The correct answer is B because:

Write an expression for the total length and the time taken for the light to travel through the device:
- $D = x_{1} + x_{2} + x_{3} + x_{4} + . . . = v_{1} Δ t + v_{2} Δ t + v_{3} Δ t + v_{4} Δ t + . . .$ where v_i is the speed of light passing through block i.
Express each speed, v_i in terms of its refractive index, n_i:
- $v_{i} = \frac{c}{n_{i}} \to v_{1} Δ t + v_{2} Δ t + v_{3} Δ t + v_{4} Δ t + . . .$
- $v_{i} = 0 + \frac{c}{n_{1}} Δ t + \frac{c}{n_{2}} Δ t + \frac{c}{n_{3}} Δ t + \frac{c}{n_{4}} Δ t + . . .$
- $v_{i} = \frac{c}{n_{1}} Δ t$
Use the relationship n_{i + 1}= 2n_i:
- $v_{i + 1} = \frac{c}{n_{1}} Δ t + \frac{c}{n_{2}} Δ t + \frac{c}{n_{3}} Δ t + \frac{c}{n_{4}} Δ t + . . .$
- $v_{i + 1} = \frac{c}{n_{1}} Δ t + \frac{c}{2 n_{1}} Δ t + \frac{c}{4 n_{1}} Δ t + \frac{c}{8 n_{1}} Δ t + . . .$
- $v_{i + 1} = \frac{c Δ t}{n_{1}} (1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + . . .) = \frac{2 c Δ t}{n_{1}}$
- $v_{i + 1} = \frac{2 c Δ t}{n_{1}}$
Obtain the ratio of the length of the block to the distance travelled by light in a vacuum in Δt:
- Ratio = $\frac{v_{i}}{v_{i + 1}} = \frac{2 c Δ t}{n_{1}} \div c Δ t = \frac{2}{n_{1}}$

A is incorrect because it is the expression obtained when dividing the ratio incorrectly or when factorising the expansion incorrectly.

C is incorrect because this is the expression obtained if the relationship n_{i + 1} = 2n was not used

D is incorrect because it is the ratio taken the wrong way around. This is a common error made by students so make sure you understand that the ratio of say x to y is equal to the value obtained by dividing x by y.

This question tested your understanding of the fact that optically refractive materials alter the speed of light passing through them. This allowed us to express the contribution of each length of the device to the total in terms of their refractive indices.

We end up with an infinite series of terms, it needed to be evaluated by tending to the limit of two. Notice that the series is an infinite geometric series from GCSE maths: $S_{\infty} = \frac{a}{1 - r} = \frac{1}{1 - \frac{1}{2}} = 2,$ where in this case the first term a = 1 and the common ratio of the terms $r = \frac{1}{2}$ .

Material	A	B	C	D	E
n	2.22	1.89	1.50	1.30	1.65

Refraction & Reflection (OCR A Level Physics)

Exam Questions