Explanation
Two identical conducting spheres $$M$$ and $$N$$ has charges $${q_m}$$ and $${q_n}$$ respectively. A third identical neutral sphere $$P$$ is brought in contact with $$M$$ and then separated. Now sphere $$P$$ is brought in contact with $$N$$ and then separated. Final charge on sphere $$P$$ will be:
The two parts of charge according to ratio are given as,
$${q_1} = \dfrac{1}{4}\;{\rm{\mu C}}$$
$${q_2} = \dfrac{3}{4}\;{\rm{\mu C}}$$
The electric force is given as,
$$F = \dfrac{1}{{4\pi {\varepsilon _0}}} \times \dfrac{{{q_1}{q_2}}}{{{r^2}}}$$
$$F = 9 \times {10^9} \times \dfrac{{\dfrac{1}{4} \times {{10}^{ - 6}} \times \dfrac{3}{4} \times {{10}^{ - 6}}}}{1}$$
$$F = 1.6875 \times {10^{ - 3}}\;{\rm{N}}$$
The small length is given as,
$$dl = Rd\theta $$
The small charge is given as,
$$dq = \lambda dl$$
$$dq = {\lambda _0}\cos {\rm{\theta }}Rd\theta $$
The position of charge is given as,
$$\vec r = \left( {R\cos {\rm{\theta }},R\sin {\rm{\theta }},0} \right)$$
The electric dipole moment is given as,
$$\vec p = \int {\vec rdq} $$
$$ = \int_0^{2\pi } {{\lambda _0}\cos {\rm{\theta }}Rd\theta \left( {R\cos {\rm{\theta }},R\sin {\rm{\theta }},0} \right)} $$
$${\vec p_x} = \int_0^{2\pi } {{\lambda _0}{R^2}{{\cos }^2}{\rm{\theta }}d\theta } $$
$$ = \frac{{{\lambda _0}{R^2}}}{2}\int_0^{2\pi } {\left( {1 + \cos 2{\rm{\theta }}} \right)d\theta } $$
$$ = \frac{{{\lambda _0}{R^2}}}{2}\left[ {\theta + \frac{1}{2}\sin 2{\rm{\theta }}} \right]_0^{2\pi }$$
$$ = {\lambda _0}{R^2}\pi $$
$${\vec p_y} = \int_0^{2\pi } {{\lambda _0}{R^2}\cos {\rm{\theta }}\sin {\rm{\theta }}d\theta } $$
$$ = \frac{{{\lambda _0}{R^2}}}{2}\int_0^{2\pi } {\sin 2{\rm{\theta }}d\theta } $$
$$ = \frac{{{\lambda _0}{R^2}}}{2}\left[ {\cos 2{\rm{\theta }}} \right]_0^{2\pi }$$
$$ = 0$$
$${\vec p_z} = 0$$
Thus, net electric dipole moment is $$\pi {R^2}{\lambda _0}$$.
We have
$$\dfrac{1}{2}m{v^2} = qEd$$
$${v^2} = \dfrac{{2qEd}}{m}$$
$$v = \sqrt {\dfrac{{2qEd}}{m}} $$
Thus the particle strikes the opposite plane with speed $$\sqrt {\dfrac{{2qEd}}{m}} $$ along $$ - {\rm{j}}$$.
If any positive charge object repel with any other charge then that charge must be of same nature as that of object
so that's why the nature of other charge is also positive.
because always like charges repel each other.
so, its negative
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