A small electric dipole \vecp_0 , having a moment of inertia I about its center, is kept at a distance r from the center of a spherical shell of radius R . The surface charge density \sigma is uniformly distributed on the spherical shell. The dipole is initially oriented at a small angle \theta as shown in the figure. While staying at a distance r , the dipole is free to rotate about its center. If released from rest, then which of the following statement(s) is(are) correct? [ \varepsilon_0 is the permittivity of free space.]

The dipole will undergo small oscillations at any finite value of r>R ., The dipole will undergo small oscillations with an angular frequency of \sqrt\frac\sigma p_0100 \epsilon_0 I at r=10 R .

The dipole will undergo small oscillations at any finite value of r .

The dipole will undergo small oscillations at any finite value of r>R .

The dipole will undergo small oscillations with an angular frequency of \sqrt\frac2 \sigma p_0\epsilon_0 I at r=2 R .

The dipole will undergo small oscillations with an angular frequency of \sqrt\frac\sigma p_0100 \epsilon_0 I at r=10 R .

Electrostatics PYQ — JEE Advanced 2024 · Shift 2 · Q39 with Solution

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Correct answer: B, D

Electric field due to uniformly charged spherical shell

A spherical shell of radius $R$ with uniform surface charge density $\sigma$ has total charge $Q = 4\pi R^2 \sigma.$

By Gauss's law:

For $r<R$ (inside the shell), $E=0.$
For $r>R$ (outside the shell), field is same as that of a point charge $Q$ at the center: $=\frac{1}{4\pi\varepsilon_0}\frac{4\pi R^2\sigma}{r^2} =\frac{\sigma R^2}{\varepsilon_0 r^2}.$$$

So,

0,& r<R,\\[4pt] \dfrac{\sigma R^2}{\varepsilon_0 r^2},& r>R. \end{cases}$$ --- 2. **Torque on the dipole** A dipole of moment $p_0$ in an electric field $E$ experiences torque $$\tau = p_0 E \sin\theta.$$ For small angular displacement $\theta$, restoring torque for stable equilibrium is $$\tau \approx -p_0 E\,\theta.$$ Hence equation of motion is $$I\ddot\theta = -p_0 E\,\theta,$$ which gives SHM with angular frequency $$\omega = \sqrt{\frac{p_0 E}{I}}.$$ This is possible only if $E\neq 0$. --- 3. **Case analysis** ### For $r<R$ Inside the shell, $$E=0.$$ Hence $$\tau=0,$$ so there is no restoring torque and therefore **no oscillation**. Thus statement **A** (small oscillations at any finite value of $r$) is **false**. ### For $r>R$ $$E=\frac{\sigma R^2}{\varepsilon_0 r^2}.$$ Therefore the dipole executes small oscillations with $$\omega = \sqrt{\frac{p_0}{I}\cdot \frac{\sigma R^2}{\varepsilon_0 r^2}} =\sqrt{\frac{\sigma p_0 R^2}{\varepsilon_0 I r^2}}.$$ Hence for every finite $r>R$, oscillation occurs. So **B is true**. --- 4. **Check option C: $r=2R$** Using $$\omega = \sqrt{\frac{\sigma p_0 R^2}{\varepsilon_0 I r^2}},$$ put $r=2R$: $$\omega = \sqrt{\frac{\sigma p_0 R^2}{\varepsilon_0 I (2R)^2}} =\sqrt{\frac{\sigma p_0}{4\varepsilon_0 I}}.$$ Option C says $$\sqrt{\frac{2\sigma p_0}{\varepsilon_0 I}},$$ which is not correct. So **C is false**. --- 5. **Check option D: $r=10R$** Again, $$\omega = \sqrt{\frac{\sigma p_0 R^2}{\varepsilon_0 I r^2}}.$$ For $r=10R$, $$\omega = \sqrt{\frac{\sigma p_0 R^2}{\varepsilon_0 I (10R)^2}} =\sqrt{\frac{\sigma p_0}{100\varepsilon_0 I}}.$$ This matches option D. So **D is true**. --- 6. **Final evaluation of options** - **A:** False - **B:** True - **C:** False - **D:** True Therefore the correct options are: $$\boxed{\text{B, D}}$$ --- 7. **Comparison with stored correct answer** Stored correct answer: **B, D** My derived answer: **B, D** They match.

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View written solutionFree

Correct answer: B, D

Electric field due to uniformly charged spherical shell

A spherical shell of radius $R$ with uniform surface charge density $\sigma$ has total charge $Q = 4\pi R^2 \sigma.$

By Gauss's law:

For $r<R$ (inside the shell), $E=0.$
For $r>R$ (outside the shell), field is same as that of a point charge $Q$ at the center: $=\frac{1}{4\pi\varepsilon_0}\frac{4\pi R^2\sigma}{r^2} =\frac{\sigma R^2}{\varepsilon_0 r^2}.$$$

So,

0,& r<R,\\[4pt] \dfrac{\sigma R^2}{\varepsilon_0 r^2},& r>R. \end{cases}$$ --- 2. **Torque on the dipole** A dipole of moment $p_0$ in an electric field $E$ experiences torque $$\tau = p_0 E \sin\theta.$$ For small angular displacement $\theta$, restoring torque for stable equilibrium is $$\tau \approx -p_0 E\,\theta.$$ Hence equation of motion is $$I\ddot\theta = -p_0 E\,\theta,$$ which gives SHM with angular frequency $$\omega = \sqrt{\frac{p_0 E}{I}}.$$ This is possible only if $E\neq 0$. --- 3. **Case analysis** ### For $r<R$ Inside the shell, $$E=0.$$ Hence $$\tau=0,$$ so there is no restoring torque and therefore **no oscillation**. Thus statement **A** (small oscillations at any finite value of $r$) is **false**. ### For $r>R$ $$E=\frac{\sigma R^2}{\varepsilon_0 r^2}.$$ Therefore the dipole executes small oscillations with $$\omega = \sqrt{\frac{p_0}{I}\cdot \frac{\sigma R^2}{\varepsilon_0 r^2}} =\sqrt{\frac{\sigma p_0 R^2}{\varepsilon_0 I r^2}}.$$ Hence for every finite $r>R$, oscillation occurs. So **B is true**. --- 4. **Check option C: $r=2R$** Using $$\omega = \sqrt{\frac{\sigma p_0 R^2}{\varepsilon_0 I r^2}},$$ put $r=2R$: $$\omega = \sqrt{\frac{\sigma p_0 R^2}{\varepsilon_0 I (2R)^2}} =\sqrt{\frac{\sigma p_0}{4\varepsilon_0 I}}.$$ Option C says $$\sqrt{\frac{2\sigma p_0}{\varepsilon_0 I}},$$ which is not correct. So **C is false**. --- 5. **Check option D: $r=10R$** Again, $$\omega = \sqrt{\frac{\sigma p_0 R^2}{\varepsilon_0 I r^2}}.$$ For $r=10R$, $$\omega = \sqrt{\frac{\sigma p_0 R^2}{\varepsilon_0 I (10R)^2}} =\sqrt{\frac{\sigma p_0}{100\varepsilon_0 I}}.$$ This matches option D. So **D is true**. --- 6. **Final evaluation of options** - **A:** False - **B:** True - **C:** False - **D:** True Therefore the correct options are: $$\boxed{\text{B, D}}$$ --- 7. **Comparison with stored correct answer** Stored correct answer: **B, D** My derived answer: **B, D** They match.

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