CBSE 2024 · Region 1 · Set 1 · Q31 · 5 marks
(i)Derive an expression for potential energy of an electric dipole $\displaystyle \overrightarrow{\mathrm{p}}$ in an external uniform electric field $\displaystyle \overrightarrow{\mathrm{E}}$. When is the potential energy of the dipole ($\displaystyle 1$) maximum, and(2)minimum ?(ii)An electric dipole consists of point charges - $\displaystyle 1.0$ pC and $\displaystyle +1 \cdot 0 \mathrm{pC}$ located at $\displaystyle ( 0,0 )$ and ( $\displaystyle 3 \mathrm{~mm}, 4 \mathrm{~mm}$ ) respectively in $\displaystyle \mathrm{x}-\mathrm{y}$ plane. An electric field $\displaystyle \overrightarrow{\mathrm{E}}=\left(\frac{1000 \mathrm{~V}}{\mathrm{~m}}\right) \hat{\mathrm{i}}$ is switched on in the region. Find the torque $\displaystyle \vec{\tau}$ acting on the dipole.(i)An electric dipole (dipole moment $\displaystyle \overrightarrow{\mathrm{p}}=\mathrm{p} \hat{\mathrm{i}}$ ), consisting of charges -q and q , separated by distance $\displaystyle 2$ a , is placed along the $\displaystyle \mathrm{x}$-axis, with its centre at the origin. Show that the potential V , due to this dipole, at a point $\displaystyle \mathrm{x},(\mathrm{x} \gg \mathrm{a})$ is equal to $\displaystyle \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\overrightarrow{\mathrm{p}} \cdot \hat{\mathrm{i}}}{\mathrm{x}^{2}}$.(ii)Two isolated metallic spheres $\displaystyle \mathrm{S}_{1}$ and $\displaystyle \mathrm{S}_{2}$ of radii $\displaystyle 1$ cm and $\displaystyle 3$ cm respectively are charged such that both have the same charge density $\displaystyle \left(\frac{2}{\pi} \times 10^{-9}\right) \mathrm{C} / \mathrm{m}^{2}$. They are placed far away from each other and connected by a thin wire. Calculate the new charge on sphere $\displaystyle \mathrm{S}_{1}$.
(i)
Derive an expression for potential energy of an electric dipole $\displaystyle \overrightarrow{\mathrm{p}}$ in an external uniform electric field $\displaystyle \overrightarrow{\mathrm{E}}$. When is the potential energy of the dipole ($\displaystyle 1$) maximum, and
(2)
minimum ?
(ii)
An electric dipole consists of point charges - $\displaystyle 1.0$ pC and $\displaystyle +1 \cdot 0 \mathrm{pC}$ located at $\displaystyle ( 0,0 )$ and ( $\displaystyle 3 \mathrm{~mm}, 4 \mathrm{~mm}$ ) respectively in $\displaystyle \mathrm{x}-\mathrm{y}$ plane. An electric field $\displaystyle \overrightarrow{\mathrm{E}}=\left(\frac{1000 \mathrm{~V}}{\mathrm{~m}}\right) \hat{\mathrm{i}}$ is switched on in the region. Find the torque $\displaystyle \vec{\tau}$ acting on the dipole.
(i)
An electric dipole (dipole moment $\displaystyle \overrightarrow{\mathrm{p}}=\mathrm{p} \hat{\mathrm{i}}$ ), consisting of charges -q and q , separated by distance $\displaystyle 2$ a , is placed along the $\displaystyle \mathrm{x}$-axis, with its centre at the origin. Show that the potential V , due to this dipole, at a point $\displaystyle \mathrm{x},(\mathrm{x} \gg \mathrm{a})$ is equal to $\displaystyle \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{\overrightarrow{\mathrm{p}} \cdot \hat{\mathrm{i}}}{\mathrm{x}^{2}}$.
(ii)
Two isolated metallic spheres $\displaystyle \mathrm{S}_{1}$ and $\displaystyle \mathrm{S}_{2}$ of radii $\displaystyle 1$ cm and $\displaystyle 3$ cm respectively are charged such that both have the same charge density $\displaystyle \left(\frac{2}{\pi} \times 10^{-9}\right) \mathrm{C} / \mathrm{m}^{2}$. They are placed far away from each other and connected by a thin wire. Calculate the new charge on sphere $\displaystyle \mathrm{S}_{1}$.
Marking-scheme solution
(a) 
(i) 
The amount of work done in rotating the dipole from \(\displaystyle \theta=\theta_{0}\) to \(\displaystyle \theta=\theta_{1}\) by the external torque
\[\begin{aligned}
& \mathrm{W}=\int_{\theta_{0}}^{\theta_{0}} \tau_{e \mathrm{x} t} d \theta \\
& \qquad \begin{aligned}
& =\int_{\theta_{0}}^{\theta_{0}} \mathrm{p} \mathrm{E} \sin \theta d \theta \\
\mathrm{~W} & =\mathrm{p} \mathrm{E}\left(\cos \theta_{0}-\cos \theta_{1}\right)
\end{aligned} \\
& \text { For } \theta_{0}=\frac{\pi}{2} \text { and } \theta_{1}=\theta \\
& =\mathrm{p} \mathrm{E}\left(\cos \frac{\pi}{2}-\cos \theta\right) \\
& \begin{aligned}
\mathrm{U}(\theta) & =-\mathrm{p} \mathrm{E} \cos \theta \\
& =-\vec{\mathrm{p}} \cdot \vec{\mathrm{E}}
\end{aligned}
\end{aligned}
\]Electrostatic Potential and CapacitancePotential Energy in an External FieldApplylong_answerhard
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CBSE Class 12 Physics past-paper question from the 2024board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.