CBSE 2025 · Region 1 · Set 2 · Q32 · 5 marks
(i)Two point charges $\displaystyle 5 \mu \mathrm{C}$ and $\displaystyle -1 \mu \mathrm{C}$ are placed at points ( -$\displaystyle 3$ cm, $\displaystyle 0,0)$ and ( $\displaystyle 3 \mathrm{~cm}, 0,0$ ) respectively. An external electric field $\displaystyle \overrightarrow{\mathrm{E}}=\frac{\mathrm{A}}{\mathrm{r}^{2}} \hat{\mathrm{r}}$ where $\displaystyle \mathrm{A}=3 \times 10^{5} \mathrm{Vm}$ is switched on in the region. Calculate the change in electrostatic energy of the system due to the electric field.(ii)A system of two conductors is placed in air and they have net charge of $\displaystyle +80 \mu \mathrm{C}$ and $\displaystyle -80 \mu \mathrm{C}$ which causes a potential difference of $\displaystyle 16$ V between them.(1)Find the capacitance of the system.(2)If the air between the capacitor is replaced by a dielectric medium of dielectric constant $\displaystyle 3$, what will be the potential difference between the two conductors?(3)If the charges on two conductors are changed to $\displaystyle +160 \mu \mathrm{C}$ and $\displaystyle -160 \mu \mathrm{C}$, will the capacitance of the system change? Give reason for your answer.(i)Consider three metal spherical shells $\displaystyle \mathrm{A}, \mathrm{B}$ and C , each of radius R . Each shell is having a concentric metal ball of radius $\displaystyle \mathrm{R} / 10$. The spherical shells $\displaystyle \mathrm{A}, \mathrm{B}$ and C are given charges $\displaystyle +6 \mathrm{q},-4 \mathrm{q}$, and $\displaystyle 14 \mathrm{q}$ respectively. Their inner metal balls are also given charges $\displaystyle -2 \mathrm{q},+8 \mathrm{q}$ and -$\displaystyle 10$ q respectively. Compare the magnitude of the electric fields due to shells $\displaystyle \mathrm{A}, \mathrm{B}$ and C at a distance $\displaystyle 3$ R from their centres.(ii)A charge $\displaystyle -6 \mu \mathrm{C}$ is placed at the centre B of a semicircle of radius $\displaystyle 5$ cm , as shown in the figure. An equal and opposite charge is placed at point D at a distance of $\displaystyle 10$ cm from B . A charge $\displaystyle +5 \mu \mathrm{C}$ is moved from point ' C ' to point ' A ' along the circumference. Calculate the work done on the charge. 
(i)
Two point charges $\displaystyle 5 \mu \mathrm{C}$ and $\displaystyle -1 \mu \mathrm{C}$ are placed at points ( -$\displaystyle 3$ cm, $\displaystyle 0,0)$ and ( $\displaystyle 3 \mathrm{~cm}, 0,0$ ) respectively. An external electric field $\displaystyle \overrightarrow{\mathrm{E}}=\frac{\mathrm{A}}{\mathrm{r}^{2}} \hat{\mathrm{r}}$ where $\displaystyle \mathrm{A}=3 \times 10^{5} \mathrm{Vm}$ is switched on in the region. Calculate the change in electrostatic energy of the system due to the electric field.
(ii)
A system of two conductors is placed in air and they have net charge of $\displaystyle +80 \mu \mathrm{C}$ and $\displaystyle -80 \mu \mathrm{C}$ which causes a potential difference of $\displaystyle 16$ V between them.
(1)
Find the capacitance of the system.
(2)
If the air between the capacitor is replaced by a dielectric medium of dielectric constant $\displaystyle 3$, what will be the potential difference between the two conductors?
(3)
If the charges on two conductors are changed to $\displaystyle +160 \mu \mathrm{C}$ and $\displaystyle -160 \mu \mathrm{C}$, will the capacitance of the system change? Give reason for your answer.
