Inverse square fields are special Let F be a radial field

QUESTION:

Inverse square fields are special Let F be a radial Held \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p}\). where p is a real number and \(\mathbf{r}=\langle x, y, z\rangle\). With p = 3, F is an inverse square field.

a. Show that the net flux across a sphere centered at the origin is independent of the radius of the sphere only for p = 3.

b. Explain the observation in pan (a) by finding the flux of \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{P}\) across the boundaries of a spherical box \(\left\{(\rho, \varphi, \theta): a \leq \rho \leq b, \varphi_{0} \leq \varphi \leq \varphi_{1}, \theta_{1} \leq \theta \leq \theta_{2}\right\}\) for various values of p.

Text Transcription:

F = r/|r|^p

r = langle x, y, z rangle

{(rho, varphi, theta): a leq rho leq b, varhpi_0 leq varphi leq varphi_1, theta_1 leq theta leq theta_2}