Physics 1 K Quiz # Solutions Summer 016 I. (18 points) A positron has the same mass as an electron, but has opposite charge. Consier a positron an an electron at rest, separate by a istance = 1.0 nm. What minimum velocity magnitue v 0 coul be given to each particle, in opposite irections, so they move apart from each other an never return? If there is no such velocity magnitue because the particles will return for any v 0, prove it............ Use the Work-Energy Theorem W ext = K + U + E th Choose a system consisting of the two particles. Once the initial velocity has been imparte, no external forces o work on the system. No non-conservative internal forces change the thermal energy of the system. The kinetic energy of the system is the sum of the kinetic energies of the two particles. The potential energy change of the system is ue to the work one by the internal conservative electric force. 0 = ( 1 m 1v1f 1 m 1v1i) ( + 1 m vf 1 m vi ) + (K q 1q K q ) 1q + 0 r f r i Note that the masses of the two particles are the same (m 1 = m = m e ). If the particles are never to return, they must reach infinite separation (r f ) with spees of at least zero. Solve for v 0. 0 = ( ) 0 1 m ev ) ( 0 + 0 1 m ev ) (+e) ( e) 0 + (0 K ( 1 m ev ) e 0 = K K v 0 = e m e = ( 1.60 10 19 C ) 8.988 10 9 N m /C (9.109 10 31 kg) (1.0 10 9 m) = 5.0 10 5 m/s Quiz # Solutions Page 1 of 7
II. (16 points) All capacitors in this circuit have ientical capacitances C. The battery has a potential ifference V = E between its terminals. What is the energy store (with respect to zero in the uncharge state) in the capacitor marke with an asterisk? Express your answer in terms of parameters efine in the problem, an physical or mathematical constants.......... Fin equivalent capacitances for combinations of capacitors until either the potential across or charge on the inicate capacitor has been etermine. The capacitors have been numbere for convenience. Capacitors 4 an 5 are in series. ( 1 C 45 = + 1 C 4 C 5 = ) 1 ( 1 C + 1 ) 1 ( ) 1 = = C/ C C Capacitors 3 an 45 are in parallel. C 345 = C 3 + C 45 = C + C/ = 3C/ Capacitors an 345 are in series. C 345 = = ( 1 + 1 ) 1 ( 1 = C C 345 C + ) 1 3C ( ) 1 5 = 3C/5 3C The potential across capacitor 345 is the emf of the battery, E, so from the efinition of capacitance the charge on capacitor 345 is Q 345 = C 345 V 345 = ( ) 3C E = 3CE/5 5 As capacitor is in series with capacitor 345, it must carry this same charge. Q = Q345 = Q 345 = 3CE/5 So the energy store in capacitor (the asteriske capacitor) is U = Q = (3CE/5) = 9 C C 50 CE 1. (5 points) In the problem above, let the energy you foun store in the asteriske capacitor be U 0. If the battery were replace with one that ha a potential ifference E between its terminal, what energy U woul now be store that capacitor? The energy in the asteriske capacitor must be proportional to E for the imensions to be correct. U = 4U 0 Quiz # Solutions Page of 7
III. (16 points) A ro of length L an cross-sectional area A is bent into a semi-circle about the origin. The ro has non-uniform volume charge ensity ρ = ρ 0 θ where ρ 0 is a positive constant, an θ is measure in raians from the +x axis towar the +y axis as shown. What is the electric potential at the origin, ue to the ro, with respect to zero at infinity? Express your answer in terms of parameters efine in the problem, an physical or mathematical constants.................. Divie the ro into point-like bits, each with charge q an volume V ol. This volume is the cross-sectional area A times a bit of arc length s. then from the efinition of the raian, s = θ. A up the contribution ue to all the bits of charge (that is, integrate). V = V = θ = KAρ 0 K q ρ r = K Vol = K ρ 0 θa s = KAρ 0 3π/ π/ = KAρ 0 [ (3π ) ] ( π ) = KAρ 0 3π/ π/ [( 9π 4 θ θ = KAρ 0 ) ( )] π 4 3π/ π/ = KAρ 0 θ θ [ ] 8π = KAρ 0 π 4. (5 points) What is the irection of the electric potential at the origin? Electric potential is a scalar! The electric potential at the origin has no irection. Quiz # Solutions Page 3 of 7
