A pair of eyeglasses is designed to allow a person with a far-point distance of 2.06 to read a road sign at a distance of 20.6. Find the focal length required of these glasses if they are to be worn 2.50 from the eyes.
To find the focal length required for the glasses, we need to use the lens formula:
1/f = 1/v - 1/u
where:
f = focal length of the lens
v = image distance (distance of the road sign from the glasses)
u = object distance (distance of the far point from the glasses)
In this case, the object distance (u) is the far point distance, which is given as 2.06. The image distance (v) is the distance of the road sign from the glasses, given as 20.6. We can substitute these values into the lens formula:
1/f = 1/20.6 - 1/2.06
Simplifying the equation:
1/f = (1 - 10)/20.6
1/f = -9/20.6
To find the value of f, we can take the reciprocal of both sides:
f = 20.6/-9
f ≈ -2.29
Since focal length cannot be negative in this context, the focal length required for the glasses should be positive. Therefore, the focal length required is approximately 2.29.
Now, the glasses are to be worn 2.50 cm away from the eyes. This distance is the distance between the glasses and the object (a person's eyes). We can consider this as the object distance (u) for the lens formula. Substituting this value into the lens formula:
1/2.29 = 1/20.6 - 1/u
Simplifying the equation:
1/2.29 = (1 - 10)/20.6
1/2.29 = -9/20.6
To find the value of u, we can take the reciprocal of both sides:
u = 20.6/-9
u ≈ -2.29
Since the distance cannot be negative, we can consider the absolute value:
u ≈ 2.29
Therefore, the glasses should be worn approximately 2.29 cm from the eyes to achieve the desired focal length.