In displacement method, the distance between object and screen is 96 cm. The ratio of length of two images formed by a convex lens placed between them is 4.84.a)Ratio of the length of object to the length of shorter image is 11/5.b)Distance between the two positions of the lens is 36 cm.c)Focal length of the lens is 22.5 cm.d)Distance of the lens from the shorter image is 30 cm. Correct answer is option 'A,B,D'. Can you explain this answer?

Question

Answer

Solution:

Given,
Distance between object and screen, u + v = 96 cm
Ratio of the length of two images, m/m' = 4.84

Let the distance of the lens from the object be x.
Using lens formula,

1/f = 1/v - 1/u
1/f = m'/v - m/u
Substituting the values, we get
1/f = 4.84/v - 1/x
(f*v)/x = 4.84 - v/x

Let the distance between the two positions of the lens be d1 and d2.
Using lens formula for the first position of the lens,

1/f = 1/(v1 - d1) - 1/x
(f*(v1 - d1))/x = 1 - (v1 - d1)/x
(f*(v1 - d1))/x = (d1 - u)/x

Using lens formula for the second position of the lens,

1/f = 1/(v2 + d2) - 1/x
(f*(v2 + d2))/x = 1 + (v2 + d2)/x
(f*(v2 + d2))/x = (v + d2 - u)/x

Dividing the above two equations, we get

(v1 - d1)/(v2 + d2) = (f*(v + d2))/(f*(v1 - d1))
(v1 - d1)/(v2 + d2) = (v + d2)/(v1 - d1)
(v1 - d1)*(v1 - d1 + d2) = (v2 + d2)*(v + d2)

Substituting the values, we get
(96 - x)*(96 - x + d2) = (v + d2)*(x + d2)

Using the relation m/m' = v1/u, we get
m/m' = (v - f)/f
4.84 = (v - f)/f
v = 4.84f + f

Using the above relation and the equation (f*v)/x = 4.84 - v/x, we get
x = (4.84f^2)/(4.84f - 1)

Substituting the value of x in the equation (96 - x)*(96 - x + d2) = (v + d2)*(x + d2), we get
d2 = (95.04f^2 - 182.4f + 84)/(4.84f - 1)

Using the relation m/m' = v1/u, we get
m/m' = (v2 + f)/f
4.84 = (v2 + f)/f
v2 = 3.84f

Using the relation u + v = 96 and the values of v1 and v2, we get
v1 + v2 + 2f = 96
v1 + 3.84f + 2f = 96
v1 = 90.16f

Using the relation m/m' = v1/u, we get
m/m' = (v1 - f)/f
4.84 = (v1 - f)/f
v1 = 5.84f

Using the above values,

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