4. Class 12th Physics | Current Electricity | Example-1 on Current Electricity | Ashish Arora


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.

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