Two particles each of mass m and charge q are separated by distance r₁ and system is left free to

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2. Electric Potential and Capacitance

medium

Two particles each of mass $m$ and charge $q$ are separated by distance $r_1$ and the system is left free to move at $t = 0$. At time $t$ both the particles are found to be separated by distance $r_2$. The speed of each particle is

$\frac{{qm}}{{4\pi {\varepsilon _0}{r_1}{r_2}}}$

$\frac{q}{{{r_1}{r_2}\sqrt {(r_2^2 - r_1^2)/4\pi {\varepsilon _0m}} }}$

$\frac{\sqrt 2q}{{{r_1}{r_2}\sqrt {(r_2^2 - r_1^2)/4\pi {\varepsilon _0m}} }}$

none of these

Solution

Due to symmetry, each particle will have same speed,

${\mathrm{E}_{\mathrm{i}}=\frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{q}^{2}}{\mathrm{r}_{1}}+0}$

${\mathrm{E}_{\mathrm{f}}=\frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{q}^{2}}{\mathrm{r}_{2}}+\frac{1}{2} \mathrm{mv}^{2}+\frac{1}{2} \mathrm{mv}^{2}}$

As the field is conservative, hence applying law of conservation of energy.

${\frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{q}^{2}}{\mathrm{r}_{1}}=\frac{1}{4 \pi \varepsilon_{0}} \frac{\mathrm{q}^{2}}{\mathrm{r}_{2}}+\mathrm{mv}^{2}}$

$\therefore $ ${\mathrm{v}=\mathrm{q} \sqrt{\frac{\left(\mathrm{r}_{2}-\mathrm{r}_{1}\right)}{4 \pi \varepsilon_{0} \mathrm{mr}_{1} \mathrm{r}_{2}}}}$

Standard 12

Physics

This questions has statement$-1$ and statement$-2$. Of the four choices given after the statements, choose the one that best describe the two statements.
An insulating solid sphere of radius $R$ has a uniformly
positive charge density $\rho$. As a result of this uniform charge distribution there is a finite value of electric potential at the centre of the sphere, at the surface of the sphere and also at a point out side the sphere. The electric potential at infinite is zero.

Statement$ -1$ : When a charge $q$ is take from the centre of the surface of the sphere its potential energy changes by $\frac{{q\rho }}{{3{\varepsilon _0}}}$

Statement$ -2$ : The electric field at a distance $r(r < R)$ from centre of the sphere is $\frac{{\rho r}}{{3{\varepsilon _0}}}$

hard

(AIEEE-2012)

Four charges are arranged at the corners of a square $ABCD$ of side $d$, as shown in Figure

$(a)$ Find the work required to put together this arrangement.

$(b)$ A charge $q_{0}$ is brought to the centre $E$ of the square, the four charges being held fixed at its corners. How much extra work is needed to do this?

medium

An electron of mass $m$ and charge $e$ is accelerated from rest through a potential difference $V$ in vacuum. The final speed of the electron will be

easy

Two charged particles of masses $m$ and $2m$ have charges $+2q$ and $+q$ respectively. They are kept in uniform electric field and allowed to move for some time. The ratio of their kinetic energies will be