Question

The respective angles of a flint glass prism and a crown glass prism are $A$ and $A^{\prime }$ and their refractive indices for the mean ray are $n$ and $n^{\prime }$ respectively. If these prism are to be combined to produce dispersion without deviation of the mean ray, then $A/A^{\prime }$ should be :

• A. $\frac{n-1}{n^{\prime }-1}$
• B. $-\left(\frac{n^{\prime }-1}{n-1}\right)$
• C. $\frac{n+1}{n^{\prime }+1}$
• D. $\frac{n^{\prime }+1}{n+1}$

Answer 1

The correct option is B $-\left(\frac{n^{\prime }-1}{n-1}\right)$

To have dispersion without deviation means that there is a dispersion of light after passing through the prism arrangement such that the yellow light ray or the mean ray experiences no deviation with respect to the incident ray i.e. is parallel to the incident beam.

To achieve dispersion without deviation, glass prisms are so adjusted that the deviation produced for the mean rays by the first prism is equal and opposite to that produced by the second prism, then the final ray will be parallel to the incident ray.

$\Rightarrow ({\delta }_y{)}_{net}=0$

$\Rightarrow \left({\delta }_y\right)+({\delta }_y{)}^{\prime }=0$

where, ${\delta }_y$ and ${\delta }_y^{\prime }$ are the deviation of the mean rays of light.

$\Rightarrow \left({\delta }_y\right)=-({\delta }_y{)}^{\prime }$

We know,

$\delta =A(\mu -1)$

$\Rightarrow A(n-1)=-A^{\prime }(n^{\prime }-1)$

$\frac{A}{A^{\prime }}=-(\frac{n^{\prime }-1}{n-1})$

The negative sign shows that the refracting angles of the two prisms are in opposite directions as shown in the figure.
Hence, the correct answer is OPTION B.