical force on object 1 due to interaction between charged objects 1 and 2: q1q2 F12 ke 2 rˆ12 r12 Work done by electrical force moving object 1 from A to B: B W1 F12 d s12 A PATH INTEGRAL 4 Concept Question: Sign of W Suppose a fixed positively charged object (charge qs > 0) is at the origin and we move a negatively charged object (charge q1 < 0) from A to B with rA < rB , where r is the distance from the origin. The work done on the negatively charged object by the electric force: 1. 2. 3. 4. is positive and we do a positive amount of work is positive and we do a negative amount of work is negative and we do a positive amount of work is negative and we do a negative amount of work 5 Concept Question Ans.: Sign of W Answer 3: The work done by the electric force W is negative and we do a positive amount of work W is the work done by the electrical force. This is the opposite of the work that we must do in order to move a charged object in an electric field due to source. The electrical force is attractive and we are moving the positively charged object away from the source (opposite the direction of the electric field). 6 Group Problem: Work Done by Electrical Force A point-like charged source object (charge qs) is held fixed. A second point-like charged object (charge q1)is initially at a distance rA from the fixed source and moves to a final distance rB from the fixed source. What is the work done by the electrical force on the moving object? Hint: What coordinate system is best suited for this problem? 7 Sign of W: Negative Work Suppose a fixed positively charged source (charge qs > 0) is at the origin and a positively charged object (charge q1 > 0) moves from A to B with rA > rB , where r is the distance from the origin, then W < 0. 1 1 1 1 rA rB 0 and qs q1 0 W ke qs q1 0 rB rA rB rA 8 Work and Change in Kinetic Energy W K 9 Group Problem: Work-Kinetic Energy In a Uniform Electric field Consider two thin oppositely uniform charged thin plates separated by a distance d. The surface charge densities on the plates are uniform and equal in magnitude. An electron with charge –e and mass m is released from rest at the negative plate and moves to the positive plate. What is the speed of the electron when it reaches the positive plate? 10 Potential Energy Difference Suppose charged object 1 is fixed and located at the origin and charge object 2 moves from an initial position A a distance rA from the origin to a final position B, a distance rB from the origin. The potential energy difference due to the interaction is defined to be the negative of the work done object 2 in moving from A to B: 1 1 U U B U A F12 d s W ke qs q1 A rB rA B 11 Potential Energy: Zero Point Choose the zero point for the potential energy at infinity. U () 0 Then set rA = ∞ and rB = r . The potential energy difference between ∞ and any point on a circle of radius r is keqs q1 U (r) U () U (r) r 12 Concept Question: Motion of Charged Objects Two oppositely charged are released from rest in an electric field. 1. 2. 3. 4. Both charged objects will move from lower to higher potential energy. Both charged objects will move from higher to lower potential energy. The positively charged object will move from higher to lower potential energy; the negatively charged object will move from lower to higher potential energy. The positively charged object will move from higher to lower potential energy; the negatively charged object will move from lower to higher potential energy. 13 Concept Q. Ans.: Motion of Charged Objects 2. Both charged objects will move from higher to lower potential energy so that U 0 14 Configuration Energy What is the potential energy stored in a configuration of charged objects? Start with all the charged objects at infinity. Choose U () 0 (1) Bring in the first charged object. U 1 0 (2) Bring in the second charged object U 2 U12 ke q1q2 / r12 (3) Bring in the third charged object U 3 U 23 U13 ke q2 q3 / r23 ke q1q3 / r13 (4) Configuration energy U U12 U 23 U13 ke q1q2 / r13 ke q2 q3 / r23 ke q1q3 / r13 15 Group Problem: Build It How much energy does it take you to assemble the charges into the configuration at left, assuming they all started out an infinite distance apart? 