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Figure shows a conductor of length l having a circular cross section. The radius of cross section of the conductor varies linearly from r_1 to r_2 along its length. IF the specific resistance of the material of the conductor be rho, find the resistance of the conductor. |
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Answer» Solution :Let the axis of the conductor be along the x axis of the coordinate system. At x=0, radius of crosssection is `r_1` and at x=1 radius of cross section is `r_2`. ltbr Radius of cross section varies along the length of the conductor,So it can be written that `(dr)/(dx)=k` ( where k is constant) Now intergrating the above equation , we get , `r=kx+c` where c is integration constant when x=0, `r=r_1` `therefore r_1=k.0+c or , c=r_1` So, `r=kx+r_1` ...(1) Again when x=l, then `r=r_2` `thereforer_2=kl+r_1` or, `k=(r_2-r_1)/l` .....(2) From equation (1) and (2) , we get `r=(r_2-r_1)/lx+r_1` At a distance x from the LEFT side of the conductor, RESISTANCE of a circular disc of THICKNESS dx is `dR=rho(dx)/(pir^2)=rho(dx)/(pi[((r_2-r_1)/l)x+r_1]^2)` `therefore` Total resistance of the conductor, `R=rho/piint_0^1dx/([((r_2-r_1)/l)x+r_1]^2}` Let `((r_2-r_1)/l)x+r_=uor,dx=(l/(r_2-r_1))du` when x=0 then `u=r_1` and when x=1 then `u=r_2`. `thereforeR=(rhol)/piint_(r_1)^(r_2)(du)/((r_2-r_1)u^2)=(rhol)/(pi(r_2-r_1))[-1/u]_(r_1)^(r_2)` `=(rhol)/(pi(r_2-r_1))[1/r_1-1/r_2]=(rhol)/(pir_1r_2)` This is the REQUIRED resistance. |
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