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.)
 

Solution:

 

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)

Solution:

 

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.

Solution:

 
Subscribe to MATHalino.com on