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What angle would the axis of a polarizing filter need to make with the direction of polarized light of intensity 1.80 kW/m2 t
One of the primary goals of the Kepler space telescope is to search for Earth-like planets. Data gathered by the telescope in
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Answer #1

1. The intensity transmitted by a polarizer is given by:

I = I_0\cos^2{\theta}

where, I0 is the incident intensity and theta is the angle between the axis of the polarizing filter and the direction of polarized light.

\therefore \cos^2{\theta} = \frac{I}{I_0}

\therefore \theta = \cos^{-1}\left(\sqrt{\frac{I}{I_0}}\right)

\therefore \theta = \cos^{-1}\left(\sqrt{\frac{30.0}{1.8\times 10^3}}\right) = 82.58\,^o

So, the correct answer is 82.58o . (Note that you need to convert the answer from radians into degrees in your calculator)

2.(a) we have:

L0 = 139 ...... proper length of the ship.

The length of the ship as measured by the observers on Earth is contracted as the ship is moving relative to them.

Therefore using the length contraction formula,

L = \frac{L_0}{\gamma} = L_0\,\sqrt{1-\frac{v^2}{c^2}}

\therefore L = 139\,m\times \sqrt{1-(0.94)^2} = 47.42\,m

So, the length of the moving sheep as measured by the observers on Earth is 47.42 m.

(b) As measured by an observer on Earth, the spaceship has to travel 115 ly with a speed of 0.94c

\therefore t = \frac{115\,ly}{0.94\,c} = \frac{115}{0.94}\,years = 122.34\,years

So, it will take 122.34 years for the spaceship to travel from Earth to Borails, as measured by an observer on Earth.

(c) An astronaut on the spaceship sees the distance between Earth and Borails as contracted.

The proper distance between Earth and Borails is:

x0 = 115 ly

The distance measured by an astronaut on the spaceship is:

x = \frac{x_0}{\gamma} =x_0\,\sqrt{1-v^2/c^2} = 115\,ly\times\sqrt{1-(0.94)^2} = 39.24\,ly

So, the time taken by the sapceship as measured by the astronaut is:

t_0=\frac{x}{v} = \frac{x_0\,\sqrt{1-v^2/c^2}}{v} = \frac{115\,ly\times \sqrt{1-(0.94)^2}}{0.94\,c} = 41.74\,years

So, it will take 41.74 years for the spaceship to travel from Earth to Borails, as measured by astronauts in the spaceship.

(Note that if you want to use time dilation formula, the time measured by the astronauts in the spaceship is the proper time and the time measured by an observer on Earth is improper time. So then you will have to calculate the proper time from the improper time, which will give the same answer. Proper time is the time duration between events which happen at the same place. For the astronauts, both the events of leaving the Earth and reaching Borails happen at the same place which is the spaceship itself. So, the time measured by the astronauts is the proper time.)

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