Chapter 1, Problem 1.1CYU

Check Your Understanding Table 1.1 shows that ethanol and fresh water have very similar indices of refraction. By what percentage do the speeds of light in these liquids differ?

To determine

The percentage difference in speed of light in ethanol and fresh water.

Answer to Problem 1.1CYU

The percentage difference in speed of light in ethanol and fresh water is $2.2\%$.

Explanation of Solution

Given:

The refractive index of ethanol is $1.361$.

The refractive index of fresh water is $1.333$.

Formula used:

The speed of light in ethanol is given by,

  $v_{\text{e}}=\frac{c}{n_{\text{e}}}$

The speed of light in fresh water is given by,

  $v_{\text{w}}=\frac{c}{n_{\text{w}}}$

The percentage difference in speed of liquid in ethanol and fresh water is given by,

  $P=\frac{v_{\text{w}}-v_{\text{e}}}{v_{\text{w}}}\times 100$

Calculation:

The speed of light is $3\times {10}^8\text{m}/\text{s}$.

The speed of light in ethanol is calculated as,

  $\begin{array}{c}{v}_{\text{e}}=\frac{c}{n_{\text{e}}}\\ =\frac{3\times {10}^8\text{m}/\text{s}}{1.361}\\ =2.2\times {10}^8\text{m}/\text{s}\end{array}$

The speed of light in fresh water is calculated as,

  $\begin{array}{c}{v}_{\text{w}}=\frac{c}{n_{\text{w}}}\\ =\frac{3\times {10}^8\text{m}/\text{s}}{1.333}\\ =2.25\times {10}^8\text{m}/\text{s}\end{array}$

The percentage difference in speed of liquid in ethanol and fresh water is calculated as,

  $\begin{array}{c}P=\frac{v_{\text{w}}-v_{\text{e}}}{v_{\text{w}}}\times 100\\ =\frac{\left(2.25\times {10}^8\text{m}/\text{s}\right)-\left(2.2\times {10}^8\text{m}/\text{s}\right)}{2.25\times {10}^8\text{m}/\text{s}}\times 100\\ =\frac{0.05}{2.25}\times 100\\ =2.2\%\end{array}$

Conclusion:

Therefore, the percentage difference in speed of liquid in ethanol and fresh water is $2.2\%$.