Fermat's Principle a. Two poles of he? ights ?? and n? are separated by a horizontal dis?tance d ? . A rope is stretched from the top of one pole to the ground and then to the top of the other pole. Show that the configuration that requires the least amount of rope occurs ? w? hen ??1 =? 2 (see figure). b. Fermat's Principle states that when light travels between two points in the same medium (at a constant speed), it travels on the path that minimizes the travel lime. Show that whe ? n light from a source A ? reflects off of a surface and is received at ? point ? , the angle of incidence equals th?e ang ? le of reflection, or ??1 =? 2 (see figure).
Solution Step 1 (a)Consider that two poles of height and are separated by a horizontal distance apart. A rope is stretched from the top of one pole to the ground and then the top of the other pole. Consider the following figure In the above figure, let,AB = m ,CD = n ,BD = d angle APB = a1d angle CPD = 2 Now the length of the rope is given by AP +CP Let BP = x then DP = dx Then, in right triangleABP By Pythagoras Theorem, 2 2 2 AP = AB +BP = m +x 2 AP = m +x2 2 Similarly, in right triangle CDP By Pythagoras Theorem, CP = CD +DP 2 2 = n +(dx) CP = n +(dx) 2 Therefore, the length of the rope is given by, say L L = AP +CP = m +x + 2 n +(dx) 2 Step 2 For the configuration requiring minimum amount of rope, differentiateLwith respect tox and equate to zero. dx = dx m +x +2 n +(dx) ]2 Equate this to zero In right trianglesABP andCDP , use the trigonometric formula of cosecant function to obtain Substitute the values in the following equation to obtain the following This implies that Therefore, the configuration that requires the least amount of rope occurs when = 1 2