JEE Advanced PYQ Physics – Miscellaneous Mixed Concepts Problems Questions

Q 1 :

A wooden block performs SHM on a frictionless surface with frequency $ν_{0}$ . The block carries a charge $+ Q$ on its surface. If now a uniform electric field $\vec{E}$ is switched-on as shown, then the SHM of the block will be [2011]

of the same frequency and with shifted mean position.
of the same frequency and with the same mean position.
of changed frequency and with shifted mean position.
of changed frequency and with the same mean position.

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(1)

Here frequency of SHM performed by the wooden block $v_{0} = \frac{1}{2 π} \sqrt{\frac{k}{m}}$

This value of frequency will remain unchanged because when the electric field is switched on, the values of $k$ and $m$ are not affected. The mean position of SHM shifts towards the right by $l = \frac{Q E}{k}$ due to the force acting $F = q E .$

Q 2 :

A uniformly charged thin spherical shell of radius $R$ carries uniform surface charge density of $σ$ per unit area. It is made of two hemispherical shells, held together by pressing them with force $F$ (see figure). $F$ is proportional to [2010]

$\frac{1}{ε_{0}} σ^{2} R^{2}$
$\frac{1}{ε_{0}} σ^{2} R$
$\frac{1}{ε_{0}} \frac{σ^{2}}{R}$
$\frac{1}{ε_{0}} \frac{σ^{2}}{R^{2}}$

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(1)

The electrostatic force per unit area, i.e., electrostatic pressure at a point on the surface of a uniformly charged sphere $= \frac{1}{2} ε_{0} E^{2} = \frac{σ^{2}}{2 ε_{0}}$

$∴$ The force on a hemispherical shell $F = \frac{σ^{2}}{2 ε_{0}} \times π R^{2}$

or, $F \propto \frac{σ^{2} R^{2}}{ε_{0}}$

Q 3 :

Four point charges, each of $+ q$ , are rigidly fixed at the four corners of a square planar soap film of side $' a'$ . The surface tension of the soap film is $γ$ . The system of charges and planar film are in equilibrium, and $a = k {(\frac{q^{2}}{γ})}^{1 / N},$ where $' k'$ is a constant. Then $N$ is [2011]

(3)

Net electrostatic force on one charge due to remaining three charges

$F_{electro} = \frac{k q^{2}}{2 a^{2}} + 2 [\frac{k q^{2}}{a^{2}} \times \frac{1}{\sqrt{2}}] = \frac{q^{2}}{a^{2}} \times constant$

Surface tension force $F_{s t} = γ a$

In equilibrium,

$\frac{q^{2}}{a^{2}} \times constant = γ a$

$∴$ $a = k {(\frac{q^{2}}{γ})}^{1 / 3}$

$∴$ $N = 3$

Q 4 :

One end of a spring of negligible unstretched length and spring constant $k$ is fixed at the origin $(0, 0)$ . A point particle of mass $m$ carrying a positive charge $q$ is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole $\vec{p}$ pointing towards the charge $q$ is fixed at the origin, the spring gets stretched to a length $l$ and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly by $Δ l ≪ l$ from its equilibrium position and released, it is found to oscillate at frequency $\frac{1}{δ} \sqrt{\frac{k}{m}} .$ The value of $δ$ is ________. [2010]

(3.14)

$Original frequency, f = \frac{2}{δ} \sqrt{\frac{K}{m}}$

$If dipole appears at equilibrium,$

$\frac{2 K P}{{(ℓ + x_{0})}^{3}} \cdot q = K x_{0} \dots (i)$

$When displaced towards right by length x_{0}$

$f_{net} = \frac{2 K P}{{(ℓ + x_{0} + x)}^{3}} \cdot q - K (x_{0} + x)$

$m a = \frac{2 K P q}{{(ℓ + x_{0})}^{3}} {[1 + \frac{x}{ℓ - x_{0}}]}^{3} - K (x_{0} + x)$

$= \frac{2 K P q}{{(ℓ + x_{0})}^{3}} {[1 - \frac{3 x}{ℓ + x_{0}}]}^{3} - K (x_{0} + x)$

$= \frac{6 K P q x}{{(ℓ + x_{0})}^{4}} k x = \frac{3 x}{ℓ + x_{0}} K x_{0} - K x = - K x [\frac{3 x_{0}}{ℓ + x_{0}} + 1]$

$As' ℓ' is negative, m a = - 4 K x$ $\Rightarrow a = \frac{- 4 K}{m} x$

$New frequency, f' = \frac{1}{2 π} \sqrt{\frac{4 K}{m}} = 2 f = \frac{1}{π} \sqrt{\frac{K}{m}}$

$∴$ $δ = π = 3.14$

Q 5 :

