question archive During the American Revolution, Paul Revere lit one lantern in the Old North Church in Boston if the British were coming by land, and two if coming by sea
During the American Revolution, Paul Revere lit one lantern in the Old North Church in Boston if the British were coming by land, and two if coming by sea
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During the American Revolution, Paul Revere lit one lantern in the Old North Church in Boston if the British were coming by land, and two if coming by sea. Assume that these two lanterns were set 3 m apart. Assume that the lanterns are yellow in color, around 550 nm.
You are in Concord, MA, around 36 km away from the lights. You are planning to use a telescope to check how many lights are lit. You want the centers of the spots to be further apart than their radii so that you can tell whether there are two or just one blurry spot. What is true about the diameter of your telescope?
(a) It must be smaller than 3.66 m.
(b) It must be larger than 0.00805 m.
(c) It must be larger than 3.66 m.
(d) It must be smaller than 0.00805 m.
(e) You will never be able to tell; the lights are too far away.
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The resolving power of a telescope is given by
R=D1.22λ?
where
λ=wavelength of lightD=diameter/aperture of telescope
Step-by-step explanation
If the separation between two object is x and the distance to the two object from observer is d, then the maximum resolving power required to view them distinctly is
Rmax?=dx?
Here
x=3 md=36 km=36000 m
Hence
Rmax?=360003?=8.33×10−5
hence, this must be the maximum resolving power for the telescope also. This gives,
Rmax?=8.33×10−5=Dmin?1.22λ?Dmin?=8.33×10−51.22×550×10−9?Dmin?=8.05×10−3mDmin?=0.00805 m
is the minimum diameter of the telescope.
Hence, the correct answer is (b) it must be larger than 0.00805m.