(i)
Consider three metal spherical shells $\displaystyle \mathrm{A}, \mathrm{B}$ and C , each of radius R . Each shell is having a concentric metal ball of radius $\displaystyle \mathrm{R} / 10$. The spherical shells $\displaystyle \mathrm{A}, \mathrm{B}$ and C are given charges $\displaystyle +6 \mathrm{q},-4 \mathrm{q}$, and $\displaystyle 14 \mathrm{q}$ respectively. Their inner metal balls are also given charges $\displaystyle -2 \mathrm{q},+8 \mathrm{q}$ and -$\displaystyle 10$ q respectively. Compare the magnitude of the electric fields due to shells $\displaystyle \mathrm{A}, \mathrm{B}$ and C at a distance $\displaystyle 3$ R from their centres.
(ii)
A charge $\displaystyle -6 \mu \mathrm{C}$ is placed at the centre B of a semicircle of radius $\displaystyle 5$ cm , as shown in the figure. An equal and opposite charge is placed at point D at a distance of $\displaystyle 10$ cm from B . A charge $\displaystyle +5 \mu \mathrm{C}$ is moved from point ' C ' to point ' A ' along the circumference. Calculate the work done on the charge.
Marking-scheme solution
(i)
$\displaystyle \vec{E}=\frac{3 \times 10^{5}}{r^{2}} \hat{r}$ (Given), $\displaystyle d V=-\vec{E} \cdot d \vec{r}$
$\displaystyle V=3 \times 10^{5} / r$
Electrostatic energy of the system in the absence of the field:
$\displaystyle U_{i}=\frac{K q_{1} q_{2}}{r_{12}}$
Electrostatic energy in the presence of the field:
$\displaystyle U_{f}=\frac{K q_{1} q_{2}}{r_{12}}+q_{1} V\left(\vec{r}_{1}\right)+q_{2} V\left(\vec{r}_{2}\right)$
$\displaystyle \Delta U=U_{f}-U_{i}=q_{1} V\left(\vec{r}_{1}\right)+q_{2} V\left(\vec{r}_{2}\right)$
$\displaystyle \Delta U=\frac{5 \times 10^{-6} \times 3 \times 10^{5}}{3 \times 10^{-2}}-\frac{1 \times 10^{-6} \times 3 \times 10^{5}}{3 \times 10^{-2}}$
$\displaystyle =40 \mathrm{~J}$
(ii)(1)
$\displaystyle C=\frac{Q}{V}=\frac{80}{16}=5 \mu \mathrm{F}$
(2)
$\displaystyle C^{\prime}=K C$
$\displaystyle =3 \times 5 \mu \mathrm{F}=15 \mu \mathrm{F}$
$\displaystyle V^{\prime}=\frac{Q}{C^{\prime}}=\frac{80 \mu \mathrm{C}}{15 \mu \mathrm{F}}=5.33 \mathrm{~V}$
(3)
No.
The capacitance of the system depends on its geometry.
(i)
Total charge for A = Total charge for B = Total charge for C $\displaystyle =+4 q$
As $\displaystyle E=\frac{k Q}{r^{2}}$
Since $\displaystyle Q=4 q$ and $\displaystyle r=3 R$:
$\displaystyle E=\frac{k(4 q)}{9 R^{2}}=\frac{4 k q}{9 R^{2}}$
$\displaystyle \therefore E_{A}=E_{B}=E_{C}$
(ii)
$\displaystyle V_{C}=\left[\frac{k \times 6 \times 10^{-6}}{5 \times 10^{-2}}-\frac{k \times 6 \times 10^{-6}}{5 \times 10^{-2}}\right]=0$
$\displaystyle V_{A}=\left[\frac{k \times 6 \times 10^{-6}}{15 \times 10^{-2}}-\frac{k \times 6 \times 10^{-6}}{5 \times 10^{-2}}\right]$
$\displaystyle =\frac{k \times 6 \times 10^{-6}}{10^{-2}}\left[\frac{1-3}{15}\right]$
$\displaystyle =-\frac{9 \times 10^{9} \times 6 \times 10^{-6} \times 2}{15 \times 10^{-2}}$
$\displaystyle =-7.2 \times 10^{5} \mathrm{~V}$
$\displaystyle W=q\left[V_{A}-V_{C}\right]$
$\displaystyle =5 \times 10^{-6}\left[-7.2 \times 10^{5}-0\right]$
$\displaystyle W=-3.6 \mathrm{~J}$
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