Q47E

Question

When a camera is focused, the lens is moved away from or toward the digital image sensor. If you take a picture of your friend, who is standing 3.90 m from the lens, using a camera with a lens with an 85-mm focal length, how far from the sensor is the lens? Will the whole image of your friend, who is 175 cm tall, fit on a sensor that is 24 mm * 36 mm?

Step-by-Step Solution

Verified

Answer

a) the sensor is +8.69 cm from the lens.

b) the image formed is 39 mm in size which will not fit on a screen of 36 mm size.

1Step 1: Basic Concepts

An optical lens is a transparent transmissive optical component that is used to either converge or diverge the light emitting from a object. These transmitted light forms an image of that object.

Thin Lens formula:

$\frac{1}{u} + \frac{1}{v} = \frac{1}{f}$

Here,

u = Object distance from lens

v = Image distance from lens

f = Focal length of the lens

Sign Convention:

Object Distance (u): (+) in front of lens; (-) in the back of lens
Image Distance (v): (-) in front of lens; (+) in the back of lens
Focal Length (f): (+) for convergent (convex) lens; (-) for divergent (concave) lens

Lateral magnification (m): It is defined as the ratio of the height of image to the height of the object viewed through a lens. It is a dimensionless quantity.

$m = - \frac{v}{u} = \frac{H_{i}}{H_{0}}$

Here,

H_o = Height of the object, and H_i = Height of the image

Sign Convention:

If m is positive – Image is erected
If m is negative – Image is inverted

2Step 2: Distance of Image

Focal length of 1^st lens, f₁ = +8.5 cm

Object distance, u = +390 cm

By using lens formula

$\Rightarrow \frac{1}{u} + \frac{1}{v} = \frac{1}{f} \Rightarrow \frac{1}{+ 390} + \frac{1}{v} = \frac{1}{+ 8.5} \Rightarrow \frac{1}{v} = \frac{1}{8.5} - \frac{1}{390} \Rightarrow \frac{1}{v} = \frac{390 - 8.5}{390 * 8.5} \Rightarrow v = + 8.69 c m$