Suppose f is a radial force field, s1 is a sphere of radius 4 centered at the origin, and the flux integral ∫∫s1f⋅ds=8.let s2 be a sphere of radius 16 centered at the origin, and consider the flux integral ∫∫s2f⋅ds.(a) if the magnitude of f is inversely proportional to the square of the distance from the origin, what is the value of ∫∫s2f⋅ds? (b) if the magnitude of f is inversely proportional to the cube of the distance from the origin, what is the value of ∫∫s2f⋅ds?

Explanation:

As it is given that F is radial, by definition,  some function A(x, y) can be rewritten as follow:

which is proportional to is given, so that for random constant :

Where the magnitude of is .

Hence,

The sphere of radius 8 and 16 can be parametrized by spherical coordinates: as follow:

For a in and b in .

From above, it is seen that , thus the partial derivative and normal vectors of equals to the twice of 's.

Moreover, x, y and z components of equals to the twice of s_8(a, b)'s, so doubling x, y, and z will rise and by a factor of .

Thus,

. (normal of . (2 * normal of . (normal of

As a result,

54 degrees incident angle = reflected angle

angle is from normal turned to line of ray