Question

Light travels with a speed of $2x{10}^{8}m{s}^{-1}$ in crown glass of refractive index $1.5$. The speed of light in dense flint glass having a refractive index of $1.8$ is

• A. $1.33x{10}^{8}m{s}^{-1}$
• B. $1.67x{10}^{8}m{s}^{-1}$
• C. $2x{10}^{8}m{s}^{-1}$
• D. $3x{10}^{8}m{s}^{-1}$

Answer 1

The correct option is B

$1.67x{10}^{8}m{s}^{-1}$

Explanation for the correct option

Option B: $1.67x{10}^{8}m{s}^{-1}$

Step 1- Given data and assumptions

Speed of light in crown glass is $v_{cg}=2x{10}^{8}m{s}^{-1}$

Refractive index of crown glass is ${\mu }_{cg}=1.5$

The refractive index of flint glass is ${\mu }_{fg}=1.8$

Step 2- Formulae used

1. When the ray travels from medium $1$ to medium $2$, the refractive index of $medium2\left({\mu }_2\right)$ with respect to $medium1\left({\mu }_1\right)$ is given by ${\mu }_{21}=\frac{{\mu }_2}{{\mu }_1}$
2. The relation between the refractive index and the velocity of light in the mediums is given by ${\mu }_{21}=\frac{{\mu }_2}{{\mu }_1}=\frac{v_1}{v_2}$where $v_1$and $v_2$ are the velocities in medium 1 and 2 respectively.
3. When a ray travels from air to crown glass, the relation between the velocity and refractive index is given by$\frac{{\mu }_{cg}}{{\mu }_a}=\frac{v_a}{v_{cg}}$
4. When a ray travels from air to flint glass, the relation between the velocity and refractive index is given by$\frac{{\mu }_{fg}}{{\mu }_a}=\frac{v_a}{v_{fg}}$
5. The expression of the velocity of light in a flint glass can be obtained by dividing the expressions obtained in steps 3 and 4 $$\begin{gathered} \frac{\frac{{\mu }_{cg}}{\cancel{{\mu }_a}}}{\frac{{\mu }_{fg}}{\cancel{{\mu }_a}}}=\frac{\frac{\cancel{v_a}}{v_{cg}}}{\frac{\cancel{v_a}}{v_{fg}}},or\frac{{\mu }_{cg}}{{\mu }_{fg}}=\frac{v_{fg}}{v_{cg}} \\ {v}_{fg}=\frac{{\mu }_{cg}}{{\mu }_{fg}}x{v}_{cg} \end{gathered}$$

Step 3- Calculations:

1. The velocity of light in a flint glass is calculated as $v_{fg}=\frac{{\mu }_{cg}}{{\mu }_{fg}}x{v}_{cg},or{v}_{fg}=\frac{1.5}{1.8}x2x{10}^{8}m{s}^{-1}=1.67x{10}^{8}m{s}^{-1}$
2. The velocity of light in flint glass is $1.67x{10}^{8}m{s}^{-1}$

The incorrect options are

Option A: $1.33x{10}^{8}m{s}^{-1}$

Option C : $2x{10}^{8}m{s}^{-1}$

Option D: $3x{10}^{8}m{s}^{-1}$

Thus, the speed of light in dense flint glass is $1.67x{10}^{8}m{s}^{-1}$

Hence, option (B) is correct.