Mid-Term Test

Thurs 28 Oct 1999, 10:00 – 11:15 am Dr. R. Egerton

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electron charge = 1.60 ´ 10^-19 C electron rest mass = 9.11 ´ 10^-31 kg

speed of light in vacuum = 3.00 ´ 10⁸ m/s

(b) What result was obtained in this experiment ?

(c) How was this result explained by Fitzgerald and Lorentz ?

(d) Why is the result consistent with the Special Theory of Relativity ?

2. A muon is created at a height of 5.8 km and travels vertically downwards (towards the earth’s surface) at a speed of 2.90 ´ 10⁸ m/s. Calculate the journey time, according to:

1. an observer stationary with respect to the earth
2. an observer riding with the moving muon.
3. What effect does this difference in time have on the number of muons reaching the earth’s surface ?

3. An electron is accelerated from rest in a uniform electric field of 5.7 ´ 10⁶ V/m

1. Sketch a graph showing (qualitatively) the acceleration of the electron as a function of time.
2. Calculate the acceleration of the electron when its speed is zero.
3. Calculate the acceleration when the electron speed is 2.0 ´ 10⁸ m/s .
4. What is the kinetic energy of the electron when its speed is 2.0 ´ 10⁸ m/s ?
5. How far does the electron travel to reach a speed of 2.0 ´ 10⁸ m/s ?

4. (a) Draw a fully-labeled diagram of an apparatus which you could use to demonstrate that the ratio of gravitational and inertial mass is the same for objects of different composition.

(b) Draw a fully-labeled diagram to illustrate an observation which verifies Einstein’s General Theory of Relativity.

(c) What connection is there between the scientific principles being investigated in (a) and in (b) above?

Ph 311 Mid-Term Test (28 Oct 1999) Answers (R. Egerton)

1. (a) The diagram should show an interferometer containing a light source, a central half-silvered mirror (at 45 degrees) splitting the beam into two perpendicular paths, two fully-silvered mirrors (perpendicular to the light beams) reflecting the light back towards the center and a detector for observing the resulting interference pattern. The apparatus is mounted on a solid base which can be rotated about the vertical axis.

2. The result was no change in the interference pattern when the apparatus was rotated about the vertical axis.
3. Fitzgerald and Lorentz explained the null result as a contraction of the apparatus in the direction of the "ether wind", perhaps due to properties of the electron.
4. The null result is consistent with the Special Theory because the latter is based upon the postulate that the speed of light is constant in all inertial frames, implying that the travel times for light in the two arms of the interferometer are always the same.

1. (a) earth journey time = distance / speed = (5.8E3)/(2.9E8) = 2.0E-5 s

2. gamma = [1 – (2.9E3/3.0E8)^2]^-1/2 = [0.0655]^-1/2 = 15.25^1/2 = 3.91

Muon journey time = contracted distance / speed = (2.0E-5)/(gamma) = 5.12E-6 s

3. (c) Reduced muon journey time means that fewer muons will decay in transit; more will reach the earth’s surface.

3. (a) The acceleration should fall asymptotically towards zero.

(b) acceleration = F/m0 = qE/m0 = (1.60E-19)(5.7E6) / (9.11E-31) = 1.00E18 m/s^2

(c) gamma = [1 - (2.0E9/3.0E9)^2]^-1/2 = [0.556]^-1/2 = 1.342

3. acceleration = F/[(gamma) (m0)] = 1.0E18/1.342 = 7.47E17 m/s^2
5. (d) K = (gamma – 1) m0 c^2 = (0.342)(9.11E-31)(3.00E8)^2 = 2.80E-14 Joule
7. (e) K = gain in kinetic energy = loss of potential energy = (charge)(potential drop)
8. = (charge)(electric field)(distance traveled),
9. so distance = (2.80E-14) / [(1.60E-19)(5.7E6)] = 0.0307 m = 3.07 cm

3. (a) The diagram could show a pendulum whose mass can be changed (by inserting different substances etc.) and whose period of oscillation is accurately measured.

(b) The diagram could show a measurement of the bending of starlight in the gravitational field of the sun, OR precession of the perihelion of the planet Mercury, OR gravitational blueshift of laser light at the bottom of a mineshaft.

(c) The General Theory of Relativity is based upon the Principle of Equivalence, which says that the effects of gravity and of acceleration are indistinguishable, and which also implies equality of the gravitational and inertial mass.