Electrostatics

Two point charges 4q and -q are fixed on the x-axis at x=-d/2 and x= +d/2 respectively ….

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Q: Two point charges 4q and -q are fixed on the x-axis at x=-d/2 and x= +d/2 respectively . If a third point charge q is taken from origin to x = d along the semicircle as shown in the figure , the energy of charge will

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(a) increase by $\displaystyle \frac{3 q^2}{4\pi\epsilon_0 d}$

(b) decrease by $\displaystyle \frac{q^2}{4\pi\epsilon_0 d}$

(c) decrease by $\displaystyle \frac{4 q^2}{3\pi\epsilon_0 d}$

(d) increase by $\displaystyle \frac{2 q^2}{3\pi\epsilon_0 d}$

Click to See Solution :
Ans: (c)

Sol: jee

Electrostatic Potential Energy when charge q is at origin O

$\displaystyle U_1 = \frac{1}{4\pi\epsilon_0} [\frac{4q \times q}{d/2} + \frac{(-q) \times q}{d/2}]$

$\displaystyle U_1 = \frac{1}{4\pi\epsilon_0} \frac{6q^2}{d} $

Electrostatic Potential Energy when charge q moved to x = d

$\displaystyle U_2 = \frac{1}{4\pi\epsilon_0} [\frac{4q \times q}{3d/2} + \frac{(-q) \times q}{d/2}]$

$\displaystyle U_1 = \frac{1}{4\pi\epsilon_0} \frac{2q^2}{3d} $

The change in Poential Energy ,

$\displaystyle \Delta U = \frac{1}{4\pi\epsilon_0} \frac{q^2}{d} (\frac{2}{3}-6) $

$\displaystyle \Delta U = – \frac{4 q^2}{3\pi\epsilon_0 d}$

 

A charged particle (mass m charge q) moves along X-axis with velocity vo . When it passes through origin it enters …

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Q: A charged particle (mass m charge q) moves along X-axis with velocity vo . When it passes through origin it enters a region having uniform electric field $\vec{E} = -E\hat{j}$ which extends up to x = d . Equation of path of electron in the region x > d is

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(a) $\displaystyle y = \frac{q E d}{m v_o^2}(x-d)$

(b) $\displaystyle y = \frac{q E d}{m v_o^2}(\frac{d}{2}-x)$

(c) $\displaystyle y = \frac{q E d}{m v_o^2} x $

(d) $\displaystyle y = \frac{q E d^2}{m v_o^2} x$

Click to See Solution :
Ans: (b)
Sol:

 

Consider two charged metallic spheres S1 and S2 of radii R1 and R2 respectively . The electric field E1 (on S1) and E2 (on S2) …..

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Q: Consider two charged metallic spheres S1 and S2 of radii R1 and R2 respectively . The electric field E1 (on S1) and E2 (on S2) on their surfaces are such that $\frac{E_1}{E_2} = \frac{R_1}{R_2}$ . Then the ratio V1 (on S1)/V2(on S2) of the electrostatic potentials on each sphere is

(a) $\displaystyle (\frac{R_1}{R_2})^3$

(b) $\displaystyle \frac{R_2}{R_1}$

(c) $\displaystyle \frac{R_1}{R_2}$

(d) $\displaystyle (\frac{R_1}{R_2})^2 $

Click to See Solution :
Ans: (d)
Sol: $\displaystyle \frac{E_1}{E_2} = \frac{R_1}{R_2} $ (given) …(i)

$\displaystyle E_1 = K\frac{Q_1}{R_1^2}$

$\displaystyle E_2 = K\frac{Q_2}{R_2^2}$

$\displaystyle \frac{E_1}{E_2} = \frac{Q_1}{Q_2} \times \frac{R_2^2}{R_1^2}$

$\displaystyle \frac{R_1}{R_2} = \frac{Q_1}{Q_2} \times \frac{R_2^2}{R_1^2}$ (from (i))

$\displaystyle \frac{Q_1}{Q_2} = \frac{R_1^3}{R_2^3} $ …(ii)

$\displaystyle V_1 = K\frac{Q_1}{R_1}$

$\displaystyle V_2 = K\frac{Q_2}{R_2}$

$\displaystyle \frac{V_1}{V_2} = \frac{Q_1}{Q_2} \times \frac{R_2}{R_1}$

$\displaystyle \frac{V_1}{V_2} = \frac{R_1^3}{R_2^3} \times \frac{R_2}{R_1}$ (from (ii))

$\displaystyle \frac{V_1}{V_2} = \frac{R_1^2}{R_2^2} $

 

In finding the electric field using Gauss law the formula E = q_enc/∈0 A is applicable ….

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Q: In finding the electric field using Gauss law the formula $|\vec{E}| = \frac{q_{enc}}{\epsilon_0 |A|}$ is applicable . In the formula ∈0 is permittivity of free space , A is the area of Gaussian surface and qenc is charge enclosed by the Gaussian surface . This equation can be used in which of the following situation ?

(a) For any choice of Gaussian surface

(b) Only when the Gaussian surface is an equipotential surface

(c) Only when the Gaussian surface is an equipotential surface and $\vec{E}$ is constant on the surface

(d) Only when $\vec{E}$ = constant on the surface

Click to See Solution :
Ans: (c)
Sol: The magnitude of electric field should be constant and surface should be equipotential surface .

 

An electric dipole of moment p = (-i-3j+2k) × 10^-29 C m is at the origin (0,0,0) …..

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Q: An electric dipole of moment $\displaystyle \vec{p} = (-\hat{i}-3\hat{j} + 2\hat{k}) \times 10^{-29} C m $ is at the origin (0,0,0) . the electric field due to this dipole at $\displaystyle \vec{r} = \hat{i} + 3 \hat{j} + 5 \hat{k}$ (note that $\vec{r}.\vec{p} = 0$) is parallel to

(a) $\displaystyle (\hat{i} – 3 \hat{j}-2\hat{k})$

(b) $\displaystyle (-\hat{i} – 3 \hat{j} + 2\hat{k})$

(c) $\displaystyle (\hat{i} + 3 \hat{j}-2\hat{k})$

(d) $\displaystyle (- \hat{i} + 3 \hat{j}- 2\hat{k})$

Click to See Solution :
Ans: (c)

Sol: $\displaystyle \vec{p} = (-\hat{i}-3\hat{j} + 2\hat{k}) \times 10^{-29} C m $

$\displaystyle \vec{r} = \hat{i} + 3 \hat{j} + 5 \hat{k}$

As , $\displaystyle \vec{r}.\vec{p} = 0$ , It means $\vec{r}$ is perpendicular to $\vec{p}$ Therefore point lies on equilateral plane . Thus electric field is anti-parallel to dipole moment or $\vec{E}|| (-\vec{p})$

$\displaystyle \vec{E}|| (-(-\hat{i}-3\hat{j} + 2\hat{k}))$

$\displaystyle \vec{E}|| (\hat{i} + 3\hat{j} – 2\hat{k}) $