A Thomson-type experiment with relativistic electrons. One of the earliest experiments to show that p = γmv (rather than p = mv) was that of Neumann. [G. Neumann, Ann. Physik 45:529 (1914)]. The apparatus shown in Figure P4.5 is identical to Thomson’s except that the source of high-speed electrons is a radioactive radium source and the magnetic field B is arranged to act on the electron over its entire trajectory from source to detector. The combined electric and magnetic fields act as a velocity selector, only passing electrons with speed v, where v = V/Bd (Equation 4.6), while in the region where there is only a magnetic field the electron moves in a circle of radius r, with r given by p = Bre. This latter region (E = 0, B = constant) acts as a momentum selector because electrons with larger momenta have paths with larger radii. (a) Show that the radius of the circle described by the electron is given by r = (l2 + y2)/2y. (b) Typical values for the Neumann experiment were d = 2.51 × 10−4 m, B = 0.0177 T, and l = 0.0247 m. For V = 1060 V, y, the most critical value, was measured to be 0.0024 ± 0.0005 m. Show that these values disagree with the y value calculated from p = mv but agree with the y value calculated from p = γmv within experimental error. (Hint: Find v from Equation 4.6, use mv = Bre or γmv = Bre to find r, and use r to find y.)
Figure P4.5 The Neumann apparatus.
(a)
To show that the radius of the circle described by th electron is given by
Answer to Problem 5P
It is showed that the radius of the circle described by th electron is given by
Explanation of Solution
The curved path of the electron is shown in figure 1.
Write the equation for the curved path of the electron.
Here,
Rewrite the above equation.
Rewrite the above equation for
Conclusion:
Therefore, it is showed that the radius of the circle described by th electron is given by
(b)
To show that the value of
Answer to Problem 5P
It is showed that the calculated that the value of
Explanation of Solution
Write the equation for the velocity of the electron.
Here,
Write the equation for the momentum of the electron.
Here,
Write the classical expression for the momentum of a particle.
Here,
Equate equations (III) and (IV) and rewrite it for
Write the relativistic equation for the momentum of the particle.
Here,
Equate equations (III) and (VI) and rewrite it for
Write the equation for the Lorentz factor.
Here,
Put the above equation in equation (VII).
Write the equation for the root of a quadratic equation
Conclusion:
The mass of the electron is
Substitute
Substitute
Substitute
Comparison of the above equation with the quadratic equation
Substitute
The value
The value
Substitute
Substitute
Comparison of the above equation with the quadratic equation
Substitute
The value
The value
Therefore, it is showed that the calculated that the value of
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Chapter 4 Solutions
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