U = 1 4 piepsilon1 0 q 1 q 2 r John Shumway Department of Physics and Astronomy Arizona State University • Tempe • Arizona http://physics.asu.edu/shumway john.shumway@asu.edu PHY 131: University Physics II Lecture 7: Electric Potential Energy Young and Freedman, Chapter 23.1 9:00–10:15, Tuesday, February 8, 2010 PSF 173 • Department of Physics 1 john.shumway@asu.edu In this lecture, we make a connection between the electrostatic forces and the work-energy theorem from PHY 121. K i + U i + W ot her = K f + U f We can always put the electric field into the work done by other forces. But, we will see that electric field is a conserved force, so we can describe it with a potential energy U. (Recall that ΔU = U f -U i = -W f , use either potential energy U or work W f to describe the force.) Recall the potential energies from conserved forces we used in PHY 121: U = mg h U = 1 2 kx 2 U = Gm 1 m 2 r Conservative forces are forces for which the work only depends on the change in position, not on the path. 2 john.shumway@asu.edu The work done by a constant electric field on a moving charge is only a function of the change in position of the charge. W = integraldisplay f i F · d r From mechanics, Using F=qE, W = q integraldisplay f i E · d r For a constant electric field in the x direction, W = qE integraldisplay f i ˆx · d r = qE integraldisplay f i dx = qE ∆ X Thus the change in potential energy is just given by the change in x-coordinate, ∆ U = U f − U i = − W = − qE integraldisplay f i dx = − qE ∆ X 3 john.shumway@asu.edu The electrostatic potential energy in a constant electric field is analogous to the gravitational potential energy near the Earth’s surface. U = mg h h =0 ⇒ U =0 E U = q E y y =0 ⇒ U =0 Recall that we can set the zero of potential energy wherever we want. 4 john.shumway@asu.edu The change in electrostatic energy only depends on the change in position, and is independent the path taken. E y =0 ⇒ U =0 U i = q E y i U f = q E y f ∆ U f = qE ∆ y 5 john.shumway@asu.edu Note: the general expression for the potential energy of a charge in an electric field is a dot product. U = − q E · r You can choose the origin (zero of energy) wherever you want. For a field in the +x direction, this becomes U = − q E x For a field in the -z direction, this becomes U = q E z 6 john.shumway@asu.edu By convention, we set the potential energy of two charges to be zero when they are infinitely far away. U ( r )= − W ∞→ r = − integraldisplay r ∞ E · d r prime = integraldisplay ∞ r 1 4 piepsilon1 0 q 1 q 2 r prime 2 dr prime = q 1 q 2 4 piepsilon1 0 integraldisplay ∞ r dr r prime 2 = q 1 q 2 4 piepsilon1 0 bracketleftbigg − 1 r prime bracketrightbigg ∞ r U ( r )= 1 4 piepsilon1 0 q 1 q 2 r Potential energy is a scalar; it is much easier to work with than a vector force! The potential energy is the negative of the work done by the electric field when bringing the two charges together from infinity, 7 john.shumway@asu.edu If we ever want the electrical force, we can just take a derivative (gradient) of the electric potential energy. ∆ U = U f − U i = − integraldisplay f i F · d r The inverse of this relationship is a gradient, F = ∇ U = parenleftbigg ∂ U ∂ x ˆx + ∂ U ∂ y ˆy + ∂ U ∂ z ˆz parenrightbigg Strategy: to find the electrostatic force on a charge, first calculate how the potential energy depends on the charges position, then take a derivative. 8 john.shumway@asu.edu The total electrostatic energy of a set of charges adds. q 1 q 2 q 3 q 4 U = summationdisplay i