`(dv)/(r^(2))=4pidr,` when `r=0, v=0`

Show that , v = (A)/(r) + B satisfies the differential equation (d^(2) v)/(dr^(2)) + (2)/(r) .(dv)/(dr) = 0

The velocity v and displacement x of a particle executing simple harmonic motion are related as v (dv)/(dx)= -omega^2 x . At x=0, v=v_0. Find the velocity v when the displacement becomes x.

The velocity v and displacement x of a particle executing simple harmonic motion are related as v (dv)/(dx)= -omega^2 x . At x=0, v=v_0. Find the velocity v when the displacement becomes x.

The velocity v and displacement x of a particle executing simple harmonic motion are related as v (dv)/(dx)= -omega^2 x . At x=0, v=v_0. Find the velocity v when the displacement becomes x.

The velocity v and displacement x of a particle executing simple harmonic motion are related as v (dv)/(dx)= -omega^2 x . At x=0, v=v_0. Find the velocity v when the displacement becomes x.

Show that,v=(A)/(r)+B satisfies the differential equation (d^(2)v)/(dr^(2))+(2)/(r)*(dv)/(dr)=0

A ball of radius R carries a positive charge throughout its volume, such that the volume density of charge depends on distance r from the ball's centre as rho = rho _(0) ( 1- ( r )/®) , where rho_(0) is a constant. Assuming the permittivity of ball to be one , find magnitude of electric field as a function of distance r, both inside and outside the ball. Strategy : The field has a spherical symmetry . For a point outside the ball ( r gt R ) phi =oint vec(E). bar(dA) = E xx 4 pi r^(2) By Gauss law , phi= ( Q)/( epsilon_(0)) implies E_(out) = (Q)/( 4pi epsilon_(0)r^(2)) , where Q is total charge For a point inside the ball ( r lt R ) phi = oint vec(E) . bar(dA) = E xx 4 pi r^(2) By Gauss law, phi = ( q_(enc))/(epsilon_(0)) implies E_("in")= ( q_(enc))/( 4pi epsilon_(0)r^(2)) where q_(enc)= charge in a sphere of radius r ( lt R) To find the enclosed charge , consider a spherical shell of radius r and thickness dr Its volume dV = 4pi r^(2) dr Charge dq = rho dV implies dq= rho _(0) (1-(r )/(R)) 4pi r^(2) dr implies q= int _(0)^(r ) rho _(0) ( 1- (r)/( R )) 4pi r^(2) dr = rho _(0)4pi [ int_(0)^(r ) r^(2) dr - int_(0)^(r ) (r^(3))/(R) dr ]= rho _(0) 4pi [ ( r^(3))/( 3) - (r^(4))/( 4R)]