JEE Advanced PYQ Physics – Electric Charges and Coulomb’s Law Questions with Solutions

Q 1 :

Two beads, each with charge $q$ and mass $m$ , are on a horizontal, frictionless, non-conducting, circular hoop of radius $R$ . One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by [ $ε_{0}$ is the permittivity of free space]. [2024]

$\frac{q^{2}}{(4 π ε_{0} R^{3} m)}$
$\frac{q^{2}}{(32 π ε_{0} R^{3} m)}$
$\frac{q^{2}}{(8 π ε_{0} R^{3} m)}$
$\frac{q^{2}}{(16 π ε_{0} R^{3} m)}$

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(2)

Here $r = 2 R \cos ϕ$ also $θ = 2 ϕ \Rightarrow θ = \frac{ϕ}{2}$

and $θ = \frac{x}{R}$

If $θ$ is the small angular displacement of free charge, then $F (ϕ) = \frac{K q^{2}}{r^{2}}$

So, restoring force towards mean position is $F_{R} = - \frac{K q^{2}}{r^{2}} \sin ϕ$

$a_{R} = \frac{F_{R}}{m} = - \frac{K q^{2}}{m r^{2}} \sin ϕ = - \frac{K q^{2} \cdot \sin ϕ}{4 m R^{2} \cos^{2} ϕ}$

$\Rightarrow a_{R} = - \frac{K q^{2}}{4 m R^{2} \cos^{2} (\frac{ϕ}{2})} \cdot \sin (\frac{ϕ}{2}) \approx - \frac{K q^{2}}{4 m R^{2}} \cdot \frac{1}{2} \cdot \frac{x}{R} = ω^{2} \cdot x$

$∴$ $ω^{2} = \frac{q^{2}}{32 π ε_{0} m R^{3}}$

[IMAGE 626]

Q 2 :

Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. The net charge on the sphere is then [2007]

negative and distributed uniformly over the surface of the sphere
negative and appears only at the point on the sphere closest to the point charge
negative and distributed non-uniformly over the entire surface of the sphere
zero

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(4)

[IMAGE 627]

When a positive point charge is placed outside a conducting sphere, a rearrangement of charge takes place on the surface as shown in the figure. But net charge on the sphere is zero as no charge has left or entered the sphere.

Q 3 :

Two identical non-conducting solid spheres of same mass and charge are suspended in air from a common point by two non-conducting, massless strings of same length. At equilibrium, the angle between the strings is $α$ . The spheres are now immersed in a dielectric liquid of density 800 kg $m^{- 3}$ and dielectric constant 21. If the angle between the strings remains the same after the immersion, then [2020]

electric force between the spheres remains unchanged
electric force between the spheres reduces
mass density of the spheres is 840 kg $m^{- 3}$
the tension in the strings holding the spheres remains unchanged

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Select one or more options

(1, 3)

[IMAGE 628]

The net electric force on any sphere is lesser but by Coulomb's law, the force due to one sphere to another remains the same.

In equilibrium

$T \cos θ = m g$

and $T \sin θ = F$

$∴$ $\tan θ = \frac{F}{m g}$ ...(i)

As force between two charged bodies according to Coulomb's law does not depend upon the medium, hence force between them remains same.

Hence after immersed in dielectric medium, as given no change in angle $θ$ .

$T' \sin θ = \frac{F}{ε_{r}}$ and $T' \cos θ = m g (1 - \frac{ρ_{l}}{ρ_{s}})$

$∴$ $\tan θ = \frac{F}{m g (1 - \frac{ρ_{l}}{ρ_{s}})} \cdot \frac{1}{ε_{r}}$ ...(ii)

Now equating Eqns. (i) and (ii),

$1 - \frac{ρ_{l}}{ρ_{s}} = \frac{1}{ε_{r}}$ $\Rightarrow 1 - \frac{800}{ρ_{s}} = \frac{1}{21}$

$∴$ $ρ_{s} = 840 {kg/m}^{3}$

Hence option (3) is correct.

$And tension, T' = \frac{T}{ε_{r}} = \frac{T}{21}$

Hence option (4) is incorrect.

Q 4 :

Four charges $Q_{1}, Q_{2}, Q_{3}$ and $Q_{4}$ of same magnitude are fixed along the $x$ -axis at $x = - 2 a, - a, + a$ and $+ 2 a$ , respectively. A positive charge $q$ is placed on the positive $y$ -axis at a distance $b > 0$ . Four options of the signs of these charges are given in List-I. The directions of the forces on the charge $q$ is given in List-II. Match List-I with List-II and select the correct answer using the code given below the lists. [2014]

List - I List - II

P.
$Q_{1}, Q_{2}, Q_{3}, Q_{4}$ all positive

[IMAGE 629]
1. $+ x$

Q. $Q_{1}, Q_{2}$ positive;
$Q_{3}, Q_{4}$ negative 2. $- x$

R. $Q_{1}, Q_{4}$ positive;
$Q_{2}, Q_{3}$ negative 3. $+ y$

S. $Q_{1}, Q_{3}$ positive;
$Q_{2}, Q_{4}$ negative 4. $- y$

Codes:

P-3, Q-1, R-4, S-2
P-4, Q-2, R-3, S-1
P-3, Q-1, R-2, S-4
P-4, Q-2, R-1, S-3

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(1)

If $Q_{1}, Q_{2}, Q_{3}$ and $Q_{4}$ are all positive, then the force will be along $+ y$ -direction as components of forces along, the $x$ -axis cancel out each other.

If $Q_{1}, Q_{2}$ are positive and $Q_{3}, Q_{4}$ are negative, the force will act along $+ x$ -direction as components of forces along the $y$ -axis cancel out each other.

If $Q_{1}, Q_{4}$ are positive and $Q_{2}, Q_{3}$ are negative, then attractive force will dominate repulsive force and the force will be along $- y$ -direction.

If $Q_{1}, Q_{3}$ are positive and $Q_{2}, Q_{4}$ are negative, components of forces along the $y$ -axis cancel out each other. So net force on charge $q$ is along the $x$ -axis.

Topic Question Set

Q 2 :

Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. The net charge on the sphere is then [2007]

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	List - I		List - II
P.	$Q_{1}, Q_{2}, Q_{3}, Q_{4}$ all positive [IMAGE 629]	1.	$+ x$
Q.	$Q_{1}, Q_{2}$ positive; $Q_{3}, Q_{4}$ negative	2.	$- x$
R.	$Q_{1}, Q_{4}$ positive; $Q_{2}, Q_{3}$ negative	3.	$+ y$
S.	$Q_{1}, Q_{3}$ positive; $Q_{2}, Q_{4}$ negative	4.	$- y$

Topic Question Set

Q 2 : Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. The net charge on the sphere is then [2007]

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Q 2 :

Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. The net charge on the sphere is then [2007]