For the thin convex lens, the radii of curvature are at $15 \mathrm{~cm}$ and $30 \mathrm{~cm}$ respectively. The focal length the lens is $20 \mathrm{~cm}$. The refractive index of the material is :

<p>To find the refractive index of the material of a thin convex lens, we can make use of the Lensmaker's Formula. The Lensmaker's formula is given by:</p> <p>$$\frac{1}{f} = \left( \frac{\mu - 1}{\mu} \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$$</p> <p>where</p> <ul> <li>$f$ is the focal length of the lens,</li> <li>$\mu$ is the refractive index of the material of the lens,</li> <li>$R_1$ is the radius of curvature of the first surface (convex side, positive),</li> <li>$R_2$ is the radius of curvature of the second surface (concave side, negative for convex lens).</li> </ul> <p>Given in the question, the radii of curvature are $15 \, \mathrm{cm}$ and $30 \, \mathrm{cm}$ respectively, and the focal length $f = 20\, \mathrm{cm}$.</p> <p>It's important to pay attention to the signs of the radii of curvature according to the lens maker's convention. For convex lenses, $R_1$ is positive and $R_2$ is negative; however, since the problem doesn't specify which curvature corresponds to which side in relation to the direction of light travel, we'll assume the light travels from left to right: thus, $R_1 = +15 \, \mathrm{cm}$ and $R_2 = -30 \, \mathrm{cm}$.</p> <p>Substituting the given values into the Lensmaker's Formula, we get:</p> <p>$\frac{1}{20} = (\mu - 1) \left( \frac{1}{15} - \frac{1}{-30} \right)$</p> <p>$\frac{1}{20} = (\mu - 1) \left( \frac{1}{15} + \frac{1}{30} \right)$</p> <p>$\frac{1}{20} = (\mu - 1) \left( \frac{2 + 1}{30} \right)$</p> <p>$\frac{1}{20} = (\mu - 1) \left( \frac{3}{30} \right)$</p> <p>$\frac{1}{20} = (\mu - 1) \left( \frac{1}{10} \right)$</p> <p>$\frac{1}{20} = \frac{\mu - 1}{10}$</p> <p>Now, solve for $\mu$:</p> <p>$\mu - 1 = \frac{10}{20}$</p> <p>$\mu - 1 = 0.5$</p> <p>$\mu = 1.5$</p> <p>Hence, the refractive index of the material of the lens is <strong>1.5</strong>, which corresponds to <strong>Option B</strong>.</p>