A production line inspector wants a mirror that produces an image that is upright with a magnification of 7.1 when it is located 10.0 mm from a machine part.
What is its radius of curvature?
To find the radius of curvature of the mirror, we can use the mirror equation:
1/f = 1/d_o + 1/d_i
where f is the focal length of the mirror, d_o is the object distance, and d_i is the image distance.
In this case, we are given the magnification (M) and the object distance, and we need to find the radius of curvature (R).
The magnification (M) is given by:
M = -d_i/d_o
Since we want an upright image with a magnification of 7.1, we have:
M = 7.1
Using the mirror equation, we can rewrite it as:
1/f = 1/d_o + 1/d_i --> 1/f = 1/(-d_i) - 1/d_o
Multiplying both sides by d_o * (-d_i), we get:
d_o * (-d_i) / f = d_o * (-d_i) / (-d_i) - d_o * (-d_i) / d_o
Simplifying further, we have:
- d_i^2 / f = - d_o * d_i + d_o * (-d_i)
d_i^2 = f * d_o * d_i - f * d_o * (-d_i)
d_i^2 + d_o * d_i * f = 0
d_i^2 + (d_i^2 / M) * f = 0
Multiplying both sides by M, we get:
M * d_i^2 + d_i^2 * f = 0
d_i^2 * (M + f) = 0
Since the image distance cannot be zero, we have:
M + f = 0
f = -M
Now, we know that the object distance (d_o) is 10.0 mm, and the magnification (M) is 7.1. Therefore:
f = -7.1
The radius of curvature (R) of the mirror is given by:
R = 2 * f
R = 2 * (-7.1)
R = -14.2 mm
So, the radius of curvature of the mirror is -14.2 mm (negative because the mirror is concave).
To find the radius of curvature of the mirror, we can use the mirror equation, which is given by:
\(\frac{1}{f} = \frac{1}{do} + \frac{1}{di}\)
Where:
- \(f\) is the focal length of the mirror
- \(do\) is the object distance (distance of the mirror from the object)
- \(di\) is the image distance (distance of the mirror from the image)
In this case, the object distance (\(do\)) is -10.0 mm, because the object is located on the same side as the mirror.
The magnification (\(M\)) is given by:
\(M = -\frac{di}{do}\)
In this case, \(M = 7.1\). Since the image produced is upright, the magnification will be positive.
Using the magnification equation, we can rewrite the image distance (\(di\)) in terms of the object distance (\(do\)) and magnification (\(M\)):
\(di = M \cdot do\)
Substituting the given values, we have:
\(di = 7.1 \cdot (-10.0) = -71.0\) mm
Now, let's substitute the values of \(do\) and \(di\) into the mirror equation to find the focal length (\(f\)):
\(\frac{1}{f} = \frac{1}{-10.0} + \frac{1}{-71.0}\)
Simplifying, we get:
\(\frac{1}{f} = \frac{-71.0 - 10.0}{-10.0 \cdot -71.0} = \frac{-81.0}{710.0}\)
Taking the reciprocal, we get:
\(f = \frac{710.0}{-81.0} = -8.76\) mm
Since the given information states that the mirror produces an upright image, the negative sign indicates a concave mirror.
Therefore, the radius of curvature (\(R\)) can be calculated as twice the absolute value of the focal length (\(|f|\)):
\(R = 2 \cdot |-8.76| = 17.52\) mm
Hence, the radius of curvature of the mirror is 17.52 mm.