Question

Use spherical coordinates to Find the average distance from a point in a ball of radius a to its center.

Answer 1

The average distance from a point in the ball to its center = $\frac{3}{4}$ A

Given the radius in the question = a

The integration of the total distance before dividing the volume will be =

⇒ $\int_0^{2\pi}\int_0^\pi\int_0^A a\cdot a^2\sin\theta\,da\,d\theta\,d\phi$

⇒ $\int_0^{2\pi}\int_0^\pi\int_0^A a^3\sin\theta\,da\,d\theta\,d\phi$

⇒ $\frac{A^{4} }{2}$ $\int_0^{2\pi}\int_0^\pi \sin\theta\,d\theta\,d\phi$

⇒ $\frac{A^{4} }{2}$ $\int_0^{2\pi}1\,d\phi$

⇒ $\pi A^{4}$

Let us divide the average distance by the ball's volume,

⇒ $\frac{\pi a^4}{(4/3)\pi a^3}$

⇒ ${\frac{3a}4}$

By the use of the Parcly Taxel's way,

Let the radius of the ball be = A

f(x) = probability density function

Distance = D from the given point to the center.

P( D < a )

$\begin{align}P(D\leq r)&= \frac{ \frac 43 \pi r^3 }{\frac 43 \pi R^3}=\frac {r^3}{R^3}, \ \ \ 0\leq r\leq R. \end{align}$$\frac{\frac{4}{3} \pi a^{3} }{\frac{4}{3} \pi A^{3} } = \frac{a^{3} }{R^{3} }$. 0 $\leq$ a $\leq$ A

Therefore, The average distance from a point in the ball to its center = $\frac{3}{4}$ A

To know more about Spherical coordinates,

https://brainly.com/question/19053603

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