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1 Umeå Universitet, Fysik Vitaly Bychkov Prov i fysik, Electrodynamics, 2013-11-01, kl 9.00-15.00 Hjälpmedel: Students may use any book(s) including the compendium. Minor notes in the books are also allowed. Good luck! 1) Consider a solenoid of length l , radius b and n N / l turns per unit length. A square loop of side a with current I I 0 exp( it ) is placed inside the solenoid as shown in Fig. 1. The loop is perpendicular to the solenoid axis. The solenoid is connected to a resistor R . Find current in the resistor, I R (t ) . For that purpose: (a) Find mutual inductance between the loop and the solenoid; (b) find self-inductance of the solenoid; (c) consider the circuit consisting of the emf due to mutual inductance, the emf due to selfinductance and the resistance. You may leave the answer in complex numbers. (4p) R b a Fig. 1. A loop in a solenoid; the solenoid is connected into a circuit with a resistor. 2) Consider an infinitely long cylinder of radius R with the magnetic field inside B aˆ B0 (r / R) 2 exp( it ) , where r is the distance from the cylinder axis and is the angle around the axis (in cylindrical coordinates). Find electric field E inside the cylinder. Assume zero electric field at the axis. (4 p) 3) An electron-positron plasma is a gas of free electrons and positrons (no ions), where a positron has the same mass m as an electron and a positive charge e . Electrons and positrons have the same concentration n (number of particles per unit volume) to provide 2 Umeå Universitet, Fysik Vitaly Bychkov zero total charge density 0 . A planar electromagnetic wave propagating in the plasma sets all particles in motion, which results in nonzero current density j determined by the electric force (you may neglect the magnetic force Fm qv B in the 2nd Newton’s law). Find the dispersion relation (k ) for the electromagnetic wave. (4 p) 4) Consider a wire of the shape shown in Fig. 2 with electric current I 0 for t 0 and I t for t 0 . Find the electric field in the coordinate origin for t b / c . (4 p) Fig. 2 5) A dipole oscillating as P aˆ z P0 exp( it ) is placed at height h above an infinitely long perfectly conducting plane (yz-plane) parallel to the plane as shown in Fig. 3. Find the resulting radiation (the electric field E ) in the direction of the x-axis in the radiation zone x h . In order to save time you may use the results derived for a single dipole. For which distance h the radiation in x-direction will be damped completely? For which distance h the power will be maximal? Hint: You can solve the problem either by using the method of images, or taking into account reflection at a perfect conductor. x P h z perfect conductor Fig. 3 (4p) 3 Umeå Universitet, Fysik Vitaly Bychkov Exam in physics, Electrodynamics, 2014-01-11, kl 9.00-15.00 Hjälpmedel: Students may use any book(s) including the compendium. Minor notes in the books are also allowed. Good luck! 1) Consider a solenoid of radius b and finite length h with totally N loops, and a small loop of radius a at the solenoid axis at the distance h from the closest solenoid end, with a b and a h , see Fig. 1. The axes of the small loop and the solenoid coincide. Find mutual inductance. Hint: Consider the solenoid as a combination of rings of small height dz . Help: You may use the result for the magnetic field of a ring at z-axis, Eq. (5.38), and the following formula of integral calculation (Beta, p. 156, Integral 111) (x 2 dx x . 2 3/ 2 2 b ) b x2 b2 (4 p) h b a z h Fig. 1. 2) An infinitely long layer of width 2a is placed perpendicular to z-axis within a z a . The magnetic field inside the layer is B aˆ x B0 ( z / a) 2 exp( it ) . Find the electric filed inside the layer, assuming that the electric field on the centre-plane z 0 is zero. (4 p) 3) A planar linearly polarized wave of amplitude E 0 and frequency is incident under angle from free space on a perfectly conducting material that is filling the region z 0 , see Fig. 2. The magnetic field of the wave is parallel to the conducting surface. Umeå Universitet, Fysik 4 Vitaly Bychkov Determine the surface charge and current densities, S and S , at the conductor. To save time, you do not have to show that the angle of reflection is equal the incident angle. Fig. 2. 4) Consider a wire of the shape shown in Fig. 3 (consisting of a semi-circle and two semiinfinite lines) with electric current I 0 for t 0 and I t for t 0 . Find the electric field in the coordinate origin for t b / c . (4 p) Fig. 3. 5) An electron rotates in an external magnetic field B with initial velocity v 0 perpendicular to the field direction. Find how radius of electron orbit reduces in time because of radiation losses. For that purpose: 1) find how kinetic energy of the electron is related to the orbit radius; 2) Find power loss per unit time due to radiation as a function of the radius; 3) Write the differential equation of energy balance; 4) Solve the equation. Assume that in one period of rotation the electron motion may be considered as approximately circular. 5 Umeå Universitet, Fysik Vitaly Bychkov Prov i fysik, Electrodynamics, 2014-08-28, kl 9.00-15.00 Hjälpmedel: Students may use any book(s) including the compendium. Minor notes in the books are also allowed. Define your notations properly. Present arguments in details. Good luck! 1) Consider a solenoid of length l , radius b and n N / l turns per unit length. A square loop of side a with fixed current I is placed inside the solenoid as shown in Fig. 1. The loop rotates around y-axis (perpendicular to the solenoid axis) with frequency . The solenoid is connected to a resistor R . Find current in the resistor, I R . Self-inductance of the solenoid has to be also found and taken into account in the solution. You may leave the answer in complex numbers. (4p) R y b a Fig. 1. 2) Consider an infinitely long cylinder of radius R with the electrical field inside E aˆ E0 (r / R) 2 exp( it ) , where r is the distance from the cylinder axis and is the angle around the axis (in cylindrical coordinates). By using Maxwell’s equations, find the current density j inside the cylinder. Hint: First, find the magnetic field. (4 p) 3) In order to understand skin-effect (minor penetration of altering em-fields) in conductors, we may consider the approximation of the quasi-stationary Maxwell equations (with neglected displacement current E / t j ). Then, using Ohm’s law Umeå Universitet, Fysik 6 Vitaly Bychkov j E , derive the equation for the magnetic field B assuming for simplicity uniform parameters of the conducting medium , , . Hint: The obtained equation will be of parabolic type instead of a hyperbolic one typical for wave equations. (4 p) 4) An infinite straight wire carries an electric current I 0 for t 0 and I t for t 0 . Find the vector potential at a distance r from the wire. (4 p) 5) A plane electromagnetic wave of amplitude E 0 and frequency propagating along the z-axis is incident on a charge q with mass m , and makes it oscillate along the x-axis (with v c ) and hence radiate. The charge is bound by a spring with eigen-frequency of intrinsic oscillations 0 , which produces the restoring force F m02 x , as shown in Fig. 2. Find the power P radiated by the charge. Fig. 2. (4 p)