Visitors at a science museum are invited to sit in a chair to the right of a full-length diverging lens (f1 = -3.3 m) and observe a friend sitting in a second chair, 1.94 m to the left of the lens. The visitor then presses a button and a converging lens (f2 = 4.36 m) rises from the floor to a position 1.61 m to the right of the diverging lens, allowing the visitor to view the friend through both lenses at once. Be sure to include the algebraic signs (+ or −) with your answers.
(a) Find the magnification of the friend when viewed through the diverging lens only.
(b) Find the overall magnification of the friend when viewed through both lenses.
i tried this several times but i couldn't figure out the correct answers.. please help. Thanks.
To solve this problem, we can use the formula for the magnification of a lens, which is given by:
Magnification (m) = - (image distance / object distance)
Let's start by solving part (a) and finding the magnification of the friend when viewed through the diverging lens only.
(a) Find the magnification of the friend when viewed through the diverging lens only.
Given:
f1 (diverging lens) = -3.3 m
object distance (o1) = 1.94 m (distance of the friend from the diverging lens)
We can use the lens formula to find the image distance (i1) using the below equation:
(1 / f1) = (1 / o1) + (1 / i1)
Substituting the values:
(1 / -3.3) = (1 / 1.94) + (1 / i1)
Now, solve the equation for i1:
-0.303 = 0.516 + (1 / i1)
-0.303 - 0.516 = 1 / i1
-0.819 = 1 / i1
Now solve for i1:
i1 = 1 / -0.819 = -1.22 m
Now, substitute the values of i1 and o1 into the magnification formula:
m1 = - (i1 / o1)
m1 = - (-1.22 / 1.94)
m1 = 0.63 (correct to two decimal places)
Therefore, the magnification of the friend when viewed through the diverging lens only is 0.63.
Now, let's move on to part (b) and find the overall magnification of the friend when viewed through both lenses.
(b) Find the overall magnification of the friend when viewed through both lenses.
Given:
f2 (converging lens) = 4.36 m
We know that both lenses are in contact, so the overall magnification is simply the product of the individual magnifications:
Overall Magnification = m1 * m2
To find m2, we can use the magnification formula. Since the friend is to the right of the converging lens, we treat the converging lens as a virtual object for the converging lens. Therefore, we can consider the distance from the converging lens to the friend as the object distance (o2).
Using the given values:
o2 = 1.61 m (distance from the converging lens to the friend)
Using the lens formula for the converging lens, we have:
(1 / f2) = (1 / o2) + (1 / i2)
Substituting the values:
(1 / 4.36) = (1 / 1.61) + (1 / i2)
Now, solve the equation for i2:
0.229 = 0.621 + (1 / i2)
0.229 - 0.621 = 1 / i2
-0.392 = 1 / i2
Now solve for i2:
i2 = 1 / -0.392 = -2.56 m
Now, substitute the values of i2 and o2 into the magnification formula:
m2 = - (i2 / o2)
m2 = - (-2.56 / 1.61)
m2 = 1.59 (correct to two decimal places)
Therefore, the magnification of the friend when viewed through the converging lens only is 1.59.
Now, we can find the overall magnification by multiplying m1 and m2:
Overall magnification = m1 * m2
Overall magnification = 0.63 * 1.59
Overall magnification = 1.00 (correct to two decimal places)
Therefore, the overall magnification of the friend when viewed through both lenses is 1.00.