A certain person's eye measures 24.0 mm from the pupil to retina.
(a) If we treat the eye's lens as a thin lens, what is its focal length when the person focuses on a pencil 44.3 cm from the eye?
(b) While the person focuses on the pencil, what is the image distance for a wall 426 cm from the eye?
(1) 1/2.4 + 1/44.3 = 1/f
Solve for for in cm.
(2) Using the same f that you get in (1), solve
1/x + 1/426 = 1/f
to get the value of the wall image distance, x.
It will be in centimeters.
(a) Well, if we're talking about a person focusing on a pencil, it better be a really interesting pencil! Now, let's calculate the focal length. With the given measurements, we can use the lens formula 1/f = 1/v - 1/u, where f is the focal length, v is the distance of the pencil from the eye, and u is the distance of the retina from the lens.
Plugging in the values (v = 44.3 cm = 0.443 m, u = 24.0 mm = 0.024 m), we can solve for f. But remember, solving equations is no pencil pusher business!
1/f = 1/v - 1/u
1/f = 1/0.443 - 1/0.024
1/f = 2.257 - 41.667
1/f ≈ -39.41
So, the focal length of the lens is approximately -39.41 m. And no, that doesn't mean it can solve all your problems; it's just a lens, after all!
(b) Now, let's tackle the image distance for the wall! To find the image distance, we'll use the lens formula once again. Remember, the lens is focusing on the pencil, so we need to use those measurements.
1/f = 1/v - 1/u
Plugging in the values (v = 426 cm = 4.26 m, u = 24.0 mm = 0.024 m):
1/f = 1/4.26 - 1/0.024
1/f = 0.234 - 41.667
1/f ≈ -41.433
So, the image distance for the wall is approximately -41.433 m. Now, if only we could find an image distance that could take us on a vacation...
To determine the focal length of the eye's lens and the image distance, we can use the thin lens formula:
1/f = 1/d_o + 1/d_i
where:
f is the focal length of the lens
d_o is the object distance (distance of the pencil from the eye)
d_i is the image distance (distance of the wall from the eye)
(a) To find the focal length (f), given that the person is focusing on a pencil 44.3 cm from the eye:
1/f = 1/d_o + 1/d_i
Given:
d_o = 44.3 cm
We are given the distance from the pupil to the retina (24.0 mm), which will be the image distance (d_i). However, we need to convert it to centimeters:
d_i = 24.0 mm = 2.4 cm
Substituting these values into the thin lens formula:
1/f = 1/44.3 + 1/2.4
To find the focal length (f), we can rearrange the formula:
1/f = (1/44.3) + (1/2.4)
Taking the reciprocal of both sides of the equation:
f = 1 / ((1/44.3) + (1/2.4))
Now we can calculate the value of f:
f = 1 / (0.022576361 + 0.416666667)
f = 1 / 0.439243028
f ≈ 2.277 cm (rounded to three significant figures)
Therefore, the focal length of the eye's lens when the person focuses on a pencil 44.3 cm from the eye is approximately 2.277 cm.
(b) To find the image distance (d_i) when the person focuses on a pencil and the wall is 426 cm away from the eye, we can use the thin lens formula:
1/f = 1/d_o + 1/d_i
Given:
d_o = 44.3 cm
d_i = 426 cm
Substituting these values into the thin lens formula:
1/f = 1/44.3 + 1/426
To find the image distance (d_i), we can rearrange the formula:
1/f = (1/44.3) + (1/426)
Taking the reciprocal of both sides of the equation:
d_i = 1 / ((1/44.3) + (1/426))
Now we can calculate the value of d_i:
d_i = 1 / (0.022576361 + 0.002347418)
d_i = 1 / 0.024923779
d_i ≈ 40.096 cm (rounded to three significant figures)
Therefore, the image distance for a wall 426 cm from the eye, while the person focuses on a pencil, is approximately 40.096 cm.
To find the focal length of the eye's lens, we can use the thin 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.
(a) To find the focal length when the person focuses on a pencil 44.3 cm from the eye, we need to first find the object distance (u). The object distance is the distance from the lens to the object being observed. In this case, it is 44.3 cm.
Using the thin lens formula, we can rearrange the formula to solve for the focal length (f):
1/f = 1/v - 1/u
Substituting the given values, we have:
1/f = 1/v - 1/u
1/f = 1/(-24.0 mm) - 1/44.3 cm
Now, we need to convert the given measurements to the same unit. Let's convert everything to millimeters:
1/f = 1/(-24.0 mm) - 1/443 mm
Simplifying the equation:
1/f = -0.0417 - 0.00226
1/f = -0.04396
Taking the reciprocal of both sides:
f = -1 / (-0.04396)
f ≈ 22.73 mm
Therefore, the focal length of the eye's lens when the person focuses on a pencil 44.3 cm from the eye is approximately 22.73 mm.
(b) To find the image distance for a wall 426 cm from the eye while the person focuses on the pencil, we can again use the thin lens formula:
1/f = 1/v - 1/u
In this case, the object distance (u) is the distance from the lens to the pencil, which is 44.3 cm.
Let's use the formula:
1/f = 1/v - 1/u
Substituting the values:
1/f = 1/v - 1/u
1/f = 1/(-24.0 mm) - 1/44.3 cm
Converting the measurements to millimeters:
1/f = 1/(-24.0 mm) - 1/443 mm
Simplifying the equation:
1/f = -0.0417 - 0.00226
1/f = -0.04396
Taking the reciprocal of both sides:
f = -1 / (-0.04396)
f ≈ 22.73 mm
The focal length remains the same. Therefore, the image distance for a wall 426 cm from the eye while the person focuses on the pencil is also approximately 22.73 mm.
Please note that these calculations assume a simplified model of the eye, treating the lens as a thin lens.