Find the focal length of a double concave lens made of glass (n=1.52), having radii of curvature of 6.5 cm and 7.5 cm.
The thickness of the lens is not given, so assume it is a thin lens.
Using the thin-lens approximation of the lens-maker's formula,
P=1/f=(n-1)(1/R1-1/R2)
where
n=refractive index of lens material in air
R1, R2 are radii of curvature of lens surfaces, using the convention that R>0 for convex, R<0 for concave, and R=0 for flat surface. Sign of back surface is reversed. (front surface faces light source).
Applying equation,
n=1.52, R1=-6.5, R2=-(-7.5)
P=1/f=(1.52-1)(-1/6.5-1/7.5)
=-0.1493
The focal length is thus the reciprocal of P.
To find the focal length of a double concave lens, you can use the lens maker's formula:
\(\frac{1}{f} = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)\)
Where:
\(f\) is the focal length of the lens,
\(n\) is the refractive index of the glass (given as 1.52),
\(R_1\) is the radius of curvature of the first surface of the lens (given as 6.5 cm),
\(R_2\) is the radius of curvature of the second surface of the lens (given as 7.5 cm).
Plugging in the values into the formula, we get:
\(\frac{1}{f} = (1.52 - 1) \left(\frac{1}{6.5} - \frac{1}{7.5}\right)\)
Simplifying further:
\(\frac{1}{f} = 0.52 \left(\frac{7.5 - 6.5}{6.5 \cdot 7.5}\right)\)
\(\frac{1}{f} = 0.52 \left(\frac{1}{6.5 \cdot 7.5}\right)\)
\(\frac{1}{f} = 0.52 \left(\frac{1}{48.75}\right)\)
\(\frac{1}{f} = 0.52 \times 0.0205\)
\(\frac{1}{f} = 0.01066\)
Finally, solving for \(f\), we obtain:
\(f = \frac{1}{0.01066}\)
Therefore, the focal length of the double concave lens is approximately 93.93 cm.
To find the focal length of a double concave lens, we can use the lens maker's formula, which is given as:
1/f = (n - 1) * ((1/R1) - (1/R2))
Where:
f = focal length of the lens
n = refractive index of the lens material (for glass, n = 1.52)
R1 = radius of curvature of the first surface of the lens
R2 = radius of curvature of the second surface of the lens
Given that the radii of curvature are R1 = 6.5 cm and R2 = 7.5 cm, and the refractive index is n = 1.52, we can substitute these values into the formula:
1/f = (1.52 - 1) * ((1/6.5) - (1/7.5))
Now, let's calculate the value of the expression on the right-hand side of the equation:
1/f = (0.52) * ((0.1538) - (0.1333))
= (0.52) * (0.0205)
= 0.01066
To find the focal length, we need to take the reciprocal of both sides of the equation:
f = 1 / (0.01066)
≈ 94 cm
Therefore, the focal length of the double concave lens is approximately 94 cm.