2023 A/L Physics Past Paper · Question 30 (English Medium)

Answer: (3) $D$

What the question asks

We want: the total deviation of the ray after it has passed through all three prisms.

What each letter means:

$D$ ($^\circ$): the minimum deviation of one prism.

$\delta_P$ ($^\circ$): the turn of the ray in $P$.

$\delta_{PQ}$ ($^\circ$): the turn of the ray in $P$ and $Q$ together.

$\delta_R$ ($^\circ$): the turn of the ray in $R$.

$\delta$ ($^\circ$): the total deviation.

Given information

What we know:

A ray of one colour meets the prism $P$ and suffers the minimum deviation $D$ there.

The prisms $P$, $Q$ and $R$ are identical equilateral prisms.

$Q$ is upside down. The faces of $P$ and $Q$ that face each other are parallel, with a thin layer of air between them. The same is true for $Q$ and $R$.

Relevant theory

The angle 'd' between the incident ray PQ and the emergent ray RS is the angle of deviation.

The graph shows that there is an angle of minimum deviation (D).

At the state of minimum deviation the ray is found to pass symmetrically through the prism.

The first quote says the deviation is the angle between the ray that goes in and the ray that comes out. The second quote says that at the minimum deviation the ray goes through the prism in a symmetrical way. The stem says $P$ is at the minimum deviation, $D$.

For example, think of walking along a road. The road bends left at one corner, then bends right at a corner of the same size, then bends left again. After the first two corners you face the same way as before, and the third corner is the turn you are left with.

The three prisms do the same to the ray. We find the turn of each part, with the right sign, and add them.

Reasoning and calculation

Push $P$ and $Q$ together in your mind. $P$ is upright and $Q$ is upside down, with their sloping faces facing each other.

The left face of $P$ and the right face of $Q$ are the two long outer faces of this block. They are parallel to each other.

A ray that goes into a block through one face and comes out of the parallel face on the other side leaves parallel to the way it went in. It is just moved to one side.

So after $P$ and $Q$ the ray points the same way as it did at the start.

$0$

$\delta_{PQ} = D - D$

The ray now points the same way as when it met $P$. The left face of $R$ is parallel to the left face of $P$, and $R$ is identical to $P$.

So $R$ meets the ray exactly as $P$ did. It is at the minimum deviation too, and it turns the ray by $D$ the same way as $P$ did.

$D$

$\delta_R = D$

Because of this, we do not have to follow the ray inside $Q$ at all. $P$ and $Q$ together give no turn, whatever the angles inside them are.

The question is left with one prism, $R$.

Required equations

$\delta_P = D$

$\delta_{PQ} = 0$

$\delta = \delta_{PQ} + \delta_R$

All the turns are angles in degrees, so they can be added. No change of units is needed.

$D$

$\delta = \delta_{PQ} + \delta_R = 0 + D$

$a + b$

Reading the options

Now match it with the five options.

(1) $\dfrac{D}{3}$ is not a sum of turns of $D$. It does not match.

(2) $\dfrac{D}{2}$ is not a sum of turns of $D$. It does not match.

(3) $D$ is what we found. This matches.

(4) $2D$ leaves one prism out. It does not match.

(5) $3D$ turns all three prisms the same way. It does not match.

Checks

Check by signs. Call a turn in the way of $P$ positive. $P$ gives $+D$, $Q$ gives $-D$ and $R$ gives $+D$. The sum is $+D$, the same as the answer from the blocks.

Does it make sense? $Q$ is upside down, so it undoes the turn of $P$. That leaves the turn of $R$, which is one prism turning the ray by $D$. The answer is not $3D$, as it would be if all three were the same way up.

Conclusion

$\delta = D$

Conclusion

The quotes give the deviation and the symmetrical path at the minimum deviation. That $P$ and $Q$ together make a block with parallel faces, so the ray comes out parallel to the way it went in, is geometry that we added.

Teacher's note

Look for parallel faces. Two prisms put together upside down make a block with parallel outer faces, and a ray goes through it with no turn.

Then count the turns with their signs.

Exam tip