Consider the x-y plane to be a mirror with a medium exhibiting a constant refrac
ID: 1884707 • Letter: C
Question
Consider the x-y plane to be a mirror with a medium exhibiting a constant refractive index above it. Set up the calculation using point A and B as the start and end points a height H above the x-y plane. Consider points A' and B' directly beneath points A and B, but lying in the x-y plane. You must consider light reflecting at point P somewhere in the x-y plane. Suppose the x coordinate of P is fixed: x=X, prove that the minimum path length (AD+ PB) for this arbitrary Xp is given by yp -0. Hint: To simplify things, no loss of generality is incurred by making points A and B directly above the x axis with point A at x=0 and point B at x-2 L from point A i.e. set Point A as (0,0,H) and Point B as (21,0, H). In this situation P is at (Xp, yp,0) where Xpis fixed. See the hint diagram (2L,0,H) 2L (0,0,HExplanation / Answer
Let the co-ordinates of P be xp and yp lying in x,y plane
x-y plane is a mirror.
we can consider the line joing A,B of height H directly above the x-axis i.e projection of this line on to x-y plane is on the x-axis. i.e the line joining A' and B' , further we choose the point A* as our origin.
This makes no difference to the solution as we have only selectd a simplified co-odinate system.
given that xp is fixed
co-ordinates of A (0,0,H) that of B (2L,0,H)
we take 2L as the distance between A and B
now light from A refelcted at P reaches B
angle of incidence = angle of reflection
APB is an isoceless traingle and AP =BP
path of the light = 2AP
A = (0,0,H) and P = (xp,yp,0)
AP2 = xp2 + yp2 + H2
in this expression for AP , xp is fixed , x co-ordinate of P
H - height of the line AB is fixed
the only variant is yp, y co-rdinated of P
AP2 is +ve and all quantites on RHS or +ve
hence AP2 is minimum for yp =0
=> for minimum path of light yp =0