An object is in front of a converging lens with a focal length of 17.0 cm. The image seen has a magnification of −2.10.(Include the sign of the values in your answers.)
(a) How far is the object from the lens?
(b) If the object has a height of −15.0cm because it points below the principal axis (it is inverted), what is the image height hi?
To solve this problem, we can use the lens formula:
1/f = 1/do + 1/di
where f is the focal length, do is the object distance, and di is the image distance.
(a) How far is the object from the lens?
Since the magnification (m) is given by the ratio of the image distance (di) to the object distance (do), we have:
m = -di/do
Substituting the given values:
-2.10 = -di/do
Now, let's solve for do:
do = -di / m
do = -di / -2.10
do = di / 2.10
Since we know the formula for the magnification is:
m = hi/ho = -di/do
We can rearrange it to solve for hi:
hi = -m * ho
hi = -(-2.10) * (-15.0 cm)
hi = 31.5 cm
Therefore, the image height (hi) is 31.5 cm.
To find the answers to the given questions, we can use the lens equation and the magnification formula.
The lens equation relates the object distance (do), the image distance (di), and the focal length (f) of a lens as:
1/f = 1/do + 1/di
The magnification formula relates the object height (ho), the image height (hi), and the magnification (m) as:
m = -di/do = hi/ho
Let's solve the two questions step by step:
(a) How far is the object from the lens?
Given:
Focal length (f) = 17.0 cm
Magnification (m) = -2.10
Using the magnification formula, we have:
m = -di/do
Rearranging the formula, we get:
do = -di / m
Substituting the values, we have:
do = -di / (-2.10) [Remember to include the sign of the magnification]
Since the image distance is positive for a real image formed by a converging lens, we can substitute the magnification value and solve for do.
(b) If the object has a height of -15.0 cm because it points below the principal axis (it is inverted), what is the image height hi?
Given:
Object height (ho) = -15.0 cm
Magnification (m) = -2.10
Using the magnification formula, we have:
m = hi / ho
Rearranging the formula, we have:
hi = m * ho
Substituting the values, we have:
hi = (-2.10) * (-15.0) [Remember to include the sign of the magnification and the object height]
Simplifying the expression, we can find the image height hi.