A converging thin lens made out of crown glass (n=1.52) has a focal length f=1.0m An identical lens (same radius for the spherical surfaces), made out of a high-index plastic with n=1.70 will be ...
A. Converging with a focal length smaller than 1.0 m
B. Diverging with a focal length smaller than 1.0 m
C. Converging with a focal length exactly equal to 1.0 m
D. Converging with a focal length larger than 1.0 m
E. Diverging with a focal length larger than 1.0 m
Look up the lens maker's formula and decide for yourself.
The high-index plastic will have a shorter focal length than the glass lens with the same shape.
Focal length is inversely proportional to n-1.
oh k. got it. thanks
To determine the answer, we need to understand the relationship between the refractive index, focal length, and lens material. The equation that relates these quantities is known as the lens maker's formula:
1/f = (n - 1) * (1/R1 - 1/R2)
Where:
- f is the focal length of the lens.
- n is the refractive index of the lens material.
- R1 is the radius of the curvature of the first lens surface.
- R2 is the radius of the curvature of the second lens surface.
In this case, we have two lenses with identical radii, so R1 = R2. The first lens is made of crown glass with a refractive index of n = 1.52 and focal length f = 1.0 m. The second lens is made of high-index plastic with n = 1.70.
For the crown glass lens, we can rearrange the lens maker's formula to solve for the radius R1:
1/R1 = (n - 1) / (n * f) = (1.52 - 1) / (1.52 * 1.0) = 0.017105
For the high-index plastic lens, we can similarly find R2:
1/R2 = (n - 1) / (n * f) = (1.70 - 1) / (1.70 * 1.0) = 0.176471
Since the radii of both lenses are the same (R1 = R2), it means the high-index plastic lens will have a larger refractive index (n) compared to the crown glass lens. We can see that 1.70 > 1.52.
Therefore, according to the lens maker's formula, the high-index plastic lens will have a smaller focal length (f) compared to the crown glass lens. Mathematically, the focal length is inversely proportional to the refractive index. This means that the focal length of the high-index plastic lens will be smaller than 1.0 m.
So, the correct answer is A. Converging with a focal length smaller than 1.0 m.