One end of a cylindrical glass rod of refractive index 1.5 is a hemispherical surface of radius of curvature 20mm. An object is placed on the axis of the rod at 80mm to the left of the vertex of the angle of the surface. Determine the position of the image
To determine the position of the image, we can use the lens formula:
1/f = 1/v - 1/u
Where:
- f is the focal length of the lens
- v is the image distance from the lens
- u is the object distance from the lens
In this case, the cylindrical glass rod is acting as a convex lens, and the hemispherical surface of radius of curvature 20mm is acting as its curved surface. The refractive index of the glass rod is given as 1.5.
Since the object is placed 80mm to the left of the vertex of the curved surface, the object distance (u) = -80mm (negative sign indicating it is on the left side).
We need to find the position of the image (v).
First, we need to find the focal length (f) of the lens using the formula:
f = R / (n - 1)
Where:
- R is the radius of curvature of the lens surface (in this case, the hemispherical surface)
- n is the refractive index of the lens material
Given:
- R = 20mm
- n = 1.5
Plugging in these values, we get:
f = 20mm / (1.5 - 1)
f = 20mm / 0.5
f = 40mm
Now, we can substitute the values of f and u into the lens formula:
1/40mm = 1/v - 1/-80mm
Multiplying through by 40v(-80) to clear the fractions, we get:
-2v(-80) = 40(-80) + 40v
160v = -3200 + 40v
Simplifying, we get:
160v - 40v = -3200
120v = -3200
v = -3200 / 120
v = -26.67 mm
Therefore, the position of the image is approximately 26.67mm to the left of the vertex of the curved surface.