Download Prov i fysik, strömningslära, 4p, 1998-06-04, kl 9

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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( it ) 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( it ) , 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.
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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
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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.
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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 .
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Fig. 2
5) A dipole oscillating as P  aˆ z P0 exp( it ) 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)
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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
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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( it ) . Find the
electric filed inside the layer, assuming that the electric field on the centre-plane z  0 is
zero.
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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 .
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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.
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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( it ) , 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.
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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.
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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.
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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  m02 x , as shown in
Fig. 2. Find the power P radiated by the charge.
Fig. 2.
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