Question

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?

Answer 1

(A) ∫∫S₂F⋅dS = 0.5

(B) ∫∫S₂F⋅dS = 1/3

(A) ∫∫S₂F⋅dS = 0.5, (B) ∫∫S₂F⋅dS = 1/3. In (A), the flux is 1/16 of that through S₁; in (B), it's 1/16³, reflecting the inverse square and cube relationships with distance, respectively.

Explanation

In part (A), when the magnitude of F is inversely proportional to the square of the distance from the origin, the flux integral ∫∫S₂F⋅dS can be determined by using the relationship between the flux and the radius of the sphere. The flux through a sphere is proportional to the inverse of the radius. Since S₂ has a radius of 16, and the flux integral is inversely proportional, the value is 1/16 of the flux through S₁. As ∫∫S₁F⋅dS is given as 8, ∫∫S₂F⋅dS would be 8/16 = 0.5.

In part (B), when the magnitude of F is inversely proportional to the cube of the distance from the origin, the relationship changes. The flux through a sphere is now inversely proportional to the cube of the radius. Therefore, for S₂ with a radius of 16, the flux is 1/16³ of the flux through S₁. As ∫∫S₁F⋅dS is given as 8, ∫∫S₂F⋅dS would be 8/(16³) = 1/3.

These results showcase the impact of the inverse relationship between the magnitude of the force field and the distance from the origin on the flux through different spheres. The mathematical calculations are based on the principles of flux through a sphere and the given conditions of the force field.

Answer 2

`4\sqrt 3/9`

Step-by-step explanation:

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

`F(x, y) = <x, y, z> * A(|<x, y, z>|) = <x, y> * A(√(x^2 + y^2))`

`|F(x, y, z)|` which is proportional to
`1/(x^2 + y^2 + z^2)` is given, so that for random constant
`t`:

`F(x, y) = t<x, y, z> / (x^2 + y^2 + z^2)^(3/2)`

Where the magnitude of
`<x, y, z>` is
`√(x^2 + y^2 + z^2)`.

Hence,

`|t<x, y, z>/(x^2 + y^2 + z^2)^(3/2)| = t\sqrt {x^2 + y^2 + z^2}/(x^2 + y^2 + z^2)^(3/2) =\\= t/(x^2 + y^2 + z^2)`

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

`i) s_8(a, b) = <8cos(a)sin(b), 8sin(a)sin(b), 8cos(b)>\\ii) s_(16)(a, b) = <16cos(a)sin(b), 16sin(a)sin(b), 16cos(b)>`

For a in
`[0, 2\pi]` and b in
`[0, 2\pi]`.

From above, it is seen that
`s_(16)(a, b) = 2s_8(a, b)`, thus the partial derivative and normal vectors of
`s_(16)(a, b)` equals to the twice of
`2s_8(a, b)`'s.

Moreover, x, y and z components of
`s_(16)(a, b)` equals to the twice of s_8(a, b)'s, so doubling x, y, and z will rise
`x/(x^2 + y^2 + z^2)^(3/2), y/(x^2 + y^2 + z^2)^(3/2),` and
`z/(x^2 + y^2 + z^2)^(3/2)` by a factor of
`2/(2^2 + 2^2 + 2^2)^(3/2) = \sqrt 3/36`.

Thus,

`F[s_(16)(a, b)]` . (normal of
`s_(16)(a, b)) = (\sqrt3/36)F[s_8(a, b)]` . (2 * normal of
`s_8(a, b)) = \sqrt 3/18 . F[s_8(a, b)]` . (normal of
`s_8(a, b))`

As a result,

`\int\int S_2 (F . n) dS = (\sqrt3/18)\int\int S_1 (F . n) dS = (\sqrt3/18)*(8) = 4\sqrt 3/9`