A spherical generator is used to produce a -15.00D surface on glass of refractive index 1.80. The diameter of the cutting tool is 80mm and the radius of the cutting surface is 4mm. What is the angle between the axis of the tool and the axis of the lens?
To find the angle between the axis of the tool and the axis of the lens, we need to determine the tangent of that angle.
We know that a spherical generator with a given radius will produce a specific lens power. In this case, the spherical generator is producing a -15.00D (diopter) surface on the glass.
The formula to calculate the lens power is:
Power (in diopters) = (n - 1) / R
Where:
- n is the refractive index of the material
- R is the radius of curvature in meters
In this case, the radius of the cutting surface is given in millimeters (mm), so we need to convert it to meters by dividing by 1000.
R = 4mm / 1000 = 0.004m
Substituting the values into the formula, we have:
-15.00D = (1.80 - 1) / 0.004
To find the tangent of the angle, we can use the inverse of the lens power formula:
Tangent (θ) = |Power| / (1 - |Power| * R)
Tangent (θ) = |-15.00| / (1 - |-15.00| * 0.004)
Now we can calculate the angle:
θ = arctan(Tangent (θ))
θ = arctan(|-15.00| / (1 - |-15.00| * 0.004))
Evaluating this expression will give us the angle between the axis of the tool and the axis of the lens.