Electrostatics & Capacitors · JEE Advanced previous year questions

1. A.P→3, Q→1, R→2, S→4
2. B.P→4, Q→5, R→3, S→1
3. C.P→2, Q→1, R→4, S→5
4. D.P→2, Q→1, R→3, S→5

1. A.$\frac{Q}{30{ϵ}_0}$
2. B.$\frac{Q}{15{ϵ}_0}$
3. C.$\frac{Q}{60{ϵ}_0}$
4. D.$\frac{Q}{120{ϵ}_0}$

A positive point charge of $10^{-8}$ C is kept at a distance of 20 cm from the center of a neutral conducting sphere of radius 10 cm. The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and subsequently the point charge is moved by a distance of 10 cm further away from the center of the sphere along the radial direction. Taking $\frac{1}{4\pi {ϵ}_0}=9\times 10^9\mathrm{N}{\mathrm{m}}^2/{\mathrm{C}}^2$ (where ${ϵ}_0$ is the permittivity of free space), which of the following statements is/are correct:

1. A.Before the grounding, the electrostatic potential of the sphere is 450 V.
2. B.Charge flowing from the sphere to the ground because of grounding is $5\times 10^{-9}$ C.
3. C.After the grounding is removed, the charge on the sphere is $-5\times 10^{-9}$ C.
4. D.The final electrostatic potential of the sphere is 300 V.

1. A.In region 4 of the configuration I, the magnitude of the electric field is zero.
2. B.In region 3 of the configuration II, the magnitude of the electric field is $\frac{{\sigma }_0}{{ϵ}_0}$.
3. C.Potential difference between the first and the last sheets of the configuration I is $5\mathrm{V}$.
4. D.Potential difference between the first and the last sheets of the configuration II is zero.

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 [${\epsilon }_0$ is the permittivity of free space.]

1. A.$q^2/(4\pi {\epsilon }_0{R}^{3}m)$
2. B.$q^2/(32\pi {\epsilon }_0{R}^{3}m)$
3. C.$q^2/(8\pi {\epsilon }_0{R}^{3}m)$
4. D.$q^2/(16\pi {\epsilon }_0{R}^{3}m)$

1. A.$\mathrm{P}\to 3;\mathrm{Q}\to 2;\mathrm{R}\to 4;\mathrm{S}\to 5$
2. B.$\mathrm{P}\to 2;\mathrm{Q}\to 3;\mathrm{R}\to 2;\mathrm{S}\to 1$
3. C.$\mathrm{P}\to 3;\mathrm{Q}\to 2;\mathrm{R}\to 4;\mathrm{S}\to 1$
4. D.$\mathrm{P}\to 3;\mathrm{Q}\to 2;\mathrm{R}\to 2;\mathrm{S}\to 5$

1. A.$\mathrm{P}\to 1;\mathrm{Q}\to 3;\mathrm{R}\to 2;\mathrm{S}\to 5$
2. B.$\mathrm{P}\to 1;\mathrm{Q}\to 2;\mathrm{R}\to 3;\mathrm{S}\to 5$
3. C.$\mathrm{P}\to 1;\mathrm{Q}\to 3;\mathrm{R}\to 2;\mathrm{S}\to 4$
4. D.$\mathrm{P}\to 2;\mathrm{Q}\to 5;\mathrm{R}\to 3;\mathrm{S}\to 4$

1. A.The dipole will undergo small oscillations at any finite value of $r$.
2. B.The dipole will undergo small oscillations at any finite value of $r>R$.
3. C.The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{2\sigma p_0}{{ϵ}_{0}I}}$ at $r=2R$.
4. D.The dipole will undergo small oscillations with an angular frequency of $\sqrt{\frac{\sigma p_0}{100{ϵ}_{0}I}}$ at $r=10R$.

1. A.$27\mathrm{pF}$
2. B.$63\mathrm{pF}$
3. C.$81\mathrm{pF}$
4. D.$135\mathrm{pF}$

1. A.$V_0/4$
2. B.$V_0/2$
3. C.$V_0/\sqrt{2}$
4. D.$3V_0/4$

The electric field associated with an electromagnetic wave propagating in a dielectric medium is given by $\vec{E}=30(2\hat{x}+\hat{y})\sin \left[2\pi \left(5\times 10^{14}t-\frac{10^7}{3}z\right)\right]\mathrm{V}{\mathrm{m}}^{-1}$. Which of the following option(s) is(are) correct? [Given: The speed of light in vacuum, $c=3\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$]

1. A.$B_x=-2\times 10^{-7}\sin \left[2\pi \left(5\times 10^{14}t-\frac{10^7}{3}z\right)\right]\mathrm{Wb}{\mathrm{m}}^{-2}$.
2. B.$B_y=2\times 10^{-7}\sin \left[2\pi \left(5\times 10^{14}t-\frac{10^7}{3}z\right)\right]\mathrm{Wb}{\mathrm{m}}^{-2}$.
3. C.The wave is polarized in the $xy$-plane with polarization angle $30^{\circ }$ with respect to the $x$-axis.
4. D.The refractive index of the medium is 2.