1karan, i would like to make friends with you, so that we can study together. :)
u seem to have some really interesting questions :)
a long round conductor of cross sectional area A is made of material whose resistivity depends on radial distance r from the axis of conductor as p=a/r^2 , a is contant What is total resistance per unit length of conductor and electric field strength in conductor due to which a current I flows in it
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2 Answers
1Well this is a very good question.
I will show you how to solve this.
Take a section of thickness dr at a distance r from the center of the wire/conductor.
Area of this section will be 2*pi*r*dr
Now the resistance of this small section per unit length will be
dR=p/(2*pi*r*dr)
dR=a/{(r*r)*(2*pi*r*dr)}
because there are infinitely many such concentric rings, which we can think of to be in parallel to each other, the total resistance per unit length is the resiprocal of the (sum or resiprocals)
by parallel rule, 1/R=1/dR1+1/dR2+...................
But the small problem here is that there are infinite such rings. So we take the integration.
So,
1/R= ∫(1/dR)
1/R=∫(r*r)*(2*pi*r*dr)/a
integrating this from r=0 to r (please note that there are 2 r's here which is technically not correct but still.. )
we get 1/R=pi*r^4/(2a)
Thus, R = 2a/(pi*r^4)