A double-convex lens is to be made of glass with an index of refraction of `1.5`. One surface is to have `2` times the radius of curvature of the other and the focal length is to be `60mm`. What is the (a) smaller and (b) larger radius?

Answer

To solve the problem step-by-step, we will use the Lensmaker's formula and the given information about the lens.

### Step 1: Understand the Problem
We have a double-convex lens made of glass with a refractive index (μ) of 1.5. One surface has a radius of curvature (R1) that is twice the radius of curvature of the other surface (R2). The focal length (f) of the lens is given as 60 mm.

### Step 2: Define the Radii
Let:
- The radius of curvature of the smaller surface (R1) be \( r \).
- The radius of curvature of the larger surface (R2) be \( 2r \).

### Step 3: Apply the Lensmaker's Formula
The Lensmaker's formula is given by:
\[ \frac{1}{f} = (μ - 1) \left( \frac{1}{R1} - \frac{1}{R2} \right) \]
Substituting the values we have:
- \( f = 60 \, \text{mm} = 0.06 \, \text{m} \)
- \( μ = 1.5 \)
- \( R1 = r \)
- \( R2 = -2r \) (negative because it is in the opposite direction)

Thus, the formula becomes:
\[ \frac{1}{0.06} = (1.5 - 1) \left( \frac{1}{r} - \frac{1}{-2r} \right) \]

### Step 4: Simplify the Equation
Calculating \( (1.5 - 1) \):
\[ \frac{1}{0.06} = 0.5 \left( \frac{1}{r} + \frac{1}{2r} \right) \]
Now, simplifying the right side:
\[ \frac{1}{0.06} = 0.5 \left( \frac{2 + 1}{2r} \right) = 0.5 \left( \frac{3}{2r} \right) = \frac{3}{4r} \]

### Step 5: Rearranging the Equation
Now we can rearrange the equation:
\[ \frac{1}{0.06} = \frac{3}{4r} \]
Taking the reciprocal gives:
\[ 0.06 = \frac{4r}{3} \]

### Step 6: Solve for r
Now, we can solve for \( r \):
\[ r = \frac{0.06 \times 3}{4} = \frac{0.18}{4} = 0.045 \, \text{m} = 45 \, \text{mm} \]

### Step 7: Find the Larger Radius
Now that we have \( r \), we can find the larger radius:
\[ R2 = 2r = 2 \times 45 \, \text{mm} = 90 \, \text{mm} \]

### Final Answers
(a) The smaller radius \( r \) is **45 mm**.
(b) The larger radius \( R2 \) is **90 mm**.