Question

A beam of light strikes a glass sphere of diameter 15 cm converging towards a point 30 cm behind the pole of the spherical surface. Find the position of the image if the refractive index of glass is $\mu = 1.5$.

Answer 1

Here in this question the beam of light that enters the bead is parallel meaning the light rays are coming from at a distance of infinity. So, the object is at infinity. Since we know the object distance and the radius of the bead, we can apply this formula to calculate the image distance.
$\dfrac{{{\mu _2}}}{v} - \dfrac{{{\mu _1}}}{u} = \dfrac{{{\mu _2} - {\mu _1}}}{R}$
Where $\mu$ is the refractive index of the medium, $R$ is the radius of the bead and are $v$ and $u$ image and object distances.

Complete answer:
The light rays move from negative to positive direction (space behind the pole is positive and in front of the pole is positive) here in this question the object rays are converged at 30cm behind the pole
So, our object distance is 30cm.
$u = 30cm$
The refractive index of air is 1 and the bead is 1.5
${\mu _1} = 1$
${\mu _2} = 1.5$
Since the diameter of the bead is 15cm cm the radius of the bead becomes 7.5 cm
$R = 7.5cm$
Now we substitute the value of the above variables into the equation
$\dfrac{{{\mu _2}}}{v} - \dfrac{{{\mu _1}}}{u} = \dfrac{{{\mu _2} - {\mu _1}}}{R}$
$\dfrac{{1.5}}{v} - \dfrac{1}{{30}} = \dfrac{{1.5 - 1}}{{7.5}}$
$\dfrac{{1.5}}{v} = \dfrac{{0.5}}{{7.5}} + \dfrac{1}{{30}}$
$\dfrac{{1.5}}{v} = \dfrac{1}{{15}} + \dfrac{1}{{30}} = \dfrac{{2 + 1}}{{30}} = \dfrac{1}{{10}}$
Now by rearranging we get
$\dfrac{{1.5}}{v} = \dfrac{1}{{10}}$
$v = 1.5 \times 10 = 15cm$
So, the distance of the image formed is 15cm from the centre of the bead.

Note:
Since we are following the sign convention (behind the pole is negative and front of the pole is positive) that light moves from the negative side to the positive side, we must stick with this for solving the entire equation. Placing a wrong sign in place may lead to serious miscalculations and we will eventually land up with the wrong answer. Proper sign convention should be followed when we are solving the questions related to lens and mirror.