Show that if two plane mirrors meet at an angle Φ, a single ra...

Question

Show that if two plane mirrors meet at an angle Φ, a single ray reflected successively from both mirrors is deflected through an angle of 2Φ independent of the incident angle. Assume Φ < 90° and that only two reflections, one from each mirror, take place.

Accepted Answer

Hi everyone. Let's take a look at this practice problem dealing with the law of reflections. So in this problem, we have two plane mirrors that meet at an angle of theta, a laser beam strikes the first mirror at an angle incident of phi one and reflects off of it and then strikes the second mirror at an angle of incident of phi two. After reflecting off both mirrors, the laser beam is deflected from its original path if the two mirror meet at an angle of data, where data is less than 90 degrees. What is the total angle of deflection from the laser beam? After the two reflections, we're given four possible choices as our answers. For choice A, we have the divided by two. For choice B, we have theta for choice C, we have two theta and for choice D we have three theta. The first thing we're going to want to do is draw a diagram of our problem. So we're gonna have one of our mirrors, we'll make it horizontal the bottom just to make things easier. And then the second mirror is going to be an angle data with respect to that So we're actually gonna have the light beam coming in, it's going to strike the first mirror and it's gonna strike that at an angle of P one with respect to the perpendicular, it's going to reflect off at that same angle. It's then going to strike the second mirror at the angle of by two. And it's going to reflect off at that angle of 52. And we need to measure the angle between the incident the initial first ray and the reflected ray. And we'll call that angle beta. We're gonna label one more angle which is gonna be alpha and that's gonna be the angle that is right next to beta that is contained in the triangle that has Phi one and Phi two in it. Now, since our answer needs to be, in terms of theta, we want to relate Phi one and Phi two to theta. And we can do that by looking at the triangle that contains the theta and we actually have two additional angles in that triangle. Now recall that the angles of a triangle have to add up to 100 80 degrees. So that's where we're actually going to start to get a relationship between theta and phi one and Phi two. We're gonna have 100 80 degrees that's gonna have to be equal to data. And then one of the angles, the one that's closest to Phi one, it's actually gonna be 90 degrees minus 51 we have plus 90 degrees minus by one. That's because by one is measured with respect to the vertical that is perpendicular to the surface of the mirror. And so that means that P one plus that angle has to be equal to 90 degrees. And so the angle ha is 90 degrees minus 51, we'll do the same thing for the angle that's closest to phi two in our triangle that contains data. And so that will be also plus 90 degrees minus Phi two. And so we can actually solve this for the, the 180 will be canceled out by the 90 plus 90 degrees on the right hand side. And so I can move the P +51 and P two over to the other side making them positive. So I'll get the is equal to P one plus by two. So we now have a relationship between Phi one, Phi two and theta. Now we need to relate the P 5152 to our angle beta that we're looking for. And then once we do that, we'll be able to relate those angles back to theta. So let's actually look at the triangle that is containing both P 5152 and alpha. And again, we're going to use the fact that the angles have to add up to 100 80 degrees of a triangle. So we'll have 100 80 degrees equal to alpha plus the angle with Phi one is going to actually be two phi one since we have the phi one that's being measured with respect to the perpendicular for the incident ray plus the P I one from the reflected ray. So that angle is actually two phi one. Similarly for phi two, it's also uh for that angle there that contains phi two, it's gonna be plus two P +52 since we have a 52 from both the incident and angle uh and reflected angles. And here I can actually factor out the factor of two. And so I'll have 180 degrees is equal to alpha plus two. Multiply the quantity of phi one plus P +52, but P one and P two is equal to theta. So I'll have 100 80 degrees is equal to alpha plus two theta. But if I look at my diagram, if I add alpha and beta together, I get 100 80 degrees. So I can replace alpha with the 180 degrees minus beta. So I have 100 80 degrees is equal to 180 degrees minus beta plus two. Theta. The 180 degrees will cancel out. I can move the beta to the other side and then I'll get beta is equal to theta. So that's the answer that I'm looking for. And this actually corresponds to answer choice C So the trick of this problem was to recall that our law, our, we had to recall our law of reflection that our instant angle was equal to our reflected angle. And we also had to use the fact that the angles of a triangle add up to 100 80 degrees. So we use those two concepts along with some geometry in order to show that our total angle of deflection was equal to two theta. So I hope that this has been useful and I'll see you in the next video.

Show that if two plane mirrors meet at an angle Φ, a single ray reflected successively from both mirrors is deflected through an angle of 2Φ independent of the incident angle. Assume Φ < 90° and that only two reflections, one from each mirror, take place.

Key Concepts

Reflection of Light

Angle of Deflection

Geometric Relationships in Reflection

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