At what position a candle of length 3 cm be placed in front of a convex lens so that it's image of length 6 cm be obtained on a screen placed at distance 30 cm behind the lens

Height of object=3cm

Height of image=-6cm (formed below principle axis )
V=30 cm
U=?
By using lens formula we get f=10cm
hi/ho=v/u
-6/3=30/u
U=-15 cm

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To determine the position where the candle should be placed in front of a convex lens, we can use the lens formula:

1/f = 1/v - 1/u,

where:
- f is the focal length of the lens,
- v is the image distance from the lens,
- u is the object distance from the lens (position of the candle).

In this case, we know:
- Length of the object (candle) = 3 cm,
- Length of the image = 6 cm,
- Distance of the screen from the lens (v) = 30 cm.

To find the image distance (v), we can use the magnification formula:

magnification (m) = -v/u,

where:
- magnification (m) = image size / object size.

Given the image size (6 cm) and object size (3 cm), we can substitute these values into the magnification formula:

m = (6 cm) / (3 cm).

Since we know that the magnification (m) is negative for a real and upright image, we have:

-2 = -v/u.

Substituting the focal length (f) of the lens into the lens formula, we have:

1/f = 1/v - 1/u.

Simplifying the lens formula, we get:

1/f = (-1/u) - (1/u) = -2/u.

Rearranging the equation, we can express u (object distance) in terms of f:

1/f = -2/u,
u = -2f.

Using the relation:

u + v = 30 cm,

we can substitute u = -2f to find v:

-2f + v = 30 cm,
v = 30 cm + 2f.

Now, we need to substitute the values into the relation:

m = -v/u.

-2 = -(30 cm + 2f) / (-2f).

Simplifying this equation, we can find the value of f. Once we have the value of f, we can substitute it in the equation v = 30 cm + 2f to find the value of v. Finally, we can determine the position of the object (candle) by subtracting u from the lens to screen distance (v).

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