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

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Q.
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]

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

2 $2 π \sqrt{\frac{m l}{q E_{0}}}$

3 $\frac{1}{2 π} \sqrt{\frac{m l}{2 q E_{0}}}$

4 $2 π \sqrt{\frac{m l}{2 q E_{0}}}$

Ans.
(4)

$τ = {PE}_{0} \sin θ$

If $θ$ is small, $τ = - ({PE}_{0}) θ$

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

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

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

Solution

Q.
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]

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

2 $2 π \sqrt{\frac{m l}{q E_{0}}}$

3 $\frac{1}{2 π} \sqrt{\frac{m l}{2 q E_{0}}}$

4 $2 π \sqrt{\frac{m l}{2 q E_{0}}}$

Ans.
(4)

$τ = {PE}_{0} \sin θ$

If $θ$ is small, $τ = - ({PE}_{0}) θ$

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

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

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

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Solution

Q. 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]

1 12π2mlqE0

2 2πmlqE0

3 12πml2qE0

4 2πml2qE0

Ans. (4) τ=PE0 sin θ If θ is small, τ=–(PE0)θ I=m(l2)2·2=ml22 T=2πml22·PE0=2πml22·qIE0 ⇒ T=2πml2qE0

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Q.
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]

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

2 $2 π \sqrt{\frac{m l}{q E_{0}}}$

3 $\frac{1}{2 π} \sqrt{\frac{m l}{2 q E_{0}}}$

4 $2 π \sqrt{\frac{m l}{2 q E_{0}}}$

Ans.
(4)

$τ = {PE}_{0} \sin θ$

If $θ$ is small, $τ = - ({PE}_{0}) θ$

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

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

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