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Standard XII

Physics

Infinite Line of Charge

Two identical...

Question

Two identical charged spheres suspended from a common point by two massless strings of length $l$ are initially a distance $d (d << ℓ)$ apart because of their mutual repulsion. The charge begins to leak from both the spheres at a constant rate. As a result the charges approach each other with a velocity $v$ . Then as a function of distance $x$ between them:

v \infty x^{- 1 / 2}

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v \infty x^{- 1}

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v \infty x^{1 / 2}

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v \infty x

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Solution

The correct option is A $v \infty x^{- 1 / 2}$
$T s i n θ = \frac{K q^{2}}{x^{2}}$ ....(i)
$T cos θ = m g$ .....(ii)
or
$t a n θ = \frac{K q^{2}}{x^{2} m g} = \frac{x}{l}$ (from the FBD); since $d >> l$
or
$x^{3} = \frac{K q^{2} ℓ}{m g}$ ( $. . . i i i$ )
$x^{3} \propto q^{2}$
$3 x^{2} \frac{d x}{d t} \propto 2 q \frac{d q}{d t}$
$x^{2} \cdot v \propto q$
Substituting the value of $q$ from equation (iii).
$v \propto x^{\frac{- 1}{2}}$

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The correct option is A v∞x−1/2Tsinθ=Kq2x2 ....(i)Tcosθ=mg .....(ii)ortanθ=Kq2x2mg=xl(from the FBD); since d>>lorx3=Kq2ℓmg (...iii) x3∝q23x2dxdt∝2qdqdtx2⋅v∝qSubstituting the value of q from equation (iii).v∝x−12