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(B) The electric field of an ideal dipole with moment $\vec{p}$ at a position $\vec{r}$ is $\vec{E} = \frac{1}{4\pi\epsilon_0} [\frac{3(\vec{p}\cdot\h

Vedclass · Accepted Answer

$P \rightarrow 2, Q \rightarrow 1, R \rightarrow 4, S \rightarrow 5$

Vedclass · Answer

(B) The electric field of an ideal dipole with moment $\vec{p}$ at a position $\vec{r}$ is $\vec{E} = \frac{1}{4\pi\epsilon_0} [\frac{3(\vec{p}\cdot\hat{r})\hat{r} - \vec{p}}{r^3}]$.$(P)$ Both dipoles are $\vec{p} = p\hat{j}$ at $x = -r$ and $x = r$. At $X(0,0)$,$\vec{r}_1 = r\hat{i}$ and $\vec{r}_2 = -r\hat{i}$. The field from both is equatorial: $\vec{E} = 2 	imes [-\frac{p}{4\pi\epsilon_0 r^3}\hat{j}] = -\frac{p}{2\pi\epsilon_0 r^3}\hat{j}$. Matches $(2)$.$(Q)$ Dipoles are $p\hat{j}$ at $-r$ and $-p\hat{j}$ at $r$. Fields at $X$ are equal and opposite,so $\vec{E} = 0$. Matches $(1)$.$(R)$ Dipole $p\hat{j}$ at $-r$ (equatorial field at $X$ is $-\frac{p}{4\pi\epsilon_0 r^3}\hat{j}$) and dipole $p\hat{i}$ at $r$ (axial field at $X$ is $\frac{2p}{4\pi\epsilon_0 r^3}\hat{i}$). Total $\vec{E} = \frac{p}{4\pi\epsilon_0 r^3}(2\hat{i}-\hat{j})$. Matches $(4)$.$(S)$ Both dipoles are $p\hat{i}$ at $x = -r$ and $x = r$. Both fields are axial: $\vec{E} = \frac{2p}{4\pi\epsilon_0 r^3}\hat{i} + \frac{2p}{4\pi\epsilon_0 r^3}\hat{i} = \frac{p}{\pi\epsilon_0 r^3}\hat{i}$. Matches $(5)$.Thus,$P ightarrow 2, Q ightarrow 1, R ightarrow 4, S ightarrow 5$.

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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