Exam 2 Solutions. R 1 a. (a) [6 points] Find the voltage drop between points a and b: Vab = Va Vb. , the current i 2 is found by:

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1 PHY Exam Solutions R a + R R 3. Consider the circuit aboe. The battery supplies an EMF of 6.0 V, and the resistances of the 3 resistors are R 00Ω, R 5Ω, and R 3 75Ω (a) [6 points] Find the oltage drop between points a and b: Vab Va Vb. Let s first find the equialent resistance of R and R 3 : + + R R R R.75Ω The total resistance of the circuit is thus Rtot R + R.75Ω The current in the circuit, by Ohm s Law, is: i A Rtot So the oltage drop is Vab ir ( )(.75) 0.95 V (b) [6 points] Find the current through resistor R. Knowing the oltage drop Vab, the current i is found by: V 0.95 V i ab 0.03 A R 5Ω b (c) [6 points] Find the total power dissipated in the circuit that must be proided by the battery. ( ) ( ) P i R tot A.75Ω Watts Page of 9

2 PHY (d) [ points] Suppose an identical circuit (with the same resistors and battery configuration) is connected to the aboe circuit at points a and b. That is, a conductor connects point a of the first circuit to point a of the second circuit, and another conductor connects point b of the first circuit to point b of the second. Calculate the oltage drop between points a and b for this combined circuit. (Hint: apply Kirchoff s laws to each loop of the circuit.) i R a a R i + i 3 R R 3 R 3 R + First find the ectie resistance of all 4 middle resistors in parallel: + + R R R R 9.375Ω Let i be the current leaing the left battery, i that leaing the second, and i 3 being that traeling down the center branch. By Kirchoff s First Law: i3 i+ i By Kirchoff s Second Law applied to each loop: ir + ir 0 ir + ir + ir 0 3 ir+ ir 0 ir+ ir + ir 0 3 These two equations can only be true if i i, as you might expect from symmetry. Soling for i gies: ir + ir 0 i R + R Now i i V 3 i R R ab 3 R+ R b b 0.95 V as before (Exactly as you would expect connecting two batteries of same oltage in parallel.) Page of 9

3 PHY R a b + C C S 3. Consider the circuit aboe. A battery supplies an EMF of.0 V, the resistance R.5 MΩ, and the capacitors hae capacitance C µ F and C 9µ F. Consider that 6-6 switch S is closed at time t 0. ( MΩ 0 Ω, µ F 0 F ). (a) [6 points] What is the time constant of the circuit that goerns how fast the current changes after the switch is closed? The RC time constant goerns how fast the current changes and the capacitors charge. The ectie capacitance of two capacitors in series is: + C 6µ F C C C C s τ RC (b) [6 points] What is the maximum charge attained on the top plate of capacitor C (i.e. at point a) after the switch is closed? The maximum oltage drop across the pair of capacitors will be when the current through the circuit stops, which implies 0 oltage drop across the resistor. Thus, q Vab ( t ) V C max ( )( ) max q V F C The same q is stored on each capacitor plate, and it is that for the equialent capacitance of capacitors in series. Page 3 of 9

4 PHY (c) [6 points] Sketch the oltage drop Vab Va Vb across the capacitors as a function of time t. Try to be accurate in your sketch. Page 4 of 9

5 PHY A new type of cannon can launch 5 kg projectiles at the high elocity of 0.3c (a) [6 points] What is the minimum amount of energy required to accelerate a projectile from rest to this final elocity? The energy required is the work done to change the kinetic energy of the mass: W K γ mc ( ) γ.043 / c ( )( ) 6 W kg 3 0 m/s.7 0 J (b) [6 points] If the cannon is mounted on the front of a spaceship traeling away from Earth at a elocity of 0.6c, what will be the elocity of the launched projectile in the reference frame of the Earth? Use the Lorentz transformation of elocity: ux + ux + u / c u x x 03. c+ 06. c a fa f c. 9 0 m/s Page 5 of 9

6 PHY A kaon is an unstable particle with a rest mass energy of 500 MeV and an aerage lifetime of τ 5. 0 s in its rest frame. Consider a beam of kaons, where each kaon 3 has a momentum of 000 MeV/c as measured in a laboratory. ( MeV.6 0 J ) (a) [6 points] What is the elocity of each kaon in the laboratory? p γm γmc c pc γ mc c 4 pc mc c c p c c m c c pc c mc pc + mc c.6 0 m/s (b) [6 points] What is the aerage distance the kaons trael at this momentum in the laboratory before decaying? γ c ( )( )( )( ) d t γτ m Page 6 of 9

7 PHY A long straight wire has an electric charge density of λ 7.0 µc / m along its length, where µ C 0 6 C, as measured in the rest frame of the wire. (a) [6 points] What is the obsered charge density of the wire as measured by an obserer traeling at a elocity of.4 0 m/s parallel to the direction of the wire? Length is contracted along the direction of motion: L, but the electric charge is γ inariant in some region. So the charge density per unit length increases: q q λ γ γλ0 L L 0 5 γ c λ µ / m.67 µ C/ m ( C ) L 0 (b) [6 points] What is the obsered charge density of the wire as measured by an obserer traeling at a elocity of.4 0 m/s perpendicular to the direction of the wire? Dimensions perpendicular to the direction of trael are unaffected by Lorentz Transformations, so the charge density remains the same. Page 7 of 9

8 PHY y E e- x [ points] A beam of electrons ( cathode rays ) is sent between two parallel electric plates separated by a distance d.5 cm with a potential difference of 300 V between them. The electric field points in the +y ˆ direction. If the electron beam traels perpendicular to the electric field in the +xˆ direction, and each electron has a kinetic energy of 5,000 ev, what magnetic field is necessary (direction and magnitude) so that the electrons continue traeling in a straight line without deflection by the electric field? The charge of the electron is 9 q e.6 0 C, the electron rest-mass energy is 5,000 ev, and 9 ev.6 0 J. Balance forces: F q( E+ B) 0 E+ B 0 E B zˆ V 300 V 4 E 0 N/C d 0.05 m K m (check that it is non-relatiistic!) ( ) K 5000 ev c ( ) c mc 5,000 ev m/s m/s (yes, it's non-rel.) 4 0 N/C V B 0.4 mt (0.4 Gauss) d Page of 9

9 PHY i A d C i B 7. Consider two parallel wires, A and B, separated by a distance d 0.5 cm as shown aboe. The current of wire A is i A 9 A, and the current of wire B is i B 3 A, both in the ˆx direction. (a) [6 points] Calculate the direction and magnitude of the magnetic field at point C, midway between the two wires in the plane of the wires. Each wire generates a magnetic field: ki B ˆ wire r r At point C, the field from A points into the plane of the paper, and the field from B out of the plane. The field from A in stronger, so the net field points into the plane of the paper: B B + B A B B k ( d /) ( i i ) A ( T m /A)( 9 3 A) B m T (b) [6 points] Find the direction and magnitude of the force per unit length (per meter) on wire B arising from the interaction of its current with the magnetic field of wire A. F i L B The force is attractie between the two wires with currents in the same direction. Per unit length, the force is: kiab i F ibba d F N ( 0 7 T m / A)( 9 A)( 3 A) m Page 9 of 9

Exam 2 Solutions. ε 3. ε 1. Problem 1

Exam 2 Solutions. ε 3. ε 1. Problem 1 Exam 2 Solutions Problem 1 In the circuit shown, R1=100 Ω, R2=25 Ω, and the ideal batteries have EMFs of ε1 = 6.0 V, ε2 = 3.0 V, and ε3 = 1.5 V. What is the magnitude of the current flowing through resistor