Total: 120%
1. A positron (+e) with kinetic energy 2.00 keV is projected into a uniform field of magnitude 0.100 T. The angle q between it velocity and is 45o. Find (a) the radius r (4%), (b) the period T (4%), and (c) the pitch p (4%) of its helical path.
2. Two infinitely long wires carry equal currents i, as shown in Fig. 1. The arcs on the wires are 90o arc of a circle with radius R. Find the field at the center of the circle. (8%)
3. In Fig. 2, a long rectangular conducting loop of width L, resistance R, and mass m, is hung in a uniform field (into the page) which exists above line aa. If the loop is dropped from rest, find its terminal speed vt. (12%)
4. As shown in Fig. 3, a coil with N2 turns is wrapped around a toroid with N1 turns. The toroid has a rectangular cross section. The inner radius is a, the outer radius is b, and the height is h. (a) Show that the mutual inductance is: (10%)
(b) if the current in toroid changes according to I=I0cos(wt), what is the induced emf in the coil? (4%)
5. Write down Maxwell’s equations in free space, including their names of laws, and explain the physical meaning of each equation. (12%)
6. Fig. 4 shows a circular region of radius R=4.00 cm in which a uniform displacement current id=1.00 A is out of the page. What is the magnitude of the induced field at radial distances (a) 2.00 cm (4%) and (b) 6.00 cm (4%)?
7. A resistor (R=1500 W), an inductor (L=5.00 mH), and a capacitor (C=1.00 mF) are connected in series with and AC source, V(t)=(218.5 V)sin(628t). Suppose the current I(t)=Imsin(628t-f). Calculate (a) total impedance (4%), (b) rms (root-mean-square) current delivered by the source (4%), and (c) phase angle between the current and voltages. Is the current leading or lagging behind the voltage? (4%) (d) What is the average power delivered in the circuit? (4%) (e) If the RLC circuit is in resonance, what is the resonant angular frequency? (4%)
8. An airplane is flying at a distance of 2.0 km from a radio transmitter, and it receives a radio signal. The intensity of the received signal is 4 mW/m2. (a) What is the amplitude of the component and component (3%) of the signal at the airplane? (b) What is the total energy density of the signal at the airplane (3%) (c) If the radio transmitter radiates uniformly over a sphere, what is the transmission power? (3%)
9. In Fig. 5, unpolarized light is sent into a system of three polarizing sheets, which transmits 0.0800 of the initial light intensity. The polarizing direction of the first and third sheets are q1=90o and q3=0o. What are the possible values of the polarizing direction of sheet 2? (8%)
10. (a) Define what are “proper time” and “proper length”? (6%)
(b) A father is 20.00 y (year) older than his daughter. In his frame, he traveled outward at a constant speed v from Earth for 2.000 y and then back to Earth for another 2.000 y. Finally, he is 20.00 y younger than his daughter. What is the value of the constant speed v? (6%)
(c) In (b), both the father and the daughter might think the other one is more aged than him(or her)self because of time dilation. What caused the paradox? (5%)