Çankaya University Department of Mathematics and Computer Sciences 2010-2011 Spring Semester PHYS 112 - General Physics for Engineering II FIRST MIDTERM 1) Two fixed particles of charges q 1 = 1.0µC and q 2 = 3.0µC are 10 cm apart. Where should a third charge be located so that no net electrostatic force acts on it? 2) A uniformly charged plastic disk of radius R carries surface charge density σ. At what distance z along the central axis is the magnitude of the electric field equal to one-half the magnitude of the field at the center of the surface of the disk? 3) a) A conducting sphere of radius 10 cm has an unknown charge. If the electric field 15 cm from the center of the sphere has the magnitude 3.0 10 3 N/C and is directed radially inward, what is the net charge on the sphere? b) Charge is distributed uniformly throughout the volume of an infinitely long cylinder of radius R. The volume charge density is ρ. Calculate the electric field at a distance r from the cylinder axis for r < R. 4) A rod of length L lies on x axis as shown in figure. Its linear charge density is λ. Find the electric potential at the point (0, Y ) as an integral. Do not evaluate the integral. y (0, Y ) L 4 λ + + + + + + + + + 3L 4 x
5) In the following circuit, C 1 = 1µF, C 2 = 6µF, C 3 = 4µF, C 4 = 4µF and the potential supplied by the battery is V = 60V. a) Find the energy stored on C 1 b) A dielectric material with dielectric constant κ = 6 is inserted between the plates of C 1. Find the charge stored on C 1 after insertion. (Battery is connected during and after insertion) C 1 C 2 C 3 C 4 V
Answers 1) Consider a line that connects q 1 and q 2 which are d = 10cm apart. Clearly, the third particle must be placed on this line to the left of q 1 a distance x. At this point, the electric field will be: E = 1 4πε 0 q 1 x 2 1 4πε 0 q 2 (x + d) 2 = 0 q 1 q 2 = 1 3 = x 2 (x + d) 2 1 3 = x 2 (x + d) 2 x x + 10 x = 10 3 1 cm = 13.66 cm 2) Electric field at the surface can be found by inserting z = 0 in the formula: E(z) = 1 2 E(0) σ 2ε 0 1 ( ) z 1 = σ z2 + R 2 4ε 0 z z2 + R 2 = 1 2 z z2 + R 2 = 1 2 4z 2 = z 2 + R 2 z = R 3 3) a) Outside the sphere, the charges act as if they are concentrated at the center. E = Q 4πε 0 r 2 Q = 4πε 0 r 2 E Q = 3 10 3 N/C 4π(0.15m) 2 8.85 10 12 C 2 /Nm 2 Q = 7.5 10 9 C = 7.5nC b) Φ = q enc ε 0 2πrLE = πr2 Lρ ε 0 E = rρ 2ε 0
4) dq = λ dx, dv = dq 4πε 0 x2 + Y, V = 2 dv V = λ 4πε 0 3L/4 L/4 dx x2 + Y 2 5) a) C 13 = 5µF, C 24 = 10µF 5V 1 = 10V 2 V 1 = 40V U = 1 2 CV 2 = 1 2 1µF (40V )2 = 800µJ b) C 1 = 6µF V 1 = 30V q 1 = V 1 C 1 = 30V 6µF = 180µC
Çankaya University Department of Mathematics and Computer Sciences 2010-2011 Spring Semester PHYS 112 - General Physics for Engineering II SECOND MIDTERM 1) a) (10 pts) A fuse in an electric circuit is a wire that is designed to melt, and thereby to open the circuit, if the current exceeds a predetermined value. Suppose that the material to be used in a fuse melts when the current density rises to 600 A/cm 2. What diameter of cylindrical wire should be used to make a fuse that will limit the current to 1.2 A? b) (10 pts) When 230 V is applied across a wire that is 10 m long and has a 0.4 mm radius, the current density is 6.3 10 4 A/m 2. Find the resistivity of the wire. 2) Calculate the current through each ideal battery in Figure. 5Ω 2Ω 6Ω 111V 20V 18V 4Ω 12Ω 3) a) (10 pts) A capacitor with capacitance C = 2µF and initial charge q 0 is discharged through a resistor of resistance R = 2Ω. After how much time will the charge be 2 3 q 0? b) (10 pts) A charged particle with mass m, charge q travels in a circular path in a uniform magnetic field with magnitude B. Find the frequency of motion.
