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## MIT Class: Electric Potential

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**Workshop: Using Visualization in Teaching Introductory**E&MAAPT National Summer Meeting, Edmonton, Alberta, Canada.Organizers: John Belcher, Peter Dourmashkin, Carolann Koleci, Sahana Murthy**Potential Energyand Potential**Start with Gravity**Gravity: Force and Work**Gravitational force on m due to M: Work done by gravity moving m from A to B: PATH INTEGRAL**Work done by gravity moving m from A to B:**Work Done by Earth’s Gravity**PRS: Sign of Wg**Thinking about the sign and meaning of this… Moving from rA to rB: • Wg is positive – we do work • Wg is positive – gravity does work • Wg is negative – we do work • Wg is negative – gravity does work • I don’t know :19**PRS Answer: Sign of Wg**Answer: 3. Wg is negative – we do work Wg is the work that gravity does. This is the opposite of the work that we must do in order to move an object in a gravitational field. We are pushing against gravity we do positive work**Work done by gravity moving m from A to B:**Work Near Earth’s Surface G roughly constant: Wg depends only on endpoints – not on path taken – Conservative Force**Potential Energy (Joules)**• U0: constant depending on reference point • Only potential difference DU has physical significance**Define gravitational potential difference:**Gravitational Potential(Joules/kilogram) That is, two particle interaction single particle effect**PRS: Masses in Potentials**Consider 3 equal masses sitting in different gravitational potentials: A) Constant, zero potential B) Constant, non-zero potential C) Linear potential (V x) but sitting at V = 0 Which statement is true? • None of the masses accelerate • Only B accelerates • Only C accelerates • All masses accelerate, B has largest acceleration • All masses accelerate, C has largest acceleration • I don’t know :19**PRS Answer: Masses in Potentials**Answer: 3. Only C (linear potential) accelerates When you think about potential, think “height.” For example, near the Earth: U = mgh so V = gh Constant potential (think constant height) does not cause acceleration! The value of the potential (height) is irrelevant. Only the slope matters**Gravity - Electrostatics**Mass M Charge q (±) Both forces are conservative, so…**Potential & Potential Energy**Units: Joules/Coulomb = Volts Change in potential energy in moving the charged object (charge q) from A to B: Joules**Potential & External Work**Change in potential energy in moving the charged object (charge q) from A to B: Joules The external work is If the kinetic energy of the charged object does not change, then the external work equals the change in potential energy 18**How Big is a Volt?**Know These! • AA, C, D Batteries 1.5 V • Car Battery 12 V • US Outlet 120 V (AC) • Residential Power Line • Our Van de Graaf • Big Tesla Coil**Potential: Summary Thus Far**Charges CREATE Potential Landscapes**Potential Landscape**Positive Charge Negative Charge**Potential: Summary Thus Far**Charges CREATE Potential Landscapes Charges FEEL Potential Landscapes We work with DU (DV) because only changes matter**11**PRS: Positive Charge Place a positive charge in an electric field. It will accelerate from • higher to lower electric potential; lower to higher potential energy • higher to lower electric potential; higher to lower potential energy • lower to higher electric potential; lower to higher potential energy • lower to higher electric potential; higher to lower potential energy**PRS Answer: Positive Charge**Answer: 2. + acc. from higher to lower electric potential; higher to lower potential energy Objects always “move” (accelerate) to reduce their potential energy. Positive charges do this by accelerating towards a lower potential**PRS: Negative Charge**Place a negative charge in an electric field. It will accelerate from • higher to lower electric potential; lower to higher potential energy • higher to lower electric potential; higher to lower potential energy • lower to higher electric potential; lower to higher potential energy • lower to higher electric potential; higher to lower potential energy 18**PRS Answer: Negative Charge**Answer: 4. Neg. acc. from lower to higher electric potential higher to lower potential energy Objects always “move” (accelerate) to reduce their potential energy. Negative charges do this by accelerating towards a higher potential:**Potential Landscape**Positive Charge Negative Charge**Potential in a Uniform Field**Just like gravity, moving in field direction reduces potential**Potential Created by Pt Charge**Take V = 0 at r = ∞:**-q**+q P PRS: Two Point Charges The work done in moving a positive test charge from infinity to the point P midway between two charges of magnitude +q and –q: • is positive. • is negative. • is zero. • can not be determined – not enough info is given. • I don’t know :16**-q**+q P PRS 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)**Potential Landscape**Positive Charge Negative Charge**Group Problem: Superposition**Consider the 3 point charges at left. What total electric potential do they create at point P (assuming V = 0)**Deriving E from V**A = (x,y,z), B=(x+Dx,y,z) Ex = Rate of change in V with y and z held constant**Deriving E from V**If we do all coordinates: Gradient (del) operator:**PRS: E from V**Consider the point charges you looked at earlier: You calculated V(P). From that can you derive E(P)? • Yes, its kQ/a2 (up) • Yes, its kQ/a2 (down) • Yes in theory, but I don’t know how to take a gradient • No, you can’t get E(P) from V(P) • I don’t know 15**PRS Answer: E from V**4. No, you can’t get E(P) from V(P) The electric field is the gradient (spatial derivative) of the potential. Knowing the potential at a single point tells you nothing about its derivative. People commonly make the mistake of trying to do this. Don’t!**PRS: E from V**larger than that for x < 0 smaller than that for x < 0 equal to that for x < 0 I don’t know The graph above shows a potential V as a function of x. The magnitude of the electric field for x > 0 is :20 43**PRS Answer: E from V**The slope is smaller for x > 0 than x < 0 Translation: The hill is steeper on the left than on the right. Answer: 2. The magnitude of the electric field for x > 0 is smaller than that for x < 0 44**PRS: E from V**Ex > 0 is > 0 and Ex < 0 is > 0 Ex > 0 is > 0 and Ex < 0 is < 0 Ex > 0 is < 0 and Ex < 0 is < 0 Ex > 0 is < 0 and Ex < 0 is > 0 I don’t know The above shows potential V(x). Which is true? 20 45**PRS Answer: E from V**E is the negative slope of the potential, negative on the left, positive on the right Translation: “Downhill” is to the left on the left and to the right on the right. Answer: 2. Ex > 0 is > 0 and Ex < 0 is < 0 46**Group Problem: E from V**A potential V(x,y,z) is plotted above. It does not depend on x or y. What is the electric field everywhere? Are there charges anywhere? What sign?**Configuration Energy**How much energy to put two charges as pictured? • First charge is free • Second charge sees first: