Electric Potential and Capacitance | Physics JEE previous year questions with Solutions

Q 11 :

Two isolated metallic solid spheres of radii $R$ and $2 R$ are charged such that both have same charge density $σ$ . The spheres are then connected by a thin conducting wire. If the new charge density of the bigger sphere is $σ'$ , the ratio $\frac{σ'}{σ}$ is: [2023]

$\frac{9}{4}$
$\frac{4}{3}$
$\frac{5}{3}$
$\frac{5}{6}$

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(4)

[IMAGE 148]

$\frac{Q_{1}'}{4 π ε_{0} R} = \frac{Q_{2}'}{4 π ε_{0} (2 R)}$

$∴$ $Q_{2}' = 2 Q_{1}'$

$Q_{1}' + Q_{2}' = Q_{1} + Q_{2}$

$∴$ $\frac{Q_{2}'}{2} + Q_{2}' = 20 π R^{2} σ$

$∴$ $\frac{3}{2} Q_{2}' = 20 π R^{2} σ$

$∴$ $\frac{Q_{2}'}{4 π {(2 R)}^{2}} = \frac{2}{3} \frac{20 π R^{2} σ}{16 π R^{2}}$

$∴$ $\frac{σ'}{σ} = \frac{5}{6}$

Q 12 :

Which of the following correctly represents the variation of electric potential $(V)$ of a charged spherical conductor of radius $(R)$ with radial distance $(r)$ from the centre? [2023]

[IMAGE 149]
[IMAGE 150]
[IMAGE 151]
[IMAGE 152]

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(3)

[IMAGE 153]

Q 13 :

Considering a group of positive charges, which of the following statements is correct? [2023]

Net potential of the system cannot be zero at a point but net electric field can be zero at that point.
Net potential of the system at a point can be zero but net electric field can't be zero at that point.
Both the net potential and the net field can be zero at a point.
Both the net potential and the net electric field cannot be zero at a point.

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(1)

$V = \frac{\sum K Q_{i}}{r_{i}}$

$Here, Q_{i} and r_{i} are positive$

$∴$ $V > 0$

Q 14 :

A point charge $2 \times 10^{- 2} C$ is moved from $P$ to $S$ in a uniform electric field of $30 N C^{- 1}$ directed along positive $x$ -axis. If coordinates of $P$ and $S$ are (1, 2, 0) m and (0,0,0) m respectively, the work done by electric field will be [2023]

1200 mJ
600 mJ
− 600 mJ
− 1200 mJ

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(3)

$W_{E} = q \vec{E} \cdot \vec{S}$

$= 2 \times 10^{- 2} [30 \hat{i} \cdot (- \hat{i})]$

$= 2 \times 10^{- 2} (- 30) = - 60 \times 10^{- 2}$

$= - \frac{60}{100} = - 0.6 J = - 600 mJ$

Q 15 :

For a uniformly charged thin spherical shell, the electric potential $(V)$ radially away from the centre $(O)$ of the shell can be graphically represented as [2023]

[IMAGE 154]

[IMAGE 155]
[IMAGE 156]
[IMAGE 157]
[IMAGE 158]

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(1)

$V_{inside} = \frac{k Q}{R}$

$V_{outside} = \frac{k Q}{r}$

Q 16 :

Electric potential at a point $' P'$ due to a point charge of $5 \times 10^{- 9} C$ is $50 V$ . The distance of $' P'$ from the point charge is

(Assume, $\frac{1}{4 π ε_{0}} = 9 \times 10^{- 9} {Nm}^{2} C^{- 2}$ ) [2023]

3 cm
90 cm
9 cm
0.9 cm

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(2)

$V_{P} = \frac{K Q}{r}$

$50 = \frac{9 \times 10^{9} \times 5 \times 10^{- 9}}{r}$

$r = \frac{45}{50} = \frac{9}{10} = 0.9 m = 90 cm$

Q 17 :

For a charged spherical ball, electrostatic potential inside the ball varies with $r$ as $V = 2 a r^{2} + b .$ Here, $a$ and $b$ are constants and $r$ is the distance from the center. The volume charge density inside the ball is $- λ a ε$ . The value of $λ$ is _________. ( $ε =$ permittivity of the medium) [2023]

0%

(12)

$E = - \frac{d V}{d r} = - 4 a r \equiv \frac{ρ r}{3 ε_{0}} (compare)$

$Result inside uniformly charged solid sphere.$

$ρ = - 12 a ε_{0}$

$λ = 12$

Q 18 :

Three concentric spherical metallic shells $X$ , $Y$ and $Z$ of radius $a$ , $b$ and $c$ respectively $(a < b < c)$ have surface charge densities $σ$ , $- σ$ and $σ$ , respectively. The shells $X$ and $Z$ are at the same potential. If the radii of $X$ and $Y$ are 2 cm and 3 cm, respectively, the radius of shell $Z$ is ________ cm. [2023]

0%

(5)

[IMAGE 159]

$q_{x} = σ 4 π a^{2}, q_{y} = - σ 4 π b^{2}, q_{z} = σ 4 π c^{2}$

$Potential at x = potential at z$

$V_{x} = V_{z}$

$\frac{q_{x}}{4 π ε_{0} a} + \frac{q_{y}}{4 π ε_{0} b} + \frac{q_{z}}{4 π ε_{0} c} = \frac{q_{x}}{4 π ε_{0} c} + \frac{q_{y}}{4 π ε_{0} c} + \frac{q_{z}}{4 π ε_{0} c}$

$\frac{σ 4 π a^{2}}{a} - \frac{σ 4 π b^{2}}{b} + \frac{σ 4 π c^{2}}{c} = \frac{4 π σ (a^{2} - b^{2} + c^{2})}{c}$

$c (a - b + c) = a^{2} - b^{2} + c^{2}$

$c (a - b) = a^{2} - b^{2}$

$\Rightarrow c = a + b \Rightarrow c = 5 cm$

Q 19 :

64 identical drops, each charged up to a potential of 10 mV are combined to form a bigger drop. The potential of the bigger drop will be ______ mV. [2023]

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(160)

[IMAGE 160]

$Let q = charge on each drop$

$V = \frac{K q}{r}$ ...(i)

$Now for combination of 64 drops$

$64 \times \frac{4}{3} π r^{3} = \frac{4}{3} π R^{3}$

$R = 4 r and Q = 64 q$

$Potential of bigger drop$ $= \frac{K Q}{R} = \frac{K (64 q)}{4 r} = 16 \frac{K q}{r}$

$= 16 \times 10 mV = 160 mV$

$Correct answer is 160$

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