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Chapter 12: Q11P (page 518)


Short Answer

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01

Expression for the linear speed and the circumference of the circle: 

02

Determine the ratio of the circumference to the diameter:

As the radius stays the same in the inertial frame of reference to another frame of reference, the radius of a circular field does not expand or contract. For considering the circumference, there should be Lorentz-contracted to a smaller value than at rest by a Lorentz factor y. Hence,

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Most popular questions from this chapter

A straight wire along thez-axis carries a charge densityฮปtraveling in the +z direction at speed v. Construct the field tensor and the dual tensor at the point role="math" localid="1654331549769" (x,0,0).

Show that the (ordinary) acceleration of a particle of mass m and charge q, moving at velocity u under the influence of electromagnetic fields E and B, is given by

a=qm1โ€u2/c2[E+uร—B-1c2uuโ€E]

[Hint: Use Eq. 12.74.]

Find the velocity of the muon in Ex. 12.8.

In a pair annihilation experiment, an electron (mass m) with momentum p6hits a positron (same mass, but opposite charge) at rest. They annihilate, producing two photons. (Why couldnโ€™t they produce just one photon?) If one of the photons emerges at 60ยฐto the incident electron direction, what is its energy?

Check Eq. 12.29, using Eq. 12.27. [This only proves the invariance of the scalar product for transformations along the x direction. But the scalar product is also invariant under rotations, since the first term is not affected at all, and the last three constitute the three-dimensional dot product a-b . By a suitable rotation, the x direction can be aimed any way you please, so the four-dimensional scalar product is actually invariant under arbitrary Lorentz transformations.]
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