Question

the focal length of a spherical mirror is given by 1 /f= 1 /v+1/u where u is the position of object from pole of the mirror and v is position of image from the pole of the mirror .if u =(30 +- 0.3 cm) and v=(60 +- 0.6 )cm the maximum percentage error in the measurement of focal length of the mirror is n% find the value of n​

Answer 1

The focal length of a spherical mirror is given by 1 /f = 1 /v + 1/u, where u is the position of object from pole of the mirror and v is position of image from the pole of the mirror

If u = (30 ± 0.3 ) cm and v = (60 ± 0.6 ) cm, the maximum percentage error in the measurement of focal length of the mirror is n% , we have to find the value of n.

Let's derive the formula to find the percentage error of focal length.

we have, 1/f = 1/v + 1/u

now differentiating both sides,

-df/f² = -dv/v² - du/u²

⇒∆f/f² = ∆v/v² + ∆u/u²

⇒∆f/f = f(∆v/v² + ∆u/u²)

⇒∆f/f × 100 = f(∆v/v² + ∆u/u²) × 100

hence percentage error of focal length = f(∆v/v² + ∆u/u²) × 100

here, ∆v = 0.6 , v = 60 , ∆u = 0.3 , u = 30

and f = uv/(u + v) = (60 × 30)/(60 + 30) = 20 cm

∴ percentage error in focal length = 20[0.6/60² + 0.3/30²] × 100

= 2000[0.6/3600 + 0.3/900]

= 2000[(0.6 + 1.2)/3600]

= 2000 × 1.8/3600

= 1 %

Therefore the percentage error in the measurement of focal length is 1 %. hence the value of n is 1.