4) a) (10 pts) A wire 2.5 m long carries a current of 8 A and makes an angle of 25 with a uniform magnetic field of magnitude B = 0.7 T. Calculate the magnetic force on the wire. b) (15 pts) A circular coil of 150 turns has a radius of 3 cm. Calculate the current that results in a magnetic dipole moment of 5A.m 2. 5) a) (15 pts) A square loop of wire of edge length a carries current i. Calculate the magnitude of the magnetic field at the center of the loop. B b) (10 pts) Evaluate d s for the path shown in figure. 2I 3I I I 3I
Answers 1) a) J = i A = i πr 2 i i r = R = 2 πj πj b) V = ir = i ρl A = JρL = 0.05cm = 0.5mm ρ = V JL = 3.65 10 4 Ω.m 2) Let i 1 be the current through 18 V battery and i 2 through the 20 V battery. Then, a current of i 1 + i 2 passes through the 111 V battery. 111 5(i 1 + i 2 ) 6i 1 18 4(i 1 + i 2 ) = 0 2i 2 20 12i 2 + 18 + 6i 1 = 0 i 1 = 5 A, i 2 = 2 A, i 1 + i 2 = 7 A 3) a) 2 3 q 0 = q 0 e t/rc ln 2 3 = t RC t = RC ln 3 2 = 1.62µs = 1.62 10 6 s b) r = mv qb, f = v 2πr = qb 2πm 4) a) F = BiL sin θ = 5.92N b) µ = NiA i = 5A.m 2 150π(0.03m) 2 = 11.8A 5) a) B = µ 0I 2 2πa 4 = 2 2µ0 I πa B b) ds = 3µ0 I
Çankaya University Department of Mathematics and Computer Sciences 2010-2011 Spring Semester PHYS 112 - General Physics for Engineering II FINAL 1) A parallel-plate capacitor has a capacitance of 100 pf, a plate area of 100cm 2 and a mica dielectric (κ = 5.4) filling the space between the plates. At 50 V potential difference, calculate a) (6 Pts.) the electric field magnitude E in the mica b) (7 Pts.) the magnitude of the free charge on the plates c) (7 Pts.) the magnitude of the induced surface charge on the mica. 2) (20 Pts.) In the given circuit, determine the value of R that maximizes the energy dissipation rate (power) at R. 24V 3Ω R 5Ω 3) a) (13 pts.) An AC generator is to be connected in series with an inductor of L = 2 mh and a capacitance C. You are to produce C by using capacitors of capacitance C 1 = 4 µf and C 2 = 6 µf either singly or together. What resonant frequencies can the circuit have, depending of how you use C 1 and C 2? b) (7 pts.) Average power in an AC circuit is 60W. The maximum value of current is 15 A and the maximum value of emf is 20V. Find the phase constant.
4) A circular loop of radius 2 cm is placed in a uniform magnetic field as seen in figure. The magnetic field is changed uniformly from 0.2 T to 0.8 T in a time interval of 1 s, beginning at t = 0. a) (5 pts.) Find the magnitude of magnetic field t = 0, t = 0.5 and t = 1 s. b) (10 pts.) What emf is induced in the loop at t = 0.5 s? c) (10 pts.) Find the direction of induced current in the loop. B 5) a) (10 Pts.) At what rate must the potential difference between the plates of a parallelplate capacitor with a 2 µf capacitance be changed to produce a displacement current of 1.5 A? b) (8 Pts.) Calculate the intensity of a plane traveling electromagnetic wave if B m is 2.0 10 4 T. c) (7 Pts.) 3G cellphones use electromagnetic waves of frequency 2100 MHz. What is the corresponding wavelength?
Answers 1) a) E = V d A C = κε 0 D d = κε 0A C E = V C κε 0 A = 1 104 V/m b) q = CV = 5 10 9 C c) q i = q ε 0 AE = 4.1 10 9 C=4.1 nc 2) 24 3i 5i Ri = 0 i = 24 8 + R P = i 2 R P = 242 (8 + R) 2 R dp dr = 0 R = 8 Ω 3) a) f 1 = f 2 = f 3 = f 4 = 1 2π LC 1 = 1.78 10 3 Hz 1 2π LC 2 = 1.45 10 3 Hz 1 2π LC 1 C 2 /(C 1 + C 2 ) = 2.3 103 Hz 1 2π L(C 1 + C 2 ) = 1.13 103 Hz b) P avg = E rms I rms cos φ cos φ = 4 5
4) a) B = 0.2 + 0.6t B(0) = 0.2T, B(0.5) = 0.5T, B(1) = 0.8T. b) E = dφ B = d (BA) = AdB dt dt dt = π(2 10 2 m) 2 0.6T/s = 7.54 10 4 V c) The magnetic field is into the page and increasing. By Lenz s law, induced current must decrease it. So the direction of the induced current is counter clock-wise. dφ E 5) a) i d = E 0 dt dea = E 0 dt = E 0A dv d dt = C dv dv dt = i d C = b) I = 1 cµ 0 E 2 rms dt 1.5 A 2 10 6 F = 7.5 105 V/s = 1 Em 2 cµ 0 2 = c Bm 2 µ 0 2 3 10 8 m/s (2 10 4 T ) 2 = 4π 10 7 T.m/a 2 = 4.777 10 6 W/m 2 c) λ = c f = 3 108 m/s = 0.14m 2100 10 6 s 1