A thin convex lens produce a real image on an a screen . if an object of height 36mm is placed 2000mm from the screen ,an image of height 684mm will be produced .determine

A) the distance of the object from the lens
B) the focal length of the lens

To determine the distance of the object from the lens and the focal length of the lens, we can use the lens formula and the magnification formula.

Given:
Height of the object (h_1) = 36mm
Distance of the object from the screen (u) = 2000mm
Height of the image (h_2) = 684mm

Step 1: Calculate the magnification (m)
The magnification (m) can be calculated using the formula:
m = h_2 / h_1

Substituting the given values:
m = 684mm / 36mm
m ≈ 19

Step 2: Use the magnification formula to calculate the distance of the object from the lens (v):
m = - v / u

Rearranging the formula:
v = - m * u

Substituting the values:
v = -19 * 2000mm
v = -38,000mm

The negative sign indicates that the image is formed on the same side as the object.

Step 3: Use the lens formula to find the focal length (f):

The lens formula is given by:
1/f = 1/v - 1/u

Substituting the values:
1/f = 1/(-38,000mm) - 1/2000mm

Simplifying the equation:
1/f = -1/38,000 - 1/2000
1/f = (-2000 - 38,000) / (38,000 * 2000)
1/f = -40,000 / 76,000,000

Taking the reciprocal of both sides:
f = 76,000,000 / -40,000
f = -1900mm

Therefore, the distance of the object from the lens is 2000mm and the focal length of the lens is -1900mm.

To determine the distance of the object from the lens and the focal length of the lens, we can use the lens formula:

1/f = (1/v) - (1/u)

where:
f = focal length of the lens
v = distance of the image from the lens (in this case, it is the distance from the screen)
u = distance of the object from the lens

We are given the following information:
- Object height (ho) = 36mm
- Image height (hi) = 684mm
- Distance from the screen (v) = 2000mm

A) To find the distance of the object from the lens (u):
We need to use the magnification formula:

Magnification (m) = |hi| / |ho| = v/u

Substituting the given values:
684 / 36 = 2000/u

Cross-multiplying:
684u = 36 * 2000

Simplifying:
684u = 72,000

Dividing both sides by 684:
u = 72,000 / 684

Calculating:
u ≈ 105.26 mm

Therefore, the distance of the object from the lens is approximately 105.26 mm.

B) To find the focal length of the lens (f):
We can rearrange the lens formula to solve for f:

1/f = (1/v) - (1/u)

Substituting the given values:
1/f = (1/2000) - (1/105.26)

Calculating:
1/f ≈ 0.0005 - 0.0095

1/f ≈ -0.00899

Taking the reciprocal to find f:
f ≈ -1 / 0.00899

Calculating:
f ≈ -111.1 mm

Therefore, the focal length of the lens is approximately -111.1 mm.

Note: The negative sign indicates that the lens is a convex lens.