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A cone made of insulating material has a...

A cone made of insulating material has a total charge Q spread uniformly over its sloping surface. Calculate the work done in bringing a small test charge q from infinity to the apex of the cone. The cone has a slope length L.

Text Solution

Verified by Experts

The correct Answer is:
B

`W=-/_\U=U_("apex")-U_oo`
`W=qV_("apex")` ………..i
`as U_prop=0`
`V_("apex") r/x=R/L `
`:. r=(R/L)x`
Surface charge density
`sigma=Q/(piRl)`
`dq=(sigma)(dA)`
`=(Q/(piRL))(2pir) (R/Xx)dx`
=`(theta/(piRL))(2pi)(R/Lx)dx`
`=((2Q)/L^2)xdx`
`Now, dV=(1/(4piepsilon_0))((dq)/x)`
`=((2Q)/(4piepsilon_0^L^(2)))dx`
`:. V=int_0^LdV=Q/(2piepsilon_0L)`
substituting in eqn i we have
`W=(Qq)/(2piepsilon_0L)`
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Knowledge Check

  • If 6W is the work done in bringing a infinitesimal small test charge 6g from infinity to a point P, then potential at point P is

    A
    `(deltaW)/(deltaq)`
    B
    `-(deltaW)/(deltaq)`
    C
    `-deltaW.deltaq`
    D
    None of these

  • Three equal charges Q are placed at the three corners of an equilateral triangle of side length a. Work done in shifting a charge q from infinity to the centroid of the triangle is

    A
    `(3Qq)/(4pi in_(0)a)`
    B
    `(3Qq)/(4pi in_(0)sqrt3a)`
    C
    `(3sqrt3Qq)/(4pi in_(0)a)`
    D
    `(sqrt3Qq)/(4pi in_(0)3a)`

  • Four equal charges q each are placed at four corners of a square of side a each. Work done in carrying a charge -q from its centre to infinity is

    A
    zero
    B
    `(sqrt2q^(2))/(pi in_(0)a)`
    C
    `(sqrt2q)/(pi in_(0)a)`
    D
    `(q^(2))/(pi in_(0)a)`