3. (5 points) A uniformly charge ring has raius an total charge Q. With respect to zero at infinite separation, what is the electric potential energy U of a system consisting of this ring an a q point charge at its center?.................. The entire charge Q on the ring is the same istance from the center. As potential is a scalar, the potential at the center of the ring is the same as it woul be if that entire charge Q were concentrate at a point. Therefore, the potential energy of this system consisting of a ring an point is the same as that of a system consisting of two points. U = K q 1q r = K Qq 4. (5 points) Current flows to the right through the wire shown. Segment b on the right has three times the iameter of segment a on the left. Segment b on the right has one half the conuctivity of segment a on the left. Compare the electric fiel magnitue in segment b with that in segment a................ The current must be the same in each segment, an can be relate to the current ensity in each segment I a = I b J a A a = J b A b Current ensity can be relate to electric fiel. ( π ) ( π ) (σ a E a ) 4 a = (σ b E b ) 4 b Solve for E b. E b = σ a σ b ( a b ) E a = σ ( ) a a 1 σ E a = a 3 a 9 E a Quiz # Solutions Page 4 of 7
5. (5 points) A one-imensional electric fiel points only in the ±x irection, with values shown in the graph. At which point in the range 0 m to 1 m oes the electric potential have its maximum value?............. Electric fiel an potential ifference are relate by V = E s so the ifference in potential is represente by the opposite of the area uner the curve on the graph. The area from 0 m to 9 m is positive. In this range, then, the potential has ecrease from the value at 0 m. The potential increases between 9 m an 1 m, where the area uner the curve is negative. However, this area has less magnitue than the area from 0 m to 9 m. By 1 m, the potential has still not increase enough to return to its value at 0 m. The maximum value, then, is foun at 0 m 6. (5 points) A one-imensional electric fiel points only in the ±x irection. The electric potential associate with the fiel is shown in the graph. At which of the inicate points oes the electric fiel point in the negative irection with greatest magnitue (that is, have its most negative value)?............. Electric fiel an potential are relate by E x = δv δx so the electric fiel will have its most negative value at the point where the graph has its most positive slope, Point ii. Quiz # Solutions Page 5 of 7
7. (5 points) Four ientical light bulbs are connecte to two ientical batteries, as shown. Which bulb(s) is brightest? The entire potential ifference create by the battery is across each of the bulbs iii an iv. However, that potential is across the two bulbs i an ii together, so only half the potential is across each of those bulbs. With more potential across a bulb, more current will flow through it, an it will be brighter. Bulbs iii an iv are equally bright, an brighter than bulbs i an ii. 8. (5 points) Four wires are all mae of the same material, but have ifferent imensions, as shown. ank the resistances between the ens of the four wires from greatest to least. esistance an resistivity are relate by = ρ l A so i = ρ L πr meaning ii = ρ L π (r) = 1 4 i iii = ρ iv > i > iii > ii L π (r) = 1 i iv = ρ L π (r) = i Quiz # Solutions Page 6 of 7
9. (5 points) A parallel plate capacitor is charge with a battery. This results in energy U 0 being store in the capacitor (with respect to zero in the uncharge state). The battery is isconnecte from the capacitor an insulating hanles are use to push the plates closer together, until they have half their original spacing. How much energy U is store in the capacitor, now? If the battery is isconnecte from the capacitor, the charge on the plates cannot change. emembering how the capacitance of a parallel plate capacitor epens on geometry, U 0 = Q C 0 = Q ϵ 0 A/ = Q ϵ 0 A So U = Q C = Q ϵ 0 A/ (/) = Q / ϵ 0 A = 1 [ Q ] ϵ 0 A U = U 0 / 10. (5 points) A parallel plate capacitor is charge with a battery. This results in energy U 0 being store in the capacitor (with respect to zero in the uncharge state). The battery remains connecte to the capacitor an insulating hanles are use to push the plates closer together, until they have half their original spacing. How much energy U is store in the capacitor, now? If the battery remains connecte to the capacitor, the potential ifference between the plates cannot change. emembering how the capacitance of a parallel plate capacitor epens on geometry, U 0 = 1 C 0 ( V ) = 1 ϵ A 0 ( V ) So [ ] U = 1 C ( V ) = 1 ϵ A 0 / ( V ) 1 = ϵ A 0 ( V ) U = U 0 Quiz # Solutions Page 7 of 7