Spherical shell
In geometry, a spherical shell is a generalization of an annulus to three dimensions. It is the region of a ball between two concentric spheres of differing radii.[1]
Volume
The volume of a spherical shell is the difference between the enclosed volume of the outer sphere and the enclosed volume of the inner sphere:
![{\displaystyle {\begin{aligned}V&={\tfrac {4}{3}}\pi R^{3}-{\tfrac {4}{3}}\pi r^{3}\\[3mu]&={\tfrac {4}{3}}\pi {\bigl (}R^{3}-r^{3}{\bigr )}\\[3mu]&={\tfrac {4}{3}}\pi (R-r){\bigl (}R^{2}+Rr+r^{2}{\bigr )}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb99f83c65f0782b2fc723b1d5e22b23f09b85b8)
where r is the radius of the inner sphere and R is the radius of the outer sphere.
Approximation
An approximation for the volume of a thin spherical shell is the surface area of the inner sphere multiplied by the thickness t of the shell:[2]
when t is very small compared to r ().
The total surface area of the spherical shell is .
See also
References
- ^ Weisstein, Eric W. "Spherical Shell". mathworld.wolfram.com. Wolfram Research, Inc. Archived from the original on 2 August 2016. Retrieved 7 January 2017.
- ^ Znamenski, Andrey Varlamov; Lev Aslamazov (2012). A.A. Abrikosov Jr. (ed.). The wonders of physics. Translated by A.A. Abrikosov Jr.; J. Vydryg; D. Znamenski (3rd ed.). Singapore: World Scientific. p. 78. ISBN 978-981-4374-15-6.