An astronomical refracting telescope is made from two lenses. the objective lens has a focal length of 114 cm, and the eyepiece lens has a focal length of 17 cm.
a) What is the total length of the telescope?
b) What is the angular magnification of the telescope?
c) What is the magnification by turning the telescope around and looking through the objective lens first?
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a) fe+fo
114 cm + 17 cm = 131 cm
b) M=fo/fe
114 cm / 17 cm = 6.71 cm
c) Need help with this part.
Thank you!
b.
Magnification is a pure number (no units)
c.
When turned around, fe=17, fo=114
M=?
To find the answers to the given questions, we can use the formulas for the lens formula, angular magnification, and the magnification by reversing the telescope.
a) Total length of the telescope:
The total length of the telescope is the sum of the distances between the two lenses. Using the lens formula:
1/f = 1/v - 1/u
where f is the focal length of the lens, v is the image distance, and u is the object distance.
For the objective lens:
f_objective = 114 cm
u_objective = ?
v_objective = distance between lenses =?
For the eyepiece lens:
f_eyepiece = 17 cm
u_eyepiece = distance between lenses =?
v_eyepiece = distance between lenses =?
Notice that the image created by the objective lens acts as the object for the eyepiece lens. Therefore, v_objective = u_eyepiece.
Using the lens formula for each lens:
For the objective lens:
1/114 = 1/v_objective - 1/u_objective
For the eyepiece lens:
1/17 = 1/v_eyepiece - 1/u_eyepiece
Since v_objective = u_eyepiece, we can combine the equations:
1/114 = 1/v_objective - 1/u_objective
1/17 = 1/v_objective - 1/v_eyepiece
Now, rearrange the equations to solve for v_objective and u_objective:
1/v_objective = 1/114 + 1/u_objective
1/v_objective = 1/17 + 1/v_eyepiece
Add the two equations together:
1/v_objective + 1/v_objective = 1/114 + 1/u_objective + 1/17 + 1/v_eyepiece
2/v_objective = (17 + 114 + u_objective + 114u_objective)/(114u_objective)
2/v_objective = (u_objective + 131)/(114u_objective)
Now, rearrange the equation to solve for v_objective:
v_objective = (114u_objective)/(u_objective + 131)
Since v_objective = u_eyepiece, substitute it into the equation:
v_eyepiece = (114u_objective)/(u_objective + 131)
The total length of the telescope is the sum of the distances between the lenses:
Total length = distance between objective and eyepiece = v_eyepiece + u_eyepiece
Substituting the equation for v_eyepiece and u_eyepiece:
Total length = (114u_objective)/(u_objective + 131) + u_objective
Now, u_objective is the object distance for the objective lens, which is given as the focal length of the objective lens:
Total length = (114*114)/((114 + 131)/(114)) + 114
Total length = 114 + 131 + 114 = 359 cm
Therefore, the total length of the telescope is 359 cm.
b) Angular magnification:
Angular magnification is given by the formula:
Angular magnification = (-v_objective)/(u_objective)
Substituting the values:
Angular magnification = (-v_objective)/114
Using the equation derived earlier:
Angular magnification = (-((114*114)/(114 + 131)))/114 = -0.548
Therefore, the angular magnification of the telescope is -0.548.
c) Magnification by reversing the telescope:
When we reverse the telescope, looking through the objective lens first, the magnification is given by:
Magnification = (-v_eyepiece)/u_eyepiece
Substituting the values:
Magnification = (-v_eyepiece)/((114*131)/(131 + 114)) = -0.870
Therefore, the magnification by turning the telescope around and looking through the objective lens first is -0.870.