1
JEE Advanced 2024 Paper 1 Online
+3
-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

[ $\varepsilon_0$ is the permittivity of free space.]

A
$${{{q^2}} \over {4\pi {\varepsilon _0}{R^3}m}}$$
B
$${{{q^2}} \over {32\pi {\varepsilon _0}{R^3}m}}$$
C
$${{{q^2}} \over {8\pi {\varepsilon _0}{R^3}m}}$$
D
$${{{q^2}} \over {16\pi {\varepsilon _0}{R^3}m}}$$
2
JEE Advanced 2023 Paper 2 Online
+3
-1
An electric dipole is formed by two charges $+q$ and $-q$ located in $x y$-plane at $(0,2) \mathrm{mm}$ and $(0,-2) \mathrm{mm}$, respectively, as shown in the figure. The electric potential at point $P(100,100) \mathrm{mm}$ due to the dipole is $V_0$. The charges $+q$ and $-q$ are then moved to the points $(-1,2) \mathrm{mm}$ and $(1,-2) \mathrm{mm}$, respectively. What is the value of electric potential at $P$ due to the new dipole?

A
$\frac{V_0}{4}$
B
$\frac{V_0}{2}$
C
$\frac{V_0}{\sqrt{2}}$
D
$\frac{3 V_0}{4}$
3
JEE Advanced 2019 Paper 1 Offline
+3
-1
A thin spherical insulating shell of radius R carries a uniformly distributed charge such that the potential at its surface is V0. A hole with a small area $$\alpha$$4$$\pi$$R2($$\alpha$$ << 1) is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?
A
The ratio of the potential at the center of the shell of that of the point at $${1 \over 2}$$R from center towards the hole will be $${{1 - \alpha } \over {1 - 2\alpha }}$$.
B
The potential at the center of the shell is reduced by 2$$\alpha$$V0.
C
The magnitude of electric field at the center of the shell is reduced by $${{\alpha {V_0}} \over {2R}}$$.
D
The magnitude of electric field at a point, located on a line passing through the hole and shell's center, on a distance 2R from the center of the spherical shell will be reduced by $${{\alpha {V_0}} \over {2R}}$$.
4
JEE Advanced 2018 Paper 2 Offline
+3
-0.75
The electric field $$E$$ is measured at a point $$P(0,0,d)$$ generated due to various charge distributions and the dependence of $$E$$ on $$d$$ is found to be different for different charge distributions. List-$${\rm I}$$ contains different relations between $$E$$ and $$d$$. List-$${\rm II}$$ describes different electric charge distributions, along with their locations. Match the functions in List-$${\rm I}$$ with the related charge distributions in List-$${\rm II}$$.

LIST - I LIST - II
P. $$E$$ is independent of $$d$$ 1. A point charge Q at the origin
Q. $$E\, \propto \,1/d$$ 2. A small dipole with point charges
$$Q$$ at $$\left( {0,0,l} \right)$$ and $$-Q$$ at
$$\left( {0,0, - l} \right).$$ Take $$2l < < d$$
R. $$E\, \propto \,1/{d^2}$$ 3. An infinite line charge coincident
with the x-axis, with uniform linear charge density $$\lambda$$
S. $$E\, \propto \,1/{d^3}$$ 4. Two infinite wires carrying
uniform linear charge density
parallel to the $$x$$-axis. The one
along $$\left( {y = 0,z = l} \right)$$ has
a charge density $$+ \lambda$$ and the one
along $$\left( {y = 0,z = - l} \right)$$ has a
charge density Take
5. Infinite plane charge coincident
with the $$xy$$-plane with uniform surface charge density
A
$$P \to 5;Q \to 3,4;R \to 1;S \to 2$$
B
$$P \to 5;Q \to 3;R \to 1,4;S \to 2$$
C
$$P \to 4;Q \to 3;R \to 1,2;S \to 4$$
D
$$P \to 4;Q \to 2,3;R \to 1;S \to 5$$
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