72 - 74 Light intensity of illumination and theory of attraction

Problem 72
A light is to be placed above the center of a circular area of radius a. What height gives the best illumination on a circular walk surrounding the area? (When light from a point source strikes a surface obliquely, the intensity of illumination is

$I = \dfrac{k \sin \theta}{d^2}$

where θ is the angle of incidence and d the distance from the source.)



Problem 73
It is shown in the theory of attraction that a wire bent in the form of a circle of radius a exerts upon a particle in the axis of the circle (i.e., in the line through the center of the circle perpendicular to the plane) an attraction proportional to

$\dfrac{h}{(a^2 + h^2)^{3/2}}$

where h is the height of the particle above the plane of the circle. Find h, for maximum attraction. (Compare with Problem 72 above)



Problem 74
In Problem 73 above, if the wire has instead the form of a square of side $2l$, the attraction is proportional to

$\dfrac{h}{(h^2 + l^2)\sqrt{h^2 + 2l^2}}$

Find h for maximum attraction.