Electrostatics (Electric Charges and Fields, Potential) questions
Questions are transcribed from NTA's official question paper. Diagrams are described in words; open the official PDF to see them. Every answer was solved twice independently and is shown only when both agree. Where they did not, we say so.
Two short electric dipoles $A$ and $B$ having dipole moment $p_1$ and $p_2$ respectively are placed with their axis mutually perpendicular as shown in the figure. The resultant electric field at a point $x$ is making an angle of $60°$ with the line joining points O and $x$. The ratio of the dipole moments $p_2/p_1$ is _________. [Figure: Two perpendicular dipoles A and B with A along vertical axis and B along horizontal axis, point x on horizontal axis]
(1) $\frac{\sqrt{3}}{2}$
(2) $2\sqrt{3}$
(3) $\frac{1}{\sqrt{3}}$
(4)
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Answer under review. Our two independent solutions did not agree, so we are not showing an answer yet. Check it against NTA's final answer key.
Key idea: Electric field due to dipole at axial and equatorial positions
Two metal plates $(A, B)$ are kept horizontally with separation of $\left(\frac{12}{\pi}\right)$ cm, with plate $A$ on the top. An atomizer jet sprays oil (density $1.5$ g/cm$^3$) droplets of radius $1$ mm horizontally. All oil droplets carry a charge $5$ nC. The potentials $V_A$ and $V_B$ are required on plates $A$ and $B$ respectively in order to ensure the droplets do not descend. The values of $V_A$ and $V_B$ are __________. (Neglect the air resistance to the droplets and take $g = 10$ m/s$^2$)
(1) $100$ V and $580$ V
(2) $580$ V and $100$ V
(3) $60$ V and $400$ V
(4) $0$ V and $-200$ V
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Answer under review. Our two independent solutions did not agree, so we are not showing an answer yet. Check it against NTA's final answer key.
Key idea: Equilibrium of charged particle in electric field - Balancing gravitational and electrostatic forces
Two point charges $8$ $\mu$C and $-2$ $\mu$C are located at $x = 2$ cm and $x = 4$ cm, respectively on the $x$-axis. The ratio of electric flux due to these charges through two spheres of radii $3$ cm and $5$ cm with their centers at the origin is __________.
(1) $4:1$
(2) $3:4$
(3) $4:3$
(4) $4:5$
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Answer: Option (3)
Key idea: Gauss's Law - Electric flux depends only on enclosed charge
Apply Gauss's Law: Electric flux $\Phi = \frac{q_{\text{enclosed}}}{\epsilon_0}$
For sphere of radius $3$ cm: Check which charges are inside. Charge at $x=2$ cm is inside (distance $2 < 3$), charge at $x=4$ cm is outside (distance $4 > 3$)
Enclosed charge in sphere 1: $q_1 = 8$ μC, so $\Phi_1 = \frac{8 \mu C}{\epsilon_0}$
For sphere of radius $5$ cm: Both charges are inside (distances $2 < 5$ and $4 < 5$)
A thin half ring of radius 35 cm is uniformly charged with a total charge of $Q$ coulomb. If the magnitude of the electric field at centre of the half ring is 100 V/m, then the value of $Q$ is ______ nC. ($\epsilon_0 = 8.85 \times 10^{-12}$ C$^2$/Nm$^2$ and $\pi = 3.14$)
(1) 2.14
(2) 2.44
(3) 3.25
(4) 0.7
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Answer: Option (1)
Key idea: Electric field at the center of a uniformly charged half ring
For a uniformly charged half ring, linear charge density $\lambda = \frac{Q}{\pi R}$ where $R = 0.35$ m
Consider element $dq = \lambda R d\theta = \frac{Q}{\pi}d\theta$ at angle $\theta$. Electric field $dE = \frac{k dq}{R^2}$
A three coulomb charge moves from the point $(0, -2, -5)$ to the point $(5, 1, 2)$ in an electric field expressed as $\vec{E} = 2x\hat{i} + 3y^{2}\hat{j} + 4\hat{k}$ N/C. The work done in moving the charge is ______J.
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Answer: 186
Key idea: Work done by electric field on a moving charge in a conservative field
Work done is $W = q \int_A^B \vec{E} \cdot d\vec{r}$ where $q = 3$ C
Check that $\nabla \times \vec{E} = 0$, confirming the field is conservative (path-independent)
Write $\vec{E} \cdot d\vec{r} = 2x dx + 3y^2 dy + 4 dz$ and integrate each component separately
Two point charges $q_1 = 3~\mu C$ and $q_2 = -4~\mu C$ are placed at points $\left(2\hat{i} + 3\hat{j} + 3\hat{k}\right)$ and $\left(\hat{i} + \hat{j} + \hat{k}\right)$ respectively. Force on charge $q_2$ is _______ N. $\left(\text{Take } \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \text{ SI Units}\right)$
