A small electric dipole vec{p}_0,having a mom | vedclass

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(D) The electric field inside a spherical shell is zero,so the dipole will not experience any torque and thus will not oscillate if $r

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$B, D$

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(D) The electric field inside a spherical shell is zero,so the dipole will not experience any torque and thus will not oscillate if $r  R$,the electric field at the position of the dipole is $E = \frac{Q}{4 \pi \varepsilon_0 r^2}$,where $Q = 4 \pi R^2 \sigma$. Thus,$E = \frac{\sigma R^2}{\varepsilon_0 r^2}$.The torque on the dipole is $	au = -p_0 E \sin 	heta$. For small $	heta$,$\sin 	heta \approx 	heta$,so $	au = -p_0 E 	heta$.Using the equation of motion $	au = I \alpha$,we get $I \frac{d^2 	heta}{dt^2} = -p_0 E 	heta$,which represents simple harmonic motion with angular frequency $\omega = \sqrt{\frac{p_0 E}{I}}$.Substituting $E = \frac{\sigma R^2}{\varepsilon_0 r^2}$,we get $\omega = \sqrt{\frac{p_0 \sigma R^2}{\varepsilon_0 I r^2}} = \frac{R}{r} \sqrt{\frac{p_0 \sigma}{\varepsilon_0 I}}$.For $r = 2R$,$\omega = \frac{R}{2R} \sqrt{\frac{p_0 \sigma}{\varepsilon_0 I}} = \frac{1}{2} \sqrt{\frac{p_0 \sigma}{\varepsilon_0 I}} = \sqrt{\frac{p_0 \sigma}{4 \varepsilon_0 I}}$. Thus,statement $(C)$ is correct.For $r = 10R$,$\omega = \frac{R}{10R} \sqrt{\frac{p_0 \sigma}{\varepsilon_0 I}} = \frac{1}{10} \sqrt{\frac{p_0 \sigma}{\varepsilon_0 I}} = \sqrt{\frac{p_0 \sigma}{100 \varepsilon_0 I}}$. Thus,statement $(D)$ is correct.Therefore,statements $(B)$,$(C)$,and $(D)$ are correct. However,based on the provided options,the best choice is $(D)$.

List-$I$	List-$II$
$(P)$ Two dipoles pointing in $+\hat{j}$ direction at $x = -r$ and $x = +r$	$(1) \ \vec{E}=0$
$(Q)$ Two dipoles pointing in $+\hat{j}$ and $-\hat{j}$ direction at $x = -r$ and $x = +r$ respectively	$(2) \ \vec{E}=-\frac{p}{2 \pi \epsilon_0 r^3} \hat{j}$
$(R)$ Two dipoles pointing in $+\hat{j}$ and $+\hat{i}$ direction at $x = -r$ and $x = +r$ respectively	$(3) \ \vec{E}=-\frac{p}{4 \pi \epsilon_0 r^3}(\hat{i}-\hat{j})$
$(S)$ Two dipoles pointing in $+\hat{i}$ direction at $x = -r$ and $x = +r$	$(4) \ \vec{E}=\frac{p}{4 \pi \epsilon_0 r^3}(2\hat{i}-\hat{j})$
	$(5) \ \vec{E}=\frac{p}{\pi \epsilon_0 r^3} \hat{i}$

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