Electric Dipole and Field due to a Dipole and Dipole in an Electric Field

Q 1 :

Two charges of $- 4 μ C$ and $+ 4 μ C$ are placed at the points A(1,0,4)m and B(2,−1,5)m located in an electric field $\vec{E} = 0.20 \hat{i} V / c m .$ The magnitude of the torque acting on the dipole is $8 \sqrt{α} \times 10^{- 5} N m,$ where $α$ = _________. [2024]

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(2)

$\vec{τ} = \vec{p} \times \vec{E}$

$\vec{p} = q \vec{ℓ}$

$\vec{E} = 0.2 \hat{i} \frac{V}{cm} = 20 \hat{i} \frac{V}{m}$

$\vec{p} = 4 \times (\hat{i} - \hat{j} + \hat{k}) = (4 \hat{i} - 4 \hat{j} + 4 \hat{k}) μ C - m$

$\vec{τ} = (4 \hat{i} - 4 \hat{j} + 4 \hat{k}) \times (20 \hat{i}) \times 10^{- 6} Nm$

$= (8 \hat{k} + 8 \hat{j}) \times 10^{- 5} \Rightarrow | \vec{τ} | = 8 \sqrt{2} \times 10^{- 5}$

$alpha equals 2$

Q 2 :

The electric field at point p due to an electric dipole is E. The electric field at point R on the equatorial line will be $\frac{E}{x}$ . The value of $x$ is _____ . [2024]

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(16)

$E_{P} = \frac{2 k p}{r^{3}} = E (at axial)$

$E_{R} = \frac{k p}{{(2 a)}^{3}} (at equational)$

$E_{R} = \frac{k p}{8 r^{3}} = \frac{2 k p}{16 r^{3}} = \frac{E}{16}$

$E_{R} = \frac{E}{16}$

$Hence x = 16$

Q 3 :

A point particle of charge Q is located at P along the axis of an electric dipole 1 at a distance r as shown in the figure. The point P is also on the equatorial plane of a second electric dipole 2 at a distance r. The dipoles are made of opposite charge q separated by a distance 2a. For the charge particle at P not to experience any net force, which of the following correctly describes the situation? [2025]

$\frac{a}{r} ~ 20$
$\frac{a}{r} ~ 10$
$\frac{a}{r} ~ 0.5$
$\frac{a}{r} ~ 3$

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(4)

$\Rightarrow \frac{1}{{(r - a)}^{2}} - \frac{1}{{(r + a)}^{2}} = \frac{2 a}{{(a^{2} + r^{2})}^{3 / 2}}$

$\frac{4 a r}{{(r^{2} - a^{2})}^{2}} = \frac{2 a}{{(a^{2} + r^{2})}^{3 / 2}}$

Q 4 :

An electric dipole of mass m, charge q, and length $l$ is placed in a uniform electric field $\vec{E} = E_{0} \hat{i}$ . When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be: [2025]

$\frac{1}{2 π} \sqrt{\frac{2 m l}{q E_{0}}}$
$2 π \sqrt{\frac{m l}{q E_{0}}}$
$\frac{1}{2 π} \sqrt{\frac{m l}{2 q E_{0}}}$
$2 π \sqrt{\frac{m l}{2 q E_{0}}}$

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(4)

$τ = {PE}_{0} \sin θ$

If $θ$ is small, $τ = - ({PE}_{0}) θ$

$I = m {(\frac{l}{2})}^{2} \cdot 2 = \frac{m l^{2}}{2}$

$T = 2 π \sqrt{\frac{m l^{2}}{2 \cdot {PE}_{0}}} = 2 π \sqrt{\frac{m l^{2}}{2 \cdot q {IE}_{0}}}$

$\Rightarrow T = 2 π \sqrt{\frac{m l}{2 q E_{0}}}$

Q 5 :

Two charges each of magnitude 0.01 C and separated by a distance of 0.4 mm constitute an electric dipole. If the dipole is placed in an uniform electric field $' \vec{E}'$ of 10 dyne/C making $30 °$ angle with $\vec{E}$ , the magnitude of torque acting on the dipole is [2023]

