Reflection, Refraction and Lenses: Question 10

Worked solution

Part (a): Angle of incidence and turn at the top mirror

A mirror tilted at $45°$ to the vertical tube is also tilted at $45°$ to a horizontal ray, since $45°$ from vertical is the same as $45°$ from horizontal. The normal to the mirror (perpendicular to its surface) therefore also makes an angle of $45°$ with the incoming horizontal ray.

So the angle of incidence is:

$i = 45°$

By the law of reflection, the angle of reflection is also $45°$. The ray is turned from its original horizontal direction through the sum of these two equal angles:

$\text{angle turned} = i + r = 45° + 45° = 90°$

The horizontal ray is therefore turned through $90°$, sending it straight down the vertical tube.

Part (b): Path between the two mirrors

After reflecting off the top mirror, the ray travels in a single straight line, vertically downward, parallel to the walls of the tube. It does not bend again until it reaches the second mirror, so it travels the full $1.2\text{ m}$ separating the two mirrors as a straight vertical path.

Part (c): Reflection at the bottom mirror

The bottom mirror is fixed at the same $45°$ tilt as the top mirror, so the now-vertical ray strikes it at an angle of incidence, measured from the normal, of:

$i = 45°$

By the law of reflection, this ray is again turned through a further $90°$. Since it arrived travelling vertically downward, this second $90°$ turn redirects it to travel horizontally once more, out of the periscope tube and into the operator’s eye, at the lower end of the tube.

Part (d): Characteristics of the final image

Because both reflections are from plane mirrors, the final image shares the usual properties of a plane-mirror image, with one extra effect from using two mirrors instead of one:

• Virtual, the image cannot be captured on a screen; the light rays only appear to come from the ship.
• Same size as the ship, plane mirrors never magnify or shrink an image (magnification $= 1$).
• Not laterally inverted. A single plane mirror would reverse the image left-to-right, but with two reflections in the periscope, this inversion is applied twice and cancels out, so the operator sees the ship the same way round as they would with the naked eye.

Final answers

• (a) Angle of incidence at the top mirror $= \boxed{45°}$; the ray is turned through $\boxed{90°}$
• (b) The ray travels in a straight vertical line down the tube to the bottom mirror
• (c) Angle of incidence at the bottom mirror $= \boxed{45°}$; the ray is turned a further $90°$ and leaves the periscope travelling horizontally
• (d) The final image is virtual, the same size as the ship, and not laterally inverted (correct way round), because the two reflections cancel out the inversion of a single mirror