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1 Umeå Universitet, Fysik 1 Vitaly Bychkov Prov i fysik, Electricity and Waves, , kl Hjälpmedel: Students can use any book. Define the notations you are using properly. Present your arguments in details. Good luck! 1) Consider a sphere of radius R with charge distributed as 0 r / R. Find electric potential in the centre of the sphere with respect to infinity. 2) Consider a cylindrical conductor consisting of two co-axial metal cylinders of resistivities 1 (at a r d ) and 2 (at Find the resistance between the surfaces d r b) and length as shown in Fig. 1. r a and r b. b d a Fig. 1.

2 Umeå Universitet, Fysik 2 Vitaly Bychkov I z y x D Fig. 2 3) An infinitely long metal plate of width D (shown in Fig. 2) carries a current I. The current is distributed over the plate as K K x / D. Here K is the surface current 0 density, with di Kdx. Find magnetic field B at the point (; 0; 0). 4) Consider the circuit of alternating current shown in Fig. 3. Find impedance. C R Fig. 3. 5) A source of light of frequency f is placed at a distance behind an obstacle with two point-holes as shown in Fig. 4. The source is placed at the same height as the lower hole. The upper hole is at the height d. The resulting picture is observed on a screen at the same distance from the obstacle. Find position of the first interference maximum along the y-axis. Find the distance y to the second maximum. To simplify the calculations

3 Umeå Universitet, Fysik 3 Vitaly Bychkov you may use the condition with d and the relation / 2 for 1. y d x Fig. 4.

4 Umeå Universitet, Fysik 4 Vitaly Bychkov Prov i fysik, Electricity and Waves, , kl Hjälpmedel: Students can use any book. Define the notations you are using properly. Present your arguments in details. Good luck! 1) Consider a charged wire of the shape shown in Fig. 1 with uniform linear charge density. Find the electric field E in the coordinate origin. b R a y b a a b x Fig. 1. Fig. 2. 2) Consider a cylindrical capacitor consisting of two co-axial metal cylinders of radii a and b and length. In between them, at radii a r R, there is a layer of dielectric with the dielectric constant as shown in Fig. 2. Find the capacitance of the system. 3) Consider an infinitely long cylinder of radius R with current along the cylinder axis distributed as j j r / R. The distribution is rotationally symmetric. Find magnetic field 0 both inside and outside the cylinder. (4 p)

5 Umeå Universitet, Fysik 5 Vitaly Bychkov 4) Consider the circuit of alternating current shown in Fig. 3. Find impedance. C R Fig. 3. 5) A fisherman (of height h ) stands by a lake of depth d. At the bottom of the lake, the fisherman sees a fish at the distance the fish, f? The refraction coefficient of water is n. I, as shown in Fig. 4. What is the real distance to h d f I Fig. 4.

6 Umeå Universitet, Fysik 6 Vitaly Bychkov Prov i fysik, Elektricity and Waves, Alexandr Talyzin, Umeå University. Define your notations clearly and explicitly. Your problem solutions must include the detailed steps (not just the final result). Though you can use the textbook, you cannot refer to intermediate results of the book when presenting your solutions. Handbooks allowed but not lecture notes. Solution of any problem should start from basic equations. Good luck! 1. Figure 1 shows three infinitely long wires with currents I 1, I 2 and I 3. Distance AB=BC= 5 cm Currents I 1 =I 2 =I and direction is away from observer (as noted in figure), current I 3 =2I and direction is towards observer. Find a point on the line AC for which the magnetic field B is equal to zero. Figure 1.

7 Umeå Universitet, Fysik 7 Vitaly Bychkov 2. Uniformly charged line with charge density λ has configurations shown in Figure 2 R is radius of arc and R is considered much smaller compared to the length of the line. Find modulus of electric field vector E in the point O shown in Figure2. Figure 2 3. ong straight wire with current I is in the same plane with contour which consists of wire with rails and sliding bar (see figure 3). The length of sliding bar is and resistance is R. The bar moves away from wire with current with velocity υ. Find current induced in the contour as a function of distance r between wire and sliding bar. Self-induction of contour and resistance of other wires in the contour can be neglected. Figure 3.

8 Umeå Universitet, Fysik 8 Vitaly Bychkov 4. Converging lens with f 1 = 15 cm is 40 cm in front of diverging lens with f 2 =-8 cm Draw rays diagrams, locate final image and find transverse magnification for object placed in front of first lens on the distance p 1 =30 cm 5. Calculate total impedance and phase angle for alternating current contour shown in Figure 4. Parameters of contour elements are: C=0.5 µf, R=10 kω, =10H. Current parameters f=60 Hz and V=120 V. Figure 4.

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