Potential

1. Potential Alan Murray

2. V 0 |E| 0 Potential … Start Simply … V Remember the capacitor E = -V/d E=-(rate of changeof V with distance) Alan Murray – University of Edinburgh

3. E = -V/d • Should really be E = -dV/dx • And if V = Mx+c, dV/dx = M = constant • Then E = -M as shown • In 3D, dV/dx becomes(dV/dx, dV/dy, dV/dz) = ÑV, so • E = -ÑV = -(dV/dx, dV/dy, dV/dz) E = -ÑV Alan Murray – University of Edinburgh

4. 300M 250M 200M 150M 100M Potential : Analogy These contour lines are lines of equal gravitational potential energy mgh Where they are close together, the effect of the gravitational field is strong The field acts in a direction perpendicular to the countoursand it points in a negative direction … (i.e. that’s the way you will fall!) Alan Murray – University of Edinburgh

5. Potential - comments • Walking around a contour expends no energy • In a perfect world • i.e. no-one moves the hill as you walk! • Walking to the top of the hill and back again expends no energy • In a perfect world • i.e. – the hill stays still and you recoup the energy you expend while climbing as you descend (using your internal generator!) Alan Murray – University of Edinburgh

6. 5V Metal electrode 5V 1V Metal electrode E-Fieldlines 2V 3V 0V 4V Electric Fields and Potentials are the Same Voltage contours Alan Murray – University of Edinburgh

7. 5V 1C 1C 1C 1C Metal electrode 5V Metal electrode E-Fieldlines 0V Potential Difference : Formal Definition (L) The Potential Difference (Voltage)between a and b is the –the workdone to move a 1C chargefrom a to b b x a x Alan Murray – University of Edinburgh

8. Potential Difference : Formal Definition (L) • The Potential Difference (Voltage)between a and b is the –the workdone to move a 1C chargefrom a to b • In 1D, Work = -Fd • In 3D, Work = -F.dl • Force = F = Q´E =+1´E = E • Work done = -E.dl • Total Work done = -òabE.dl Alan Murray – University of Edinburgh

9. E E E E E E E E E E E E E dl dl dl dl dl dl dl dl dl dl dl dl dl Line integral …revision E Alan Murray – University of Edinburgh

10. Potential Difference = -òabE.dl • òabis a line integral • In general mathematics, the value of a line integral depends upon the path dl takes from a to b • In this potential calculation, the path does not matter • So : choose a “convenient” path Alan Murray – University of Edinburgh

11. 1C Q E Potential Difference : Worked Example – Point charge Q (b) Place a 1C charge at (a) Move it to (b) Work done in this movementis the potential difference(voltage) between (a) and (b) (a) Alan Murray – University of Edinburgh

12. 1C (a) E E dr dl E Potential Difference : Worked Example – Point charge Q Choose this path From (a)-(c), no workis done (ra=rc) E and dl perpendicular (b) x From (c)-(b), work isdone E and dr parallel E = Qâr 4pe0r2 E (a) (c) x Q Alan Murray – University of Edinburgh

13. Potential Difference : Worked Example – Point charge Q Correct! – if Q>0, we have moved a positive charge along a field line – work is done by the electric field, so Vab<0 Alan Murray – University of Edinburgh

14. Potential Difference (PD) in an Electric Field : Procedure • Write an expression for the electric field E • Define a path between the points whose PD you want to calculate • Select a “sensible” path • Write down -òabE.dlfor this path and field • If you have chosen a sensible path, this will not be a tricky integral Alan Murray – University of Edinburgh

15. Potential - Footnote • Once you have decided where the “zero” of potential is, all other potentials are calculated with respect to that zero • cf – choice of “zero” of height • One conventional physicist’s choice is to set point (a) to¥ • Vab is then the energy expended in moving a 1C charge from ¥ to point b, and Vb is then the absolute potential at point b Alan Murray – University of Edinburgh