  # Problems in Lenses and Mirrors - Problem L1

Physics

## example

### Image distance or object distance for refraction of rays at spherical surfaces

Where would an object be placed in a medium of refractive index , so that its real image is formed at equidistant from sphere of radius R and refractive index   which is also placed in the medium of refractive index as shown in figure?
We know,
For 1, we get:
,   object distance
or,
For 2, we get:

or,
or,
or,
So,
or,

## formula

### Refraction of light at spherical surfaces

Let,

- refractive index of medium from which rays incident.

- refractive index of another medium.

u - distance of object from pole of spherical surface

v - distance of image from pole of spherical surface

Now, for NOC, i is the exterior angle.

.........(1)

Similarly,

.........(2)

Now by using snells law we get

Or for small angles

Substituting i and r from eq. (1) and (2), we get

As,

OM = -u, MI = +v, MC = + R

Hence equation becomes

## example

### Minimum distance between object and its real image for a convex lens

Example: In a convex lens of focal length , find the minimum distance between an object and its real image.

Solution:
Let image distance be , object distance be and focal length be .
Let distance between object and image be .

Now,
or,
or,
or,
For point of minima,
or,
or,
Therefore,

## definition

### Magnification

Magnification of a lens is defined as:

Note:
Sign convention must be followed while using formula for magnification. Hence, it can be positive or negative.
image is magnified.
image is same size as object.
image is diminished.

## example

### Calculate Focal Length of Lenses using lens maker's formula

Example: A plano-convex lens has thickness . When placed on a horizontal table with the curved surface in contact with it, the apparent depth of the bottom-most point of the lens is found to be . If the lens is inverted such that the plane face is in contact with the table, the apparent depth of the centre of the plane face of the lens is found to be . Find the focal length of the lens.

Solution:
When the curved surface of the lens (refractive index ) is in contact with the table, the image of the bottom-most point of lens (in glass) is formed due to refraction at plane face.
The image of O appears at .
Here, , , and
gives,
................ (i)
When the plane surface of the lens in contact with the table, the image of center of the plane face is formed due to refraction at curved surface. The image of O is formed at .
Here, , and
gives.

From Eq. (i), we get:
, therefore this equation gives

or
This gives
The focal length () of plano-convex lens and is

## example

### Calculate magnification of lens

Problem:
The focal length of a thin biconvex lens is . When an object is moved from a distance of in front of it to , the magnification of its image changes from to . The ratio is :
Solution:
When object is at

,

.....(1)

When object is at

,

.....(2)

We have to find

## example

### The minimum distance between an object and its real image in a convex lens

Distance between object and image