- #1

Neolight

## Homework Statement

Suppose we have a Electic field, E (vector) = kr

^{2}r(vector). Find the total charge contained in the sphere of radius R centered in the origin.

solution-

here E can also be written as kr

^{3}r

^{∧}, where r

^{∧}is the unit vector

this is the given question , so obiously the best way to solve this is to use Gauss Law but while using it i have a confusion on which surface area vector to choose..

since gauss law is

∮E.ds = Q

_{enclosed}

__/ε__

_{°}this is where my confussion starts ,ds(surface area vector in spherical polar ) = r

^{2}sin(θ )drdθdΦ

but here the area element and E are in dot product so i have to use vector form of ds

since this is a sphere the area vector will be directed along the radial line so

ds(vector) = r

^{2}sinθdθdΦ r

^{∧}

so after putting this in the equation and doing dot product i get

∮(kr

^{3})( r

^{2}sinθdθdΦ =Q

_{enclosed}

__/__ε

_{°}

therefore

therefore

Q

_{enclosed}= ε

_{°}{ ∮ kr

^{5}sinθdθdΦ

so after integration we get

Q

_{enclosed}= 4πε

_{°}kr

^{5}

but somehow in the answer given in the book it is

4πε

_{°}kR

^{5}

what am i doing wrong here ? please help

is there a mistake in the selection of area vector?