16 Electric Potential Difference Change in potential energy per test object in moving the test object (charge qt) from A to B: U F V d s E d s qt q A t A B B Units: Joules/Coulomb = Volts 17 Demonstration Van de Graaf D29 Breakdown of dry air 33 kV/cm Video of Tesla Coil http://www.youtube.com/watch?v=FY-AS13fl30 18 How Big is a Volt? AA Batteries 1.5 V High Voltage Transmission Lines 100 kV-700 kV Car Batteries 12 V Van der Graaf 300 kV US Outlet (AC) 120 V Tesla Coil 500 kV Distribution Power Lines 120 V- 70 kV Lightning 10-1000 MV 19 E Field and Potential: Effects If you put a charged particle, (charge q), in a field: F qE To move a charged particle, (charge q), in a field and the particle does not change its kinetic energy then: U qV 20 Concept Question: Motion of Charged Objects Two oppositely charged are released from rest in an electric field. 1. 2. 3. 4. Both charged objects will move from lower to higher electric potential. Both charged objects will move from higher to lower electric potential. The positively charged object will move from higher to lower electric potential; the negatively charged object will move from lower to higher electric potential. The positively charged object will move from higher to lower electric potential; the negatively charged object will move from lower to higher electric potential. 21 Concept Q. Ans.: Motion of Charged Objects Two oppositely charged are released from rest in an electric field. 3. The positively charged object will move from higher to lower electric potential; the negatively charged object will move from lower to higher electric potential. For the positively charged object: V 0 U qV 0 For the negatively charged object: V 0 U qV 0 22 Potential & External Work Change in potential energy in moving the charged object (charge q) from A to B: U qV Conservation of Energy Law: Wext K U 23 Demonstration: Kelvin Water Drop (32-082) or Wimshurst Machine (26-152) 24 Potential Created by Pt Charge B V VB VA E ds A B dr rˆ kQ 2 d s kQ 2 A A r r 1 1 kQ rB rA B Take V = 0 at r = ∞: kQ VPoint Charge (r) r rˆ E kQ 2 r ds dr rˆ r d ˆ 25 Concept Question: Two Point Charges The work done in moving a positively charged object that starts from rest at infinity and ends at rest at the point P midway between two charges of magnitude +Q and –Q 1. 2. 3. 4. is positive. is negative. is zero. can not be determined – not enough info is given. 26 Concept Question Answer: Two Point Charges 3. Work from ∞ to P is zero. The potential at ∞ is zero. The potential at P is zero because equal and opposite potentials are superimposed from the two point charges (remember: V is a scalar, not a vector) 27 Potential Landscape Positive Charge Negative Charge 28 Continuous Charge Distributions 29 Continuous Charge Distributions Break distribution into infinitesimal charged elements of charge dq. Electric Potential difference between infinity and P due to dq. dq dV Vdq ( P) Vdq () ke r dq Superposition Principle: V ( P) V () ke r source Reference Point: V () 0 V ( P) ke dq r source 30 Calculating Electric Potential Difference for Continuous Distributions 1. Choose V () 0 2. Choose integration r variables d s 3. Identify dq da d v 4. Choose field point variables r 5. Calculate source to field dq V (r ) V () ke point distance r - r r - r source 6. Define limits of integral 7. Integrate 31 Worked Example Consider a uniformly charged ring with total charge Q. Find the electric potential difference between infinity and a point P along the symmetric axis a distance z from the center of the ring. 32 Worked Example: Charged Rod Q / 2 R dq ds Rd Choose V () 0 r z kˆ r Rrˆ r - r R 2 z 2 dq 1 Rd dV 4 0 r - r 4 0 R2 z 2 1 V (z) ke 2 Rd ke R R2 z 2 R2 z 2 ke 2 R k eQ V (z) R2 z 2 R2 z 2 0 b a R d 33 Group Problem A thin rod extends along the x-axis from x = -l /2 to x = l/2 . The rod carries a uniformly distributed positive charge +Q. Calculate the electric potential difference between infinity and at a point P along the x-axis. 34