Question

The headlights of a car emit light of wavelength 400 nm and are separated by 1.2 m. The headlights are viewed by an observer whose eye has an aperture of 4.0 mm. The observer can just distinguish the headlights as separate images. What is the distance between the observer and the headlights?A. 8 kmB. 10 kmC. 15 kmD. 20 km

Answer 1

Answer:

The most correct option is;

B. 10 km

Explanation:

$L = \frac{y \times d}{1.22 \times \lambda} = \frac{1.2 \times 0.004}{1.22 \times 400 \times 10^{-9}} = 9836.066 \ km$

Where:

y = Distance between the two headlights

d = Aperture of observers eye

λ = Wavelength of light

L = Distance between the observer and the headlight

Therefore, from the above solution, the distance between the observer and the headlights is 9386.066 km which is approximately 10 km.

Also we have

sinθ = y/L = 1.22 (λ/d)  

$= 1.22 \times \frac{400 \times 10^{-9}}{0.004}$

sinθ = 1.22×10⁻⁴ rad

Answer 2

Answer:

L = 9836.1 m ≈ 10 km

Explanation:

Given:-

- The separation between head-lights, s = 1.2 m

- The wavelength of light emitted, λ = 400 nm

- The aperture of an eye, d = 4.0 mm

Find:-

What is the distance between the observer and the headlights?

Solution:-

- We will assume the observer is located in between two headlights and the distance between the observer an each headlight is same and equal to (L).

- We will apply the results of Young's split ( interference experiment ). Where the angle of separation between interference pattern form ( θ ). Also the angle of separation between observer and head light. is related to the wavelength and slit opening.

                        sin ( θ ) = 1.22*λ / d

- Determine the angle of separation θ :

                       θ = arc sin ( 1.22*(00*10^-9 / 0.004) )

                       θ = arc sin (0.000122)

                      θ =  0.000122 rads

- Using trigonometric ratios we can determine the distance between the headlights and the observer:

                      sin ( θ ) = s / L

                      sin ( 0.000122 ) = s / L

                      0.000122 = 1.2 / L

                      L = 1.2 / 0.000122

                      L = 9836.1 m ≈ 10 km