Solving Optics Problem: Position of Image of Face Due to Reflection

[EngageEngage]

Homework Statement

Consider a bathroom mirror with a glass sheet of thickness d and an index of refraction n. Your face is located at a distance h from the front surface of the galss sheet. a
a. find the position of the image of your face that you see due to the reflection from the front surface of the glass
b. find the position of the image of your face that you see due to the reflection from the mirror surface behind the glass. Take into accound the effects of refraction at the glass-air interface.

Homework Equations

na/s + nb/s' = (nb - na)/R
1/s + 1/s' = 1/f

The Attempt at a Solution

for part a, we know that the image is a distance of h behind the glass, thus s' = -h. However, I'm not sure what to do for the second part.
the answer to part b is h+2d/n, but I'm not sure how to get there. If someone could help me that would be much appreciated. Thank you

 

[EngageEngage]

I tried to do something else and I got close to the answer, but still not it:

first i did:

1/h+ n/s' = 0 (because R --> infinity for plane mirror)
s' = -nh
then I took the distance from the surface of the mirror to the glass, d, and subtracted s' from it to find the object distance for the mirror:

s2 = d+nh.
After reflection, we have s2' = -d-nh. Then using the 'relevant equation' to find the final image, i get

n/(-d-nh) =-1/(finalim)

final I am = (d+nh)/n => d/n + h.

So, I am off by a factor of two on the (d/n) term. I'm not sure what it is that I'm doing wrong. if someone could please help me out, that would be much appreciated.

 

[Moetasim]

.

To solve part b, we can use the refraction equation, which relates the angles of incidence and refraction at an interface between two different media. In this case, we have air and glass, so the equation becomes:

n_air * sin(theta_air) = n_glass * sin(theta_glass)

Where n_air and n_glass are the refractive indices of air and glass, respectively, and theta_air and theta_glass are the angles of incidence and refraction, respectively.

Since we are interested in the position of the image of our face, we can use the thin lens equation, which relates the object distance (h) and image distance (s') to the focal length (f) of the lens:

1/h + 1/s' = 1/f

Combining these equations, we can solve for the image distance (s'):

1/h + 1/s' = 1/f
1/h + 1/(h+2d) = 1/f
1/(h+2d) = 1/f - 1/h
1/(h+2d) = (h-f)/(hf)
(h+2d)/(h-f) = hf
s' = (h+2d)[(hf)/(h-f)]

Therefore, the position of the image of our face due to the reflection from the mirror surface behind the glass is (h+2d)[(hf)/(h-f)].

I hope this helps. Let me know if you have any other questions.