Electric Potential & Potential Energy I) ELECTRIC POTENTIAL ENERGY of a POINT CHARGE Okay, remember from your Mechanics: Potential Energy (U) is gaine when you o work against a fiel (like lifting a weight, for example). Work is (+) whenever U is lost an K is gaine (remember, W = K). As a result, U is gaine whenever work is (-) (i.e., whenever, K is lost). More remembering: (a) The expression for electrostatic force, aka Coulomb's Law, is given by F el = kqq'/r 2 (b) Since this is a force that varies with position, the work one by this force in moving a point charge q from to a istance r away from a charge q is given by (compliments of the Work/Energy & Universal Gravitation Notes): W = kqq'/x 2 x U el = -kqq' x/x 2 r U el = kqq'/r Base on the expression above for U el, we can efine the electric potential energy (Uel) relative to infinity between two point charges q an q' that are separate by a istance r as U el = kqq'/r 2.1 - For the charge configuration shown below, fin the total electric potential energy. Assume q = 1.0 x 10-7 C an a = 10 cm. -4q a a +q a +2q
II) ELECTRIC POTENTIAL (V) at a point, POTENTIAL DIFFERENCE, & WORK Electric Potential Due to a Point Charge Electric potential represents the number of Joules of electric potential energy (U el ) per Coulomb of point charge q that is in an electric fiel cause by a charge q'. Therefore, by using the expression for U el, we get V = kq'/r (where q' causes an electric fiel at a istance r away) The unit for potential, V, is the Volt: 1 Volt = 1Joule/1 Coul There are 2 qualitative relationships between potential V an electric fiel, E: (1) The value for V ecreases if you move in the irection of the electric fiel lines. + - Here, V becomes less (+) as you move out along the fiel lines. Becoming less (+) is a ecrease. Here, V becomes more (-) as you move in along the fiel lines. Becoming more (-) is a ecrease. (2) The lines of equipotential are always perpenicular to the electric fiel lines + - The ashe lines are lines of equipotential. At any point along these lines, the value for V is the same. An, the electric fiel, as you can see, is always to the equipotential lines. This is true at any point, regarless of the shape of the fiel.
Potential Difference (Voltage) A given value for V, by itself, oesn t tell us much it s more like a reference point. We nee it to calculate something that s of greater use the potential ifference. The potential ifference ( V), or voltage, in moving charge q from point A to point B while it is uner the influence of charge(s) q' is foun by V = V AB = V B - V A (Remember, the unit is Volts) Change in Potential Energy: The External Work Done in Moving a Charge Potential ifference can be a slippery slope since you can have a loss of potential for a charge that gains potential energy, an a gain of potential for a charge that loses potential energy. This is where the external work one in moving a charge comes in. The external work one in moving a charge is like the work one by an applie force in lifting or lowering a weight: it represents the change in its potential energy. It s (-) when it s move the way it wants to go naturally, an (+) when it s move the way it oesn t want to go naturally. It involves both the sign of the charge being move an the sign of the potential ifference. When we combine these two factors, we get a result (for external work) that makes sense. The external work one in moving a charge q from point A to point B is foun by: See if these situations makes sense : W = q( V) = U + V is (-), W is (-), U is (-) & K is (+) + E - V is (-), W is (+), U is (+) & K is (-) + - E - V is (+), W is (+), U is (+) & K is (-) V is (+), W is (-), U is (-) & K is (+) What about if you re aske to fin the work one by the electric fiel? It s just the opposite of the external work. So, if the external work ( U) is -5 J, then the work one by the fiel is + 5 J.
2.2 - For the charge istribution shown below, fin: (a) the work one in moving the +2 C charge from to the position below (b) the work one by the fiel +2 C 10 cm 10 cm +1 C +1 C 10 cm 2.3 - For the charge istribution shown, Fin (a) the potential at point A (b) the work one in moving a charge of q = 1 C from point A to point B, locate miway between the charges. A 3 cm 4 cm +4 C -6 C B
III) ELECTRIC KINETIC ENERGY When a charge is release in an electric fiel, it unergoes a change in its potential an therefore a change in its potential energy (remember the gravity analog... it's like ropping a ball or tossing it up in the air; there's a change in U g ). This change in U el "turns into" a change in K (remember the Conservation of Energy a loss of U el becomes an equal gain in K). Therefore, use the Conservation of Energy to fin the spee of a charge particle that unergoes a change in its electric potential energy. Since Then U el = qv (from Section II) U el = q V K = q V or (1/2)mv f 2 (1/2)mv 0 2 = q V In using this relationship, remember that if q V is (+), then K will be (-); if q V is (-), then K will be (+). A gain in potential energy correspons to a loss of kinetic energy, an vice-versa. You make the sign ajustment. If you on t like worrying about signs, you can use conservation of energy In particular U0 + K0 = Uf + Kf qv0 + K0 = qvf + Kf 2.4 2.2 revisite Suppose the + 2 C is release from rest. (a) Describe its subsequent motion (the AP folks like to ask this ) (b) Fin its spee as r (i.e., when it s really far away).
IV) CONNECTING the FIELD E to the POTENTIAL V So far, we ve ealt only with point charges. But if the charges are istribute over a surface or shape, then we must integrate to fin V or V. Since U = - F(x) x, then U/q = V = - (F/q) x = - E x. Possibility 1: The electric fiel is uniform (this inclues the sheet/plate of charge case) V = - E x = Ex V = E 0 0 2.5 - A positive charge +q moves (as shown) through a isplacement from point A to point B in a uniform electric fiel E. Fin its potential ifference an the work one by the fiel. E A +q B These are equipotential lines. The value for V is the same at any point along them, AND they re perpenicular to the electric fiel lines. 2.6 - Repeat the previous problem, but consier a negative charge, -q. E A -q B 2.7 - Fin the potential ifference in moving a (+) test charge a istance "" from left to right between the oppositely charge parallel plates, as shown below. Both plates have a charge istribution. (Remember, from Gauss' Law, that fiel E = for a charge conucting plates) E +q + -