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Chapter 30: Problem 58

Show that a three-dimensional corner reflector (three mutually perpendicular mirrors, or a solid cube in which total internal reflection occurs) turns an incident light ray through \(180^{\circ} .\) (Hint: Let \(\vec{q}=q_{x} \hat{\imath}+q_{y} \hat{\jmath}+q_{z} \hat{k}\) be a vector in the propagation direction. How does this vector get changed on reflection by a mirror in a plane defined by two of the coordinate axes?)

Short Answer

Expert verified
After successive reflections off of three mutually perpendicular mirrors, the original light-ray vector, \(\vec{q}=q_{x} \hat{\imath}+q_{y} \hat{\jmath}+q_{z} \hat{k}\), becomes \(\vec{q'''}=(-q_{x}) \hat{\imath}+(-q_{y}) \hat{\jmath}+(-q_{z}) \hat{k}\), indicating that the light ray turns an angle of \(180^{\circ}\).

Step by step solution

01

Understand the Reflection in One Mirror

Firstly, consider how one mirror changes the direction of the vector. We know that reflection off a mirror changes the sign of the component of a vector perpendicular to the mirror while leaving the parallel components unaffected. If we assume a mirror is aligned with the xy-plane, then the vector \(\vec{q}=q_{x} \hat{\imath}+q_{y} \hat{\jmath}+q_{z} \hat{k}\) becomes \(\vec{q'}=(-q_{x}) \hat{\imath}+q_{y} \hat{\jmath}+q_{z} \hat{k}\) after reflection.
02

Reflection in Second Mirror

Next, consider how a second mirror perpendicular to the first (assume it's aligned with the xz-plane) changes the direction of the vector \(\vec{q'}\). The vector becomes \(\vec{q''}=(-q_{x}) \hat{\imath}+(-q_{y}) \hat{\jmath}+q_{z} \hat{k}\) after reflection.
03

Reflection in Third Mirror

Finally, reflect the vector \(\vec{q''}\) in a third mirror perpendicular to the other two (assume it's aligned with the yz-plane). The vector becomes \(\vec{q'''}=(-q_{x}) \hat{\imath}+(-q_{y}) \hat{\jmath}+(-q_{z}) \hat{k}\) after reflection.
04

Analyze the Result

Now, observe that \(\vec{q'''}\) is a vector that points in exactly the opposite direction as \(\vec{q}\). This means that the light ray has been turned through an angle of \(180^{\circ}\).

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