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Question
I have another practice test problem I'm having difficulty working. It reads as follows:
Given a 1Gev electron traveling in space with a magnetic field B = 0.00005 T, find:
a)the frequency generated
b)the radius of the electron’s motion in the B-field
c)the angle theta of the cone (centered on the direction of the instantaneous velocity)
d)the length of time for a pulse received from the electron by an outside observer.
Thanks for any help in solving this!
Hello,
)
Again, this one isn't terribly difficult but you need to be able to craft the problem in its correct context to infer or derive the correct expressions (which I will largely leave up to you!) and obtain the answers.
A diagram I have included ought to help shed light. This diagram shows the orbital plane of the (relativistic) electron's motion, as well as the orientations of the magnetic induction (B) - out of the plane (toward you), and the radiated E-field which is polarized parallel to the orbital plane. (E with arrow direction in the diagram). The electron velocity, v, is also shown (tangent to the orbit at the position) and the angle subtended by the radiation cone.
Now, note that a 1 Gev (10^9 ev) electron will be marginally “relativistic” and its velocity can be worked out – using:
E = mc_o^2/ SQRT {1 – (v/c)^2}
Where v is the velocity, c is the speed of light and m_o the electron rest mass (in kg).
You should be able to show from this that v = 6.7 x 10^6 m/s
a)the frequency generated can then be obtained from:
f = (1/2pi) eB/ m_o SQRT[1 – (v/c)^2]
where e = 1.6 x 10^-19 C, the unit of electron charge
so f will be approximately, 1.4 MHz
b)the radius of the electron’s orbital plane (see diagram included) will be obtained from:
R = c/ 2pi(f) = 34.1 m
c)the angle subtended by the radiation cone, in which bulk of radiation is concentrated (see diagram) will be:
theta = (1.2 x 10^19) [m_oc^2/ E(ev)]
where E(ev)= 10^9, the energy in electron volts.
You will obtain theta = 9.8 x 10^-4 rad (radians) or about 0.001 rad.
To get degrees, multiply this by 57.3 (deg/rad) to get 0.057 deg
Which – multiplied by 60’ (per degree) yields 3.4 minutes of arc
d)the pulse duration (t) will be obtained using:
t = R(theta)/ e [1 – (v/c)^2]
= 2.2 x 10^-7 sec or about 22 microseconds each
Hope this helps!
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Philip A. Stahl
Expertise
I specialize in stellar and solar astrophysics. Can answer any questions pertaining to these areas, the spectroscopic analysis of stars – as well as the magneto-hydrodynamics of sunspots and solar flares. Sorry – No homework problems done or research projects! I will provide hints on solutions.
Experience
Have published papers on the relationship between sunspot morphology and solar flares; discovery of SID flares related to this, constructed computerized stellar models; MHD research.
Organizations
American Astronomical Society (Solar physics and Dynamical astronomy divisions), American Geophysical Union, American Mathematical Society, Intertel.
Publications
Solar Physics, Journal of the Royal Astronomical Society of Canada, Journal of the Barbados Astronomical Society, Meudon Solar Flare Proceedings (Meudon, France). Books: 'Selected Analyses in Solar Flare Plasma Dynamics', 'Physics Notes for Advanced Level'.
Education/Credentials
B.A. degree in Astronomy; M.Phil. degree in Physics - specializing in solar physics.
Awards and Honors
Postgraduate research award- Barbados government; Studentship Award in Solar Physics - American Astronomical Society
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