Question 643: Electrostats & Capacitance

Question

Two identical charged spheres suspended from a common point by two massless strings of length $$l$$ are initially a distance $$d\left( {d <  < 1} \right)$$ apart because of their mutual repulsion. The charge begins to leak from both the spheres at a constant rate. As a result charges approach each other with a velocity $$v$$. Then as a function of distance $$x$$ between them,

Accepted Answer

At any instant
$$T\cos 	heta  = mg\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( i ight)$$
$$T\sin 	heta  = {F_e}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...\left( {ii} ight)$$
$$ \Rightarrow {{\sin 	heta } \over {\cos 	heta }} = {{{F_e}} \over {mg}} \Rightarrow {F_e} = mg\,	an 	heta $$
$$ \Rightarrow {{k{q^2}} \over {{x^2}}} = mg\,	an 	heta  \Rightarrow {q^2} \propto {x^2}	an 	heta $$
$$\sin 	heta  = {	extstyle{x \over {2l}}}$$
For small $$	heta ,\,\sin 	heta  \approx 	an 	heta $$
$$	herefore$$ $${q^2} \propto {x^3}$$
$$ \Rightarrow q{{dq} \over {dt}} \propto {x^2}{{dx} \over {dt}}$$
$$	herefore$$ $${{dq} \over {dt}} = const.$$
$$	herefore$$ $$q \propto {x^2}.v \Rightarrow {x^{3/2}}\alpha {x^2}.v\,\,$$ $$\,\,\,\,\,$$ $$\left[ {\,\,} ight.$$ as $$\left. {{q^2} \propto {x^3}\,\,} ight]$$
$$ \Rightarrow v \propto {x^{ - 1/2}}$$

Answer

$$v\, \propto \,{x^{ - 1}}$$

Answer

$$y\, \propto \,{x^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}}}$$

Answer

$$v\, \propto \,x$$

Answer

$$v\, \propto \,{x^{ - {\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}}}$$

Electrostats & Capacitance

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