The figures below depict two situations in which two infinitely long static line charges of constant positive line charge density $λ$ are kept parallel to each other. In their resulting electric field, point charges q and −q are kept in equilibrium between them. The point charges are confined to move in the $x$ direction only. If they are given a small displacement about their equilibrium positions, then the correct statement(s) is(are) [2015]

Both charges execute simple harmonic motion
Both charges will continue moving in the direction of their displacement
Charge +q executes simple harmonic motion while charge −q continues moving in the direction of its displacement
Charge −q executes simple harmonic motion while charge +q continues moving in the direction of its displacement

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(3)

$Net force on charge q when it is given a small displacement x$

$F_{net} = F_{1} - F_{2}$

$= \frac{1}{2 π ε_{0}} \frac{λ}{d - x} - \frac{1}{2 π ε_{0}} \frac{λ}{d + x}$

$∴$ $F_{net} = \frac{λ}{2 π ε_{0}} [\frac{d + x - d + x}{d^{2} - x^{2}}]$

$∴$ $F_{net} = \frac{λ}{2 π ε_{0}} \frac{2 x}{d^{2} - x^{2}}$

$When x ≪ d,$

$F_{net} = \frac{λ}{π ε_{0}} x$ $i.e., F \propto x$

$displacement therefore the charge (+ q) will execute SHM.$

$In case of charge (- q), F_{2} > F_{1},$ $i.e., charge (- q) is unstable; therefore the charge (- q) continues to move in the direction of its displacement.$

Q 6 :

Under the influence of the Coulomb field of charge $+ Q$ , a charge $- q$ is moving around it in an elliptical orbit. Find out the correct statement(s). [2009]

The angular momentum of the charge $- q$ is constant
The linear momentum of the charge $- q$ is constant
The angular velocity of the charge $- q$ is constant
The linear speed of the charge $- q$ is constant

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(1)

According to the question, under the influence of the Coulomb field of charge $+ Q$ , a charge $- q$ is moving around it in an elliptical orbit. This situation is shown in the figure which is similar to a planet revolving around the Sun.

The distance of $- q$ from $+ Q$ is changing; therefore, the force between the charges will change. The speed of the charge $- q$ will be greater when the charge is nearer to $+ Q$ as compared to when it is far. Hence, the angular velocity of charge $- q$ is not constant.

The direction of the velocity changes continuously; therefore, the linear momentum is not constant. The angular momentum of charge $(- q)$ about $Q$ is constant because the torque about charge $+ Q$ is zero.

Topic Question Set

Q 1 :

A wooden block performs SHM on a frictionless surface with frequency $ν_{0}$ . The block carries a charge $+ Q$ on its surface. If now a uniform electric field $\vec{E}$ is switched-on as shown, then the SHM of the block will be [2011]

Q 2 :

A uniformly charged thin spherical shell of radius $R$ carries uniform surface charge density of $σ$ per unit area. It is made of two hemispherical shells, held together by pressing them with force $F$ (see figure). $F$ is proportional to [2010]

Q 6 :

Under the influence of the Coulomb field of charge $+ Q$ , a charge $- q$ is moving around it in an elliptical orbit. Find out the correct statement(s). [2009]

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Topic Question Set

Q 1 : A wooden block performs SHM on a frictionless surface with frequency ν0. The block carries a charge +Q on its surface. If now a uniform electric field E→ is switched-on as shown, then the SHM of the block will be [2011]

Q 2 : A uniformly charged thin spherical shell of radius R carries uniform surface charge density of σ per unit area. It is made of two hemispherical shells, held together by pressing them with force F (see figure). F is proportional to [2010]

Q 3 : Four point charges, each of +q, are rigidly fixed at the four corners of a square planar soap film of side 'a'. The surface tension of the soap film is γ. The system of charges and planar film are in equilibrium, and a=k(q2γ)1/N, where 'k' is a constant. Then N is [2011]

Q 6 : Under the influence of the Coulomb field of charge +Q, a charge −q is moving around it in an elliptical orbit. Find out the correct statement(s). [2009]

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Q 1 :

A wooden block performs SHM on a frictionless surface with frequency $ν_{0}$ . The block carries a charge $+ Q$ on its surface. If now a uniform electric field $\vec{E}$ is switched-on as shown, then the SHM of the block will be [2011]

Q 2 :

A uniformly charged thin spherical shell of radius $R$ carries uniform surface charge density of $σ$ per unit area. It is made of two hemispherical shells, held together by pressing them with force $F$ (see figure). $F$ is proportional to [2010]

Q 6 :

Under the influence of the Coulomb field of charge $+ Q$ , a charge $- q$ is moving around it in an elliptical orbit. Find out the correct statement(s). [2009]