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Flux across concentric spheres Consider the radial fields
Chapter 11, Problem 65E(choose chapter or problem)
Consider the radial fields \(\mathbf{F}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{p / 2}}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}}\), where p is a real number. Let S consist of the spheres A and B centered at the origin with radii \(0<a<b\), respectively. The total outward flux across S consists of the outward flux across the outer sphere B minus the inward flux across the inner sphere A.
a. Find the total flux across S with p = 0. Interpret the result.
b. Show that for p = 3 (an inverse square law), the flux across S is independent of a and b.
Questions & Answers
QUESTION:
Consider the radial fields \(\mathbf{F}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{p / 2}}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}}\), where p is a real number. Let S consist of the spheres A and B centered at the origin with radii \(0<a<b\), respectively. The total outward flux across S consists of the outward flux across the outer sphere B minus the inward flux across the inner sphere A.
a. Find the total flux across S with p = 0. Interpret the result.
b. Show that for p = 3 (an inverse square law), the flux across S is independent of a and b.
ANSWER:Solution 65E
Step 1:
Given that
Consider the radial fields where p is a real number. Let S consist of the spheres A and B centered at the origin with radii 0<a, respectively. The total outward flux across S consists of the flux out of S across the outer sphere B minus the flux into S across the inner sphere A.
