lecture8_electric_potential_and_energy_072011

lecture8_electric_potential_and_energy_072011 - 7/20/2011...

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Unformatted text preview: 7/20/2011 Physics 142 Fields Inside Conductors Electrostatic Electrostatic equilibrium – Outside influences not changing – Charges » have had time to redistribute themselves » No longer moving Inside Inside a conductor in electrostatic r equilibrium E =0 – If not, charges would be moving » It’s a conductor! Physics 142 Summer 2011 Charges in a Conductor Since Since the E-field is zero, there can be no E(unbalanced) charges inside a conductor in equilibrium – Surround the hypothetical charge by a Gaussian surface and use Gauss’ law All All the (unbalanced) charges must be on the surface Surface Surface field is determined by surface charge density Physics 142 Summer 2011 Example: Planar Conductor Surface Physics 142 Summer 2011 Dr. Nick Cummings 1 7/20/2011 Physics 142 Field at the Surface When When you’re close enough to a surface it always looks like a plane, so…. Close Close enough to a conducting surface the field is approximately perpendicular to the surface with magnitude E= η ε 0 (like the plane) Surface Surface charge density η not constant on larger scale, though Physics 142 Summer 2011 Work and Energy r r ∆ (1 mv 2 ) = F net || ⋅ ∆r = W net 2 Note Note each force in the problem might contribute to the work. If If the force changes over the course of the motion f W net rr = ∫ F ⋅ ds i Physics 142 Summer 2011 Potential Energy If If the work done by a force depends only on the endpoints not the path, then it is called conservative For For a conservative force you can define a potential energy: ∆ (potential energy) = (potential energy final) – potential energy initial = - (work done by forces on object as it is moved from initial to final position) Physics 142 Summer 2011 Dr. Nick Cummings 2 7/20/2011 Physics 142 Potential Energy Mathematically Mathematically this is defined by the equation r r2 rr r r U (r2 ) − U (r1 ) = − ∫ F ⋅ ds r r1 Physics 142 Summer 2011 Potential Energy ∆U equals the amount of work one mustrdo r against F in order to move a body from r1 to r 2 – That’s the negative of the work done by F on the body This This is the amount of energy that would be r r gained back by going from r 2 to r1 ∆U matters but where you put the zero doesn’t Physics 142 Summer 2011 Example: Gravity Remember Remember how work and potential energy relate in the case of gravity ( ) r r ∆ 1 mv 2 = F net ⋅ ∆r 2 rr = mg ⋅ ∆r = −mg∆h ∆U grav = mg∆h Physics 142 Summer 2011 Dr. Nick Cummings 3 7/20/2011 Physics 142 Electric Potential Energy The The electric force is a conservative force We We can define an electric potential energy from the work the electric force does The The amount of energy will depend on all the charges involved, not just one charge – Because the force on each body depends on all the charges Conventionally Conventionally we set U=0 when all charges are far apart Physics 142 Summer 2011 Potential Energy of Two Point Charges Physics 142 Summer 2011 The Potential Energy of Point Charges Consider two point charges, q1 and q2, separated by a distance r. The electric potential energy is This is explicitly the energy of the system, not the energy of just q1 or q2. Note that the potential energy of two charged particles approaches zero as r → ∞. Physics 142 Summer 2011 Dr. Nick Cummings 4 7/20/2011 Physics 142 Electric Potential In In the same way that we “removed the test charge” from Coulomb’s law to define the electric field, we “remove the test charge” from the electric potential energy to create the electric potential, V. r rF E= q V= U q Physics 142 Summer 2011 Electric Potential: The Math r r r rr r r 1rr V (r ) = − ∫ F ⋅ ds = − ∫ E ⋅ ds q∞ ∞ r For any point in space, r , the electric potential at the point is the negative of the work required to bring a test charge, q, r from ∞ to r divided by q. Physics 142 Summer 2011 Electric Potential: Meaning Just Just like the electric field, the electric potential can be defined at any point in space. The The electric potential is a scalar (has no direction) so it’s easier to work with than the E-field. E– Adding the influence of multiple sources just means adding the voltages » No vector addition Physics 142 Summer 2011 Dr. Nick Cummings 5 7/20/2011 Physics 142 Analog for gravity What What would the gravitational analog be for “potential”? r r Fg g= m Vgrav = U grav m = gh So So electric potential is analogous (up to a constant) to “height” for gravity. Physics 142 Summer 2011 Not equal! Equipotentials Often Often we plot potentials by drawing lines of equal potential energy which are called equipotentials equipotentials In In the case of gravity, the level curves on a topographic map are equipotentials Physics 142 Summer 2011 Equipotentials & Field Strength On On the topographic map, which part is the steepest? Field Field is stronger when equipotential curves are closer together Physics 142 Summer 2011 Dr. Nick Cummings 6 7/20/2011 Physics 142 A change in potential energy leads to force rr ∆U = − F ⋅ ∆r ∆U F =− ∆r (pointing in the direction of steepest descent) Constant Constant PE no force no Change Change in PE force (pointing downhill) force Physics 142 Summer 2011 Equipotentials & Field Direction Along Along an equipotential the field does no work – Otherwise it wouldn’t be an equipotential The The field always points perpendicular to the equipotential Physics 142 Summer 2011 Electric Potential: Units The The unit of electric potential is the volt – Sometimes it is just called voltage V= U q [V ] = E Q 1V = 1 J C Physics 142 Summer 2011 Dr. Nick Cummings 7 7/20/2011 Physics 142 Electric Potential and Electric Field rr ∆U = − F ⋅ ∆s r ∆U F r r r ∆V = = ⋅ ∆s = E ⋅ ∆s q q [E ] = V m The electric field tells us how the potential changes. Physics 142 Summer 2011 Electric Potential and Electric Field Once Once we know the voltage we can get back the electric field r Given Given some path with element ds the component of electric field along that path is dV ds rr dV = E ⋅ ds Fs = − since Physics 142 Summer 2011 When a positive charge is released from rest in a uniform electric field what happens to the electric potential energy of the positive charge? 0% 0% 0% 0% 0% 1. It will increase because the charge will move in the direction of the electric field. 2. It will decrease because the charge will move in the direction opposite to the electric field. 3. It will decrease because the charge will move in the direction of the electric field. 4. It will remain constant because the electric field is uniform. 5. It will remain constant because the charge remains at rest. Physics 142 Summer 2011 Dr. Nick Cummings 8 7/20/2011 Physics 142 When a positive charge is released from rest in a uniform electric field what happens to the electric potential energy of the negative charge? 1. 0% 0% 2. 0% 3. 0% 4. 0% 5. It will increase because the charge will move in the direction of the electric field. It will decrease because the charge will move in the direction opposite to the electric field. It will decrease because the charge will move in the direction of the electric field. It will remain constant because the electric field is uniform. It will decrease because the charge will move in the direction of the electric field. Physics 142 Summer 2011 When a positive charge is released from rest in a uniform electric field what happens to the electric potential (voltage) of the negative charge? 0% 1. 0% 2. 0% 3. 0% 4. 0% 5. It will increase because the charge will move in the direction of the electric field. It will decrease because the charge will move in the direction opposite to the electric field. It will decrease because the charge will move in the direction of the electric field. It will remain constant because the electric field is uniform. It will decrease because the charge will move in the direction of the electric field. Physics 142 Summer 2011 Explore Potential with PhET Simulations Charges Charges and Fields Electric Electric Field Hockey Physics 142 Summer 2011 Dr. Nick Cummings 9 7/20/2011 Physics 142 A positive charge might be placed at one of three spots in a region where there is a uniform electric field. How do the electric potential, V, on the charge at positions 1, 2, and 3 compare? 0% 1. 0% 2. 0% 3. 0% 4. 0% 5. V is greatest at 1 V is greatest at 2 V is greatest at 3 V is 0 at all 3 spots V is = at all 3 spots but not = 0. Physics 142 Summer 2011 Two sheets of charge are separated by a distance d , small compared to the size of the sheets. The charge per unit area on the sheets are + η and -η. Select one graph below that could describe the electrostatic potential. 1. 2. 3. 4. 5. 6. 7. 8. 9. One Two Three Four Five Six Seven Eight Nine Physics 142 Summer 2011 Potential a Point Charge Physics 142 Summer 2011 Dr. Nick Cummings 10 7/20/2011 Physics 142 Physics 142 Summer 2011 Dr. Nick Cummings 11 ...
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