A converging lens (f = 14.6 cm) is located 39.0 cm to the left of a diverging lens (f = -4.57 cm). A postage stamp is placed 34.2 cm to the left of the converging lens. (a) Locate the final image of the stamp relative to the diverging lens. (b) Find the overall magnification.
Use the iimage location of the converging lens as the object location for the diverging lens. That will give you the final image location.
So 34.2/14.6?
To answer this question, we can use the lens equation and the magnification formula. The lens equation is given by:
1/f = 1/v - 1/u,
where f is the focal length of the lens, v is the image distance from the lens, and u is the object distance from the lens. The magnification formula is given by:
magnification = - v/u,
where the negative sign signifies that the image formed is virtual and upright.
Let's solve each part of the problem step by step:
(a) To locate the final image of the stamp relative to the diverging lens, we need to calculate the image distance (v) formed by the converging lens and use that as the object distance for the diverging lens. The object distance (u) for the converging lens is given as 34.2 cm.
Using the lens equation for the converging lens, we have:
1/f1 = 1/v1 - 1/u1.
Substituting the values, we get:
1/14.6 = 1/v1 - 1/34.2.
Now, we can solve for v1:
1/v1 = 1/14.6 + 1/34.2.
Taking the reciprocal of both sides, we get:
v1 = 21.594 cm (rounded to three decimal places).
Now, the image distance (v2) formed by the diverging lens can be found using the lens equation for the diverging lens:
1/f2 = 1/v2 - 1/u2,
where the object distance (u2) is the same as the image distance (v1) we found from the converging lens. Therefore, u2 = v1 = 21.594 cm.
Substituting the values, we have:
1/-4.57 = 1/v2 - 1/21.594.
Now, solve for v2:
1/v2 = -1/4.57 + 1/21.594.
Taking the reciprocal of both sides, we get:
v2 = -21.066 cm (rounded to three decimal places).
As for the sign convention, a negative value of v2 indicates that the image formed by the diverging lens is virtual.
So, the final image of the stamp relative to the diverging lens is located 21.066 cm to the right of the diverging lens.
(b) To find the overall magnification, we can calculate the individual magnifications of both lenses and multiply them. The magnification for each lens can be calculated using the magnification formula:
magnification = -v/u.
For the converging lens:
magnification1 = -v1/u1 = -21.594/34.2 = -0.631 (rounded to three decimal places).
For the diverging lens:
magnification2 = -v2/u2 = -(-21.066)/21.594 = 0.975 (rounded to three decimal places).
Now, to find the overall magnification, we multiply the individual magnifications:
overall magnification = magnification1 * magnification2 = -0.631 * 0.975 = -0.615 (rounded to three decimal places).
The negative sign indicates that the overall magnification is upright and virtual.
Therefore, the overall magnification is approximately -0.615.