in this example we are given that in a long

wire of round cross section of radius r. current density varies with the radial distance from

axis of wire x as j is equal to c x square. and we are required to find the total, current

flowing in the wire. it is also given that c is a positive constant. in this situation

if we draw the physical situation here we can see. if this is a long cylinder and, this

is the central axis of the cylinder. here we are given that current density is varying

with the distance x. then on its, circular cross section of the wire which is of radius

r. we are required to find the total current and in this situation. we consider an elemental.

ring. or you can say elemental strip. of radius x, and width d-x. and this d-x is so small

that we can consider, in this current density remains almost uniform at c x square. so here

we can find. current in. elemental strip. is. here this can be written as d-i which

is the current flowing in the elemental strip which can be written as, j d-s. here we wont

take dot product because current density and, the area vector of this strip are in same

direction. so here this d-i we can write as, j-j we are given as c x square. and the area

of this strip d-s we can write as 2 pie x d-x. so this can be multiplied as 2 pie x

d-x. so here we can calculate the total current as. i which is given by integration of d-i

we integrate it-it’ll be 2 pie c x cube d-x. and we integrate it within limit from

zero to r, to cover the holes. cross section in which the current is flowing. so in this

situation on integrating we are getting 2 pie c is a constant. and integration of x-cube

is x four by 4 and we apply the limits from zero to r. so here this 2 gets cancelled out

and on. applying limits from zero to r we are getting it is half pie c. r to power 4

that will be the answer to this problem it is current flowing, in this wire of, cross

sectional radius r.