The diffusion equation in spherical coordinates for a spherically symmetric concentration ( C(r,t) ), where ( r ) is the radial distance and ( t ) is time, can be written as:<\/p>\n\n\n\n
[
\\frac{\\partial C(r,t)}{\\partial t} = D \\left( \\frac{\\partial^2 C(r,t)}{\\partial r^2} + \\frac{2}{r} \\frac{\\partial C(r,t)}{\\partial r} \\right)
]
where ( D ) is the diffusion coefficient.<\/p>\n\n\n\n
To solve this, we use separation of variables by assuming a solution of the form:
[
C(r,t) = R(r)T(t)
]
Substituting this into the diffusion equation, we get:
[
T'(t)R(r) = D \\left( R”(r)T(t) + \\frac{2}{r} R'(r)T(t) \\right)
]
Dividing both sides by ( D R(r) T(t) ), we obtain:
[
\\frac{T'(t)}{D T(t)} = \\frac{R”(r)}{R(r)} + \\frac{2}{r} \\frac{R'(r)}{R(r)}
]
Since the left side depends only on ( t ) and the right side only on ( r ), both must equal a constant, say ( -\\lambda^2 ).<\/p>\n\n\n\n
Now, solving for ( R(r) ), we get the ordinary differential equation:
[
r^2 R”(r) + 2r R'(r) + \\lambda^2 r^2 R(r) = 0
]
This is a standard form of the spherical Bessel differential equation, with general solutions for ( R(r) ) being the spherical Bessel functions of the first and second kinds, ( J_\\lambda(r) ) and ( Y_\\lambda(r) ), respectively. For a physical problem with boundary conditions (e.g., ( C(a,t) = 0 ) for a sphere of radius ( a )), we choose the solution that satisfies the boundary conditions, which generally leads to using only the spherical Bessel function of the first kind, ( J_\\lambda(r) ).<\/p>\n\n\n\n
The time dependence ( T(t) ) leads to the solution:
[
T(t) = e^{-D\\lambda^2 t}
]
Thus, the general solution for the concentration is:
[
C(r,t) = \\sum_{n=1}^{\\infty} A_n J_{\\lambda_n}(r) e^{-D \\lambda_n^2 t}
]
where ( \\lambda_n ) are the roots of the equation ( J_\\lambda(a) = 0 ), which ensures the boundary condition at ( r = a ) is satisfied.<\/p>\n\n\n\n
The spherical Bessel functions of the first kind are crucial in solving diffusion problems in spherical geometries, especially with specific boundary conditions.<\/p>\n\n\n\n Here is the plot showing the spherical Bessel functions ( J_n(r) ) for different modes ( n = 1, 2, 3 ). The radial distance ( r ) is on the x-axis, and the corresponding function values are on the y-axis. This visual representation helps in understanding how these functions behave, which is important when solving diffusion problems in spherical geometries.<\/p>\n","protected":false},"excerpt":{"rendered":" Diffusion in a sphere of radius a on the basis of spherical coordinates. Develop the solution for concentration as a function of radial distance and time in terms of the spherical Bessel functions. The correct answer and explanation is : The diffusion equation in spherical coordinates for a spherically symmetric concentration ( C(r,t) ), where […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-198221","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/198221","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=198221"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/198221\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=198221"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=198221"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=198221"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/03\/image-652.png\")
<\/figure>\n\n\n\n