A 6 cm high object is placed 43ncm from a converging lens of focal length 24 cm. Find the image’s position and draw a ray diagram approximately to scale.
B) To answer this question you use 1/f = 1/v + 1/u where f is the focal length, v is the image distance, and u is the object distance
v = 1/(1/24 - 1/43) = 54.3cm
Note that since the object distance exceeds the focal length, the image will be real and inverted.
To construct the ray diagram draw a horizontal line OLB representing the optical axis. At L (about 1/3 the way from O, erect a perpendicular CLD representing the lens. Erect another shorter perpendicular OO' at O representing the object. Mark the focal point (F) of the lens at a suitable point on LB scaled relative to the object distance OL.
From O' draw a line parallel to AB striking CLD at X. From X draw a line XF and project it beyond F. From O' draw another line O'L and project it beyond L until it intersects XF (projected) at I'. I' is the point on the image which corresponds to O' on the image. Drop a perpendicular from I' to AB and mark the point I where this line intersects AB. I is the point on the image which corresponds to O on the object
To find the position of the image formed by a converging lens, we can use the lens formula:
1/f = 1/v - 1/u
where:
f = focal length of the lens
v = image distance from the lens (positive for real images and negative for virtual images)
u = object distance from the lens (always positive)
Given:
Object height, h = 6 cm
Object distance, u = -43 cm (as the object is placed in front of the lens)
We need to find the image distance, v. Let's substitute the known values into the lens formula:
1/24 = 1/v - 1/-43
To simplify the equation, we'll first find the least common denominator:
1/24 = (-43 + v) / (v * (-43))
Multiplying both sides of the equation by 24v*(-43), we get:
v * (-43) = 24 * (-43) + 24v
-43v = -1032 + 24v
48v = 1032
v = 1032 / 48
v ≈ 21.5 cm
Therefore, the image is formed approximately 21.5 cm behind the lens.
To draw a ray diagram approximately to scale, follow these steps:
1. Start by drawing a horizontal line to represent the principal axis of the lens.
2. Draw a vertical line, labeled "A," on the left side of the principal axis to represent the object.
3. Measure the object distance (u = -43 cm) from the lens and mark it on the principal axis.
4. Using a ruler, draw a line from point A towards the lens.
5. Draw a line parallel to the principal axis that passes through the lens. This represents the refracted ray.
6. Measure the focal length (f = 24 cm) from the lens and mark it on the principal axis as point F.
7. Draw a line from point F through the lens. This represents the second refracted ray.
8. The point where these two rays intersect after passing through the lens represents the approximate position of the image. Mark this point as "B."
9. Measure the image distance (v ≈ 21.5 cm) from the lens and mark it on the principal axis as point B.
10. Draw a line from point B towards the right side of the principal axis. This shows the direction and approximate size of the image.
Remember, this ray diagram is approximate and does not include all the details, but it should give a general idea of the image formation by a converging lens.