$4.0 \times 10^{- 10} Nm$
$2.0 \times 10^{- 10} Nm$
$1.0 \times 10^{- 8} Nm$
$1.5 \times 10^{- 9} Nm$

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(2)

$| \vec{P} |$ $= q d = 0.04 \times 0.4 \times 10^{- 3} = 4 \times 10^{- 6}$

$| \vec{τ} |$ $= P E \sin θ = 4 \times 10^{- 6} \times 10 \times 10^{- 5} \times \sin 30 °$

$= 4 \times 10^{- 6 - 5 + 1} \times \frac{1}{2} = 2 \times 10^{- 10}$

Q 6 :

The electric field due to a short electric dipole at a large distance $(r)$ from the center of the dipole on the equatorial plane varies with distance as [2023]

$r$
$\frac{1}{r}$
$\frac{1}{r^{3}}$
$\frac{1}{r^{2}}$

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(3)

$Electric field due to a dipole at a point on its axis is E = \frac{2 k p}{r^{3}}$

Q 7 :

An electric dipole of dipole moment is $6.0 \times 10^{- 6} Cm$ placed in a uniform electric field of $1.5 \times 10^{3} {NC}^{- 1}$ in such a way that the dipole moment is along the electric field. The work done in rotating the dipole by $180 °$ in this field will be _______ mJ. [2023]

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(18)

$The work done in rotating the electric dipole = Δ U$

$= U_{f} - U_{i}$

$= (- p E \cos 180 °) - (- p E \cos 0 °)$

$= p E + p E = 2 p E$

$= 2 \times 6 \times 10^{- 6} \times 1.5 \times 10^{3} = 18 mJ$

Q 8 :

A thin infinite sheet charge and an infinite line charge of respective charge densities $+ σ$ and $+ λ$ are placed parallel at 5 m distance from each other. Points $' P'$ and $' Q'$ are at $\frac{3}{π} m$ and $\frac{4}{π} m$ perpendicular distance from the line charge towards the sheet charge, respectively. $' E_{P}'$ and $' E_{Q}'$ are the magnitudes of resultant electric field intensities at point $' P'$ and $' Q'$ , respectively. If $\frac{E_{P}}{E_{Q}} = \frac{4}{a}$ for $2$ $| σ |$ $=$ $| λ |$ , then the value of $a$ is ________. [2023]

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(6)

$E_{A} = \frac{λ}{2 π ε_{0} r_{A}} - \frac{σ}{2 ε_{0}} {r_{A} = \frac{3}{π}} = \frac{1}{2 ε_{0}} [\frac{λ}{3} - σ]$

$E_{B} = \frac{λ}{2 π ε_{0} r_{B}} - \frac{σ}{2 ε_{0}} {r_{B} = \frac{4}{π}} = \frac{1}{2 ε_{0}} [\frac{λ}{4} - σ]$

$\frac{E_{A}}{E_{B}} = \frac{4}{3} (\frac{λ - 3 σ}{λ - 4 σ}) = \frac{4}{3} [\frac{2 σ - 3 σ}{2 σ - 4 σ}] = \frac{4}{3} [\frac{- σ}{- 2 σ}] = \frac{4}{6}$

Q 9 :

Three charges $+ 2 q$ , $+ 3 q$ and $- 4 q$ are situated at $(0, - 3 a)$ , $(2 a, 0)$ and $(- 2 a, 0)$ respectively in the $x y$ plane. The resultant dipole moment about origin is _____. [2026]

$2 q a (3 \hat{j} - 7 \hat{i})$
$2 q a (7 \hat{i} - 3 \hat{j})$
$2 q a (3 \hat{i} - 7 \hat{j})$
$2 q a (3 \hat{j} - \hat{i})$

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(2)

