1
JEE Advanced 2022 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2

In the figure, the inner (shaded) region $A$ represents a sphere of radius $r_{A}=1$, within which the electrostatic charge density varies with the radial distance $r$ from the center as $\rho_{A}=k r$, where $k$ is positive. In the spherical shell $B$ of outer radius $r_{B}$, the electrostatic charge density varies as $\rho_{B}=$ $\frac{2 k}{r}$. Assume that dimensions are taken care of. All physical quantities are in their SI units.

Which of the following statement(s) is(are) correct?

A
If $r_{B}=\sqrt{\frac{3}{2}}$, then the electric field is zero everywhere outside $B$.
B
If $r_{B}=\frac{3}{2}$, then the electric potential just outside $B$ is $\frac{k}{\epsilon_{0}}$.
C
If $r_{B}=2$, then the total charge of the configuration is $15 \pi k$.
D
If $r_{B}=\frac{5}{2}$, then the magnitude of the electric field just outside $B$ is $\frac{13 \pi k}{\epsilon_{0}}$.
2
JEE Advanced 2022 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
In Circuit-1 and Circuit- 2 shown in the figures, $R_{1}=1 \,\Omega, R_{2}=2 \,\Omega$ and $R_{3}=3 \,\Omega$.

$P_{1}$ and $P_{2}$ are the power dissipations in Circuit-1 and Circuit-2 when the switches $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ are in open conditions, respectively.

$Q_{1}$ and $Q_{2}$ are the power dissipations in Circuit-1 and Circuit-2 when the switches $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ are in closed conditions, respectively.

Which of the following statement(s) is(are) correct?
A
When a voltage source of $6 V$ is connected across $\mathrm{A}$ and $\mathrm{B}$ in both circuits, $P_{1} < P_{2}$.
B
When a constant current source of $2 \mathrm{Amp}$ is connected across A and B in both circuits, $P_{1}>P_{2}$.
C
When a voltage source of $6 V$ is connected across $\mathrm{A}$ and $\mathrm{B}$ in Circuit-1, $Q_{1}>P_{1}$.
D
When a constant current source of $2 \mathrm{Amp}$ is connected across A and $\mathrm{B}$ in both circuits, $Q_{2} < Q_{1}$.
3
JEE Advanced 2022 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
A bubble has surface tension $S$. The ideal gas inside the bubble has ratio of specific heats $\gamma=$ $\frac{5}{3}$. The bubble is exposed to the atmosphere and it always retains its spherical shape. When the atmospheric pressure is $P_{a 1}$, the radius of the bubble is found to be $r_{1}$ and the temperature of the enclosed gas is $T_{1}$. When the atmospheric pressure is $P_{a 2}$, the radius of the bubble and the temperature of the enclosed gas are $r_{2}$ and $T_{2}$, respectively.

Which of the following statement(s) is(are) correct?
A
If the surface of the bubble is a perfect heat insulator, then $\left(\frac{r_{1}}{r_{2}}\right)^{5}=\frac{P_{a 2}+\frac{2 S}{r_{2}}}{P_{a 1}+\frac{2 S}{r_{1}}}$.
B
If the surface of the bubble is a perfect heat insulator, then the total internal energy of the bubble including its surface energy does not change with the external atmospheric pressure.
C
If the surface of the bubble is a perfect heat conductor and the change in atmospheric temperature is negligible, then $\left(\frac{r_{1}}{r_{2}}\right)^{3}=\frac{P_{a 2}+\frac{4 S}{r_{2}}}{P_{a 1}+\frac{4 S}{r_{1}}}$.
D
If the surface of the bubble is a perfect heat insulator, then $\left(\frac{T_{2}}{T_{1}}\right)^{\frac{5}{2}}=\frac{P_{a 2}+\frac{4 S}{r_{2}}}{P_{a 1}+\frac{4 S}{r_{1}}}$.
4
JEE Advanced 2022 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
A disk of radius $\mathrm{R}$ with uniform positive charge density $\sigma$ is placed on the $x y$ plane with its center at the origin. The Coulomb potential along the $z$-axis is

$$V(z)=\frac{\sigma}{2 \epsilon_{0}}\left(\sqrt{R^{2}+z^{2}}-z\right) .$$

A particle of positive charge $q$ is placed initially at rest at a point on the $z$ axis with $z=z_{0}$ and $z_{0}>0$. In addition to the Coulomb force, the particle experiences a vertical force $\vec{F}=-c \hat{k}$ with $c>0$. Let $\beta=\frac{2 c \epsilon_{0}}{q \sigma}$.

Which of the following statement(s) is(are) correct?
A
For $\beta=\frac{1}{4}$ and $z_{0}=\frac{25}{7} R$, the particle reaches the origin.
B
For $\beta=\frac{1}{4}$ and $z_{0}=\frac{3}{7} R$, the particle reaches the origin.
C
For $\beta=\frac{1}{4}$ and $z_{0}=\frac{R}{\sqrt{3}}$, the particle returns back to $z=z_{0}$.
D
For $\beta>1$ and $z_{0}>0$, the particle always reaches the origin.
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