Question 414: Electrostatics & Capacitance

Question

A particle of charge q and mass m is subjected to an electric field
E = E
0
(1 – $$a$$x
2
) in the x-direction,
where $$a$$ and E
0
are constants. Initially the particle was at rest at x = 0. Other than the initial
position the kinetic energy of the particle becomes zero when the distance of the particle from the
origin is :

Accepted Answer

$$W = \Delta KE$$
As inital and final kinetic energy both are zero so $$\Delta KE$$ = 0
$$ 	herefore $$ W = 0
$$ \Rightarrow $$ $$\int\limits_0^x {Fdx}  = 0$$
$$ \Rightarrow $$ $$q\int\limits_0^x {{E_0}\left( {1 - a{x^2}} ight)dx}  = 0$$
$$ \Rightarrow $$ $$q{E_0}\left[ {\int\limits_0^x {dx - a} \int\limits_0^x {{x^2}dx} } ight] = 0$$
$$ \Rightarrow $$$$q{E_0}\left[ {x - {{a{x^3}} \over 3}} ight] = 0$$
$$ \Rightarrow $$ $$x\left( {1 - {{a{x^2}} \over 3}} ight) = 0$$
$$ \Rightarrow $$ $$x = 0,$$ $${1 - {{a{x^2}} \over 3}}$$ = 0
$$ \Rightarrow $$ $${{{a{x^2}} \over 3}}$$ = 1
$$ \Rightarrow $$ $${x = \sqrt {{3 \over a}} }$$

Answer

$$a$$

Answer

$$\sqrt {{2 \over a}} $$

Answer

$$\sqrt {{3 \over a}} $$

Answer

$$\sqrt {{1 \over a}} $$

Electrostatics & Capacitance

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