$\vec{p} = q_{1} {\vec{r}}_{1} + q_{2} {\vec{r}}_{2} + q_{3} {\vec{r}}_{3}$

$\vec{p} = (2 q) (- 3 a) \hat{j} + (3 q) (2 a) \hat{i} + (- 4 q) (- 2 a) \hat{i}$

$\vec{p} = 2 q a (7 \hat{i} - 3 \hat{j})$

Q 10 :

Two short dipoles (A,B), A having charges $\pm 2 μ C$ and length $1 cm$ and B having charges $\pm 4 μ C$ and length 1 cm are placed with their centres 80 cm apart as shown in the figure. The electric field at a point P, equi-distant from the centres of both dipoles is ______ N/C. [2026]

$\frac{9}{16} \sqrt{2} \times 10^{4}$
$\frac{9}{16} \sqrt{2} \times 10^{5}$
$4.5 \sqrt{2} \times 10^{4}$
$9 \sqrt{2} \times 10^{4}$

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(1)

${\vec{E}}_{2} = - \frac{K P_{2}}{r^{3}}; {\vec{E}}_{1} = - \frac{2 K P_{1}}{r^{3}}$

$P_{1} = 2 \times 10^{- 6} \times 10^{- 2} = 2 \times 10^{- 8}$

$P_{2} = 4 \times 10^{- 6} \times 10^{- 2} = 4 \times 10^{- 8}$

${\vec{E}}_{net} = \frac{2 \times 9 \times 10^{9} \times 2 \times 10^{- 8}}{{(0.4)}^{3}} \hat{i} - \frac{9 \times 10^{9} \times 4 \times 10^{- 8}}{{(0.4)}^{3}} \hat{j}$

${\vec{E}}_{net} = \frac{9 \times 10^{9} \times 4 \times 10^{- 8}}{{(0.4)}^{3}} (\hat{i} - \hat{j})$

$| {\vec{E}}_{net} |$ $= \frac{9 \times 10^{4}}{16} \sqrt{2}$

Topic Question Set

Q 1 :

Two charges of $- 4 μ C$ and $+ 4 μ C$ are placed at the points A(1,0,4)m and B(2,−1,5)m located in an electric field $\vec{E} = 0.20 \hat{i} V / c m .$ The magnitude of the torque acting on the dipole is $8 \sqrt{α} \times 10^{- 5} N m,$ where $α$ = _________. [2024]

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Q 2 :

The electric field at point p due to an electric dipole is E. The electric field at point R on the equatorial line will be $\frac{E}{x}$ . The value of $x$ is _____ . [2024]

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Q 4 :

An electric dipole of mass m, charge q, and length $l$ is placed in a uniform electric field $\vec{E} = E_{0} \hat{i}$ . When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be: [2025]

Q 5 :

Two charges each of magnitude 0.01 C and separated by a distance of 0.4 mm constitute an electric dipole. If the dipole is placed in an uniform electric field $' \vec{E}'$ of 10 dyne/C making $30 °$ angle with $\vec{E}$ , the magnitude of torque acting on the dipole is [2023]

Q 6 :

The electric field due to a short electric dipole at a large distance $(r)$ from the center of the dipole on the equatorial plane varies with distance as [2023]

Q 7 :

An electric dipole of dipole moment is $6.0 \times 10^{- 6} Cm$ placed in a uniform electric field of $1.5 \times 10^{3} {NC}^{- 1}$ in such a way that the dipole moment is along the electric field. The work done in rotating the dipole by $180 °$ in this field will be _______ mJ. [2023]

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Q 9 :

Three charges $+ 2 q$ , $+ 3 q$ and $- 4 q$ are situated at $(0, - 3 a)$ , $(2 a, 0)$ and $(- 2 a, 0)$ respectively in the $x y$ plane. The resultant dipole moment about origin is _____. [2026]

Q 10 :

Two short dipoles (A,B), A having charges $\pm 2 μ C$ and length $1 cm$ and B having charges $\pm 4 μ C$ and length 1 cm are placed with their centres 80 cm apart as shown in the figure. The electric field at a point P, equi-distant from the centres of both dipoles is ______ N/C. [2026]

