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5. Class 12th Physics | Current Electricity | Solved Example-2 on Current Electricity | Ashish Arora

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in this example we are given that in a long
wire of square cross section of side length l. current density varies with distance from
one edge of cross section as j is equal to ay e to power b-x amp-ere per meter square.
where ay and b are positive constant and we are required to find the current flowing in
the wire. in this situation if we just draw the cross section of this wire we are given
that the wire has square cross section. which is of side length l. and from 1, edge of the
wire or from one side of the, wire cross section the current density varies with distance as
ay e to power b x. so to calculate the total current in the wire we consider an elemental
strip at a distance x, from. the end of the wire or, the edge of the wire this, strip
we consider is of width d-x. so obviously, the area of this elemental strip we can write
as d-s. which is l d x. now we can directly find out the current in. elemental strip is.
this current can be written as d-i which we can write as j d-s. or it’ll be j multiplied
by. l d x. and we can substitute the value of j as ay e to power b-x. so here total current.
in wire can be given as i is equal to integration of d-i. which is integration of, ay. e to
power b x multiplied by l d-x and we integrate the value of x from, zero to l. so in this
situation the current will be here ay and l are constant which can be taken out of the
sign of integration it is integration from zero to l. e to power b-x, d x. on integrating
here you can see this will be ay l. integration of e to power d x will be e to power b x by
b. where we apply the limits from zero to l. and, on substituting limits here you can
see it is ay l by b, b is a constant again we can take it out. and on substituting limits
at x equal to l it’ll be e to power b-l. minus if we substitute x equal to zero it’ll
be e to power zero is 1. so here you can say we are getting the final current ay l by b,
e to power b-l minus 1 that will be the answer to this problem.

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