In a thin prism of glass (refractive index 1.5), which of the following relations between the angle of minimum deviations δₘ and angle of prism r will be correct?

Steps to Solve

1. Understanding the relationship between angle of minimum deviation and prism angle

For a thin prism, the relationship between the angle of minimum deviation ($\delta_m$) and the angle of the prism ($r$) can be derived from Snell's Law and the geometry of the prism.

2. Applying the formula for minimum deviation

The formula that relates the refractive index ($n$), angle of prism ($r$), and angle of minimum deviation ($\delta_m$) for a thin prism is:

$$ n = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} $$

For small angles, we can approximate $\sin x \approx x$, leading us to:

$$ n \approx \frac{\frac{A + \delta_m}{2}}{\frac{A}{2}} $$

From here, we simplify it to obtain a relation for $\delta_m$ in terms of $r$.

3. Calculating the angle of minimum deviation

Since $A \approx r$ for a thin prism, we can interpret this as:

$$ \delta_m \approx (n - 1)r $$

Given $n = 1.5$ for glass, we substitute that in:

$$ \delta_m \approx (1.5 - 1)r = 0.5r $$

4. Determining the final relation

From the earlier steps, we have derived that:

$$ \delta_m = \frac{r}{2} $$

This relation indicates the correct answer among the options given.