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1 Part I: Solve the following 10 problems 1. An electron escapes from the surface of a conducting sphere with radius 5 cm and charge -5 µC. What is the highest acceleration which the electron experiences? (2 points) F= a= 2. kQe R2 F = 3.2 ×1018 ms − 2 m What is the electric flux through the top face of the cube shown? The side of cube is a = 15 cm and the electric field, in N/C, is E = −500iˆ + 700 ˆj − 200 kˆ . (2 points) ( = ) φ = ∫ E . dA = ∫ − 500 iˆ + 700 ˆj − 200 kˆ ⋅ kˆ dA z top face y = − 200 a 2 = − 4.5 N ⋅ m 2 / C a x 1 2 3. In the figure below, the ring with radius R = 6 cm is in the x-y plane and is centered at the origin. The ring carries a uniformly distributed charge of +2 nC. An electron is released from rest along the z-axis at z = 8 cm. What is the kinetic energy of the electron, in eV, as it passes through the origin? (3 points) z y R ∆K = − ∆U → kQ kQ K =e − R R2 + z2 ( as Kin=0 ) x =1.92 ×10 −17 J 1 2 K =120 eV 4. The electric potential V in a region of space is given by V = x2 – 3 y2 + z2 where V is in volts and x, y and z are in meters. What is the magnitude of the electric field at a point with coordinates x = y = z = 1 m? (3 points) Ei = − ∂V so ∂xi ( E = − 2 x iˆ − 6 y ˆj + 2 z kˆ ) E ( 1,1,1) = − 2 iˆ + 6 ˆj − 2 kˆ E = 2 2 + 6 2 + 2 2 N / C = 6.6 N / C 5. The capacitors C1 = 3.00 µF, C2 = 6.00 µF and C3 = 4.00 µF are fully charged; the capacitor C2 has a plate charge of 18.0 µC. What is the voltage across the capacitor C3? (3 points) 2 3 C1 C2 Q2 =18 µC →V2 = V1 = 3V and Q1 = 9 µC C3 Q3 = Q12 = 27 µC V3 = Q3 C 3 = 6.75V 6. In the circuit below, ε1 = 24.0 V, R = 12.0 Ω and r = 4.00 Ω. What is ε2 if I1 = 3.50 A? (3 points) ε From upper loop ε 1 −12 I 2 = 0 → I 2 = 2 A I1 Junction rule I3 I 3 = I 1 − I 2 → I 3 = 1.5 A From big loop 1 I2 R r ε 2 ε 1 − ε 2 − rI 3 = 0 → ε 2 =18V 7. A source of emf, a 10-kΩ resistor and 8-µF capacitor are connected in series. How long does it take the energy stored in the capacitor to reach 80% of its final value? (4 points) 3 4 −t q = Cε 1 − e τ → U =U f −t 0.8U f = U f 1 − e τ −t 1 − e τ 2 2 and t = 2.25τ = 2.25 RC = 0.18 s 8. A proton enters a region of uniform magnetic field (B = 0.5 T) with an initial velocity (v = 105 m/s) which makes an angle θ = 20° with the field. What is the pitch of the helical path of the proton? (3 points) v θ B v11 = v cos 20° = 9.4 ×10 4 m / s T= 2πm = 1.31×10 −7 s eB and p = T v11 = 0.012 m 9. A wire bent as shown carries current I = 20 A perpendicular to the magnetic field B = 0.3 T of a solenoid. If R = 12 cm and θ = 120°, what B is the magnitude of the net R magnetic force exerted on the wire? (3 points) . R θ 4 I 5 F = I L' B with L' = 2 R sin 60° = 0.208 m F =1.25 N 10. A pair of point charges q1 = 5 µC and q2 = -5 µC are moving with identical speeds v1 = v2 = 105 m/s in the directions shown. When the charges are at the locations shown what are the magnitude and direction of the net magnetic field produced at the origin? (4 points) y q1 v1 v2 0.2 m 0.4 m B1 = q2 x µ o q1 v1 = 1.25 µT 4π ( 0.2 ) 2 B2 = µ o q2 v2 = 0.313 µT 4π ( 0.4 ) 2 Direction for both: − k̂ so Bnet = − 1.56 µT kˆ Part II: Conceptual Questions Tick the most appropriate answer (each question carries 1 point) 1. Point charges +4q and -2q are held in the X-Y plane as shown. A free charge Q with coordinates xo and yo is in the same plane and in equilibrium. Then 5 6 a) xo < 0 and yo = 0 b) 0 < xo < a and yo = 0 c) xo > a and yo = 0 d) yo > a and xo = 0 2. y -2q +4q x a A positive charge is distributed uniformly within a non-conducting spherical object. If the magnitude of the electric field and the electric potential (with respect to infinity) at the center of the object are denoted by E and V, respectively, then a) E ≠ 0 and V = 0 b) E ≠ 0 and V > 0 c) E = 0 and V = 0 d) E = 0 and V > 0 3. Four point charges are held as shown. A, B and C are the mid points on three sides of the square and D is the center point. A charge Q can be moved with constant speed from one of these points to the other one while the net work performed is zero. These two points are A -q a) A and D b) A and C c) C and D d) D and B 4. D +q -q C B +q Cylindrical wires 1 and 2 shown below are made of the same material and have the same length L. If I1 = 2I2, then: a) The electric fields in wires 1 and 2 are equal. wire 2 b) The current densities in wires 1 and 2 are equal. 2r c) The charge carrier concentrations in wires 1 and 2 are equal. d) The drift velocities in wires 1 and 2 are equal. r wire 1 I2 I1 5. In the single loop circuit below, source of emf S is connected to load D by wires bc and ad. The conventional current I is shown. The electric potential V around the circuit is such that: I a) Vb = Vc > Va = Vd. c b b) Vd > Va > Vb > Vc. S D c) Va = Vb = Vc = Vd. d) Vc > Vb > Va > Vd. d a 6. A current I flows from b to a through the real source of emf shown below. b 2 a) Energy is transferred to the charge at the rate εI + rI . b) Energy is transferred from the charge at the rate εI + rI2. c) Energy is transferred to the charge at the rate εI − rI2. d) Energy is transferred from the charge at the rate εI − rI2. ε r 6 a 7 7. The figure shows the paths of two charged particles A and B in a mass spectrometer. The semicircular paths have radii RA and RB (RA = 2RB). The mass and charge of the particles are identical. The transit times of the particles in the mass spectrometer are denoted by tA and tB. a) tA > tB b) tA < tB c) tA = 2tB d) tA = tB 8. A B The figure shows a long wire and a rectangular loop, both carrying current I in the directions shown. The direction of the net magnetic force acting on the loop is a) Upward b) Downward c) To the right d) To the left 9. I I Currents I1, I2 and I3 carried by three wires surrounded by the closed loop C are 1 B.dl around loop C is equal to shown below. The value of the line integral µ0 ∫ I3 I1 a) I1 - I2 + I3 b) -I1 + I2 - I3 I2 c) I1 + I2 + I3 d) -I1 - I2 - I3 C dl 10. A ring with radius R carries current I and produces a magnetic field 0.3 mT at its center. The same current I is passing in the wire shown below where the semicircular section has the same radius R. The magnetic field at point P, the center of the semicircular section, is y a) -0.30 mT k̂ x b) +0.60 mT k̂ c) +0.15 mT k̂ I . d) -1.60 mT k̂ 2R P 2R 7