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Topic Question Set

Q 1 : Two charges of -4μC and +4μC are placed at the points A(1,0,4)m and B(2,−1,5)m located in an electric field E→=0.20i^V/cm. The magnitude of the torque acting on the dipole is 8α×10-5Nm, where α = _________. [2024]

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Q 2 : The electric field at point p due to an electric dipole is E. The electric field at point R on the equatorial line will be Ex. The value of x is _____ . [2024]

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Q 4 : An electric dipole of mass m, charge q, and length l is placed in a uniform electric field E→=E0i^. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be: [2025]

Q 5 : Two charges each of magnitude 0.01 C and separated by a distance of 0.4 mm constitute an electric dipole. If the dipole is placed in an uniform electric field 'E→' of 10 dyne/C making 30° angle with E→, the magnitude of torque acting on the dipole is [2023]

Q 6 : The electric field due to a short electric dipole at a large distance (r) from the center of the dipole on the equatorial plane varies with distance as [2023]

Q 7 : An electric dipole of dipole moment is 6.0×10-6 Cm placed in a uniform electric field of 1.5×103 NC-1 in such a way that the dipole moment is along the electric field. The work done in rotating the dipole by 180° in this field will be _______ mJ. [2023]

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Q 9 : Three charges +2q, +3q and -4q are situated at (0,-3a), (2a,0) and (-2a,0) respectively in the xy plane. The resultant dipole moment about origin is _____. [2026]

Q 10 : Two short dipoles (A,B), A having charges ±2 μC and length 1 cm and B having charges ±4 μC and length 1 cm are placed with their centres 80 cm apart as shown in the figure. The electric field at a point P, equi-distant from the centres of both dipoles is ______ N/C. [2026]

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Q 1 :

Two charges of $- 4 μ C$ and $+ 4 μ C$ are placed at the points A(1,0,4)m and B(2,−1,5)m located in an electric field $\vec{E} = 0.20 \hat{i} V / c m .$ The magnitude of the torque acting on the dipole is $8 \sqrt{α} \times 10^{- 5} N m,$ where $α$ = _________. [2024]

Q 2 :

The electric field at point p due to an electric dipole is E. The electric field at point R on the equatorial line will be $\frac{E}{x}$ . The value of $x$ is _____ . [2024]

Q 4 :

An electric dipole of mass m, charge q, and length $l$ is placed in a uniform electric field $\vec{E} = E_{0} \hat{i}$ . When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be: [2025]

Q 5 :

Two charges each of magnitude 0.01 C and separated by a distance of 0.4 mm constitute an electric dipole. If the dipole is placed in an uniform electric field $' \vec{E}'$ of 10 dyne/C making $30 °$ angle with $\vec{E}$ , the magnitude of torque acting on the dipole is [2023]

Q 6 :

The electric field due to a short electric dipole at a large distance $(r)$ from the center of the dipole on the equatorial plane varies with distance as [2023]

Q 7 :

An electric dipole of dipole moment is $6.0 \times 10^{- 6} Cm$ placed in a uniform electric field of $1.5 \times 10^{3} {NC}^{- 1}$ in such a way that the dipole moment is along the electric field. The work done in rotating the dipole by $180 °$ in this field will be _______ mJ. [2023]

Q 9 :

Three charges $+ 2 q$ , $+ 3 q$ and $- 4 q$ are situated at $(0, - 3 a)$ , $(2 a, 0)$ and $(- 2 a, 0)$ respectively in the $x y$ plane. The resultant dipole moment about origin is _____. [2026]

Q 10 :

Two short dipoles (A,B), A having charges $\pm 2 μ C$ and length $1 cm$ and B having charges $\pm 4 μ C$ and length 1 cm are placed with their centres 80 cm apart as shown in the figure. The electric field at a point P, equi-distant from the centres of both dipoles is ______ N/C. [2026]