Tag: torque on a dipole in a uniform electric field

Questions Related to torque on a dipole in a uniform electric field

An electric dipole is placed in an electric field of a point charge then...........

torque on a dipole in a uniform electric field electric dipole electric charges and fields electrostatics physics

  1. Force is always zero.

  2. Torque is always zero.

  3. Force is may be zero.

  4. Torque may be zero.

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Correct Option: D

Four equal positive charges each of magnitude $q$ are placed at the respective vertices of a square of side length $l$. A point charge $Q$ is placed at the centre of the square. Then

torque on a dipole in a uniform electric field electric dipole electric charges and fields electrostatics physics

  1. $Q$ must not be in equilibrium

  2. $Q$ must be in stable equilibrium

  3. $Q$ must be in neutral equilibrium

  4. $Q$ must be in unstable equilibrium

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Correct Option: B

If we rotate the dipole of moment $p$ placed in an electric field $E$ from an $\theta _1$ to $\theta _2$, the work done by the external force is

torque on a dipole in a uniform electric field electric dipole electric charges and fields electrostatics physics

  1. $pE(\cos \theta _2 - \cos \theta _1)$

  2. $pE(\cos \theta _1 - \cos \theta _2)$

  3. $pE(\sin \theta _2 - \sin \theta _1)$

  4. $pE(\sin \theta _1 - \sin \theta _2)$

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Correct Option: B
Explanation:
Given dipole of dipole moment $p$ in an electric field $E$. It is rotated from $\theta _{1}$ to $\theta _{2}$. We have to find the work done by external force.
When a dipole of dipole moment  $p$ is placed in electric field, work done in rotated the dipole by angle $\theta$ is
$W=-pE \cos{\theta _{1}}$
Now work done in rotating dipole by $\theta _{1}$ is 
$W _{2}=-pE\cos{\theta _{2}}$
Work done in rotating the dipole from $\theta _{1}$ to $\theta _{2}$ is
$W=W _{2}-W _{1}$
$=-pE \cos{\theta _{2}}-(-pE \cos{\theta _{1}})$
$=pE(\cos{\theta _{1}}-\cos{\theta _{2}})$

An electric dipole of dipole moment $p$ is placed in a uniform electric field $E$ in stable equilibrium position. Its moment of inertia about the centroidal axis is $I$. If it is displaced slightly from its mean position find the period of small oscillations.

torque on a dipole in a uniform electric field electric dipole electric charges and fields electrostatics physics

  1. $2\pi \sqrt{\dfrac{I}{2pE}}$

  2. $2\pi \sqrt{\dfrac{2I}{pE}}$

  3. $2\pi \sqrt{\dfrac{I}{pE}}$

  4. $\pi \sqrt{\dfrac{2I}{pE}}$

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Correct Option: C
Explanation:
Dipole moment $=p$
electric field $=E$
centroid axis $=I$
Explanation
When displaced at an angle $\theta $ from its mean position the magnitude of restoring torque is $T=-psin\theta $
For small angular displacement $\sin\theta \approx \theta $
$T=-pE\theta $
$\alpha =\dfrac { T }{ I } =-\left( \dfrac { PE }{ I }  \right) \theta $
    $={ -w }^{ 2 }\theta $
${ w }^{ 2 }=\dfrac { PE }{ I } $
$T=2\pi \sqrt { \dfrac { I }{ PE }  } $
($P.E=$ moment in electric field)

In a certain region of space, electric field is along the z-direction throughout. The magnitude of electric field is, however not constant but increases uniformly along the positive z-direction at the rate ${10^5}\,V/m.$ The force and the torque experienced by a system having a total dipole moment equal to ${10^{ - 7}}C - m$ in the negative z-direction is given by respectively.

torque on a dipole in a uniform electric field electric dipole electric charges and fields electrostatics physics

  1. 0.01,0

  2. 0.02,0

  3. 0,0.01

  4. None of the above

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Correct Option: A
Explanation:

$z$ direction positive rate $={ 10 }^{ 5 }V/m$

torque $=$ M $\times$ $E$
            $={ 10 }^{ 5 }\times { 10 }^{ -7 }$
            $=0.01cm$

 An electric dipole consist of two opposite charges each of magnitude $1\mu C$ separated by a distance of $2\,cm.$ The dipole is placed in an external field of ${10^5}{\text{N/C}}$.The maximum torque on the dipole is:

torque on a dipole in a uniform electric field electric dipole electric charges and fields electrostatics physics

  1. $2 \times {10^{ - 4}}J$

  2. $2 \times {10^{ - 3}}J$

  3. $4 \times {10^{ - 3}}J$

  4. ${10^{ - 3}}N\,m$

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Correct Option: C
Explanation:
An electric dipole consist at two opposite charge each of magnitude $=1\mu C=1\times { 10 }^{ -6 }C$
distance $=2cm$
Exter field $={ 10 }^{ 5 }N/C$
maximum torque on the dipole $=?$
$q=1\times { 10 }^{ -6 }C,\quad 2a=2cm$
                                or,  $=0.02cm$
$\therefore$    $P=q\times 2a$
           $=\left( 1\times { 10 }^{ -6 } \right) \times 0.02$
           $=2\times { 10 }^{ -8 }cm$
Intensity of the external electric field, $E=1.0\times { 10 }^{ 5 }N/C$
(i) ${ Z } _{ max }=pE=\left( 2\times { 10 }^{ -8 } \right) \left( 10\times { 10 }^{ 5 } \right) =2\times { 10 }^{ -3 }N-m$
(ii) Net work done in turning the dipole from ${ 0 }^{ 0 }$ to ${ 180 }^{ 0 }$
i.e  $W=\int _{ { 0 }^{ 0 } }^{ { 180 }^{ 0 } }{ \overline { r }  } d\theta =\int _{ { 0 }^{ 0 } }^{ { 180 }^{ 0 } }{ pE\sin\theta  } d\theta $
           $=pE{ \left[ -cos\theta  \right]  } _{ { 0 }^{ 0 } }^{ { 180 }^{ 0 } }$
           $=-pE\left( { \cos180 }^{ 0 }-\cos{ 0 }^{ 0 } \right) $
           $=2pE$
           $=2\times \left( 2\times { 10 }^{ -8 } \right) \left( 1\times { 10 }^{ 5 } \right) J$
           $=4\times { 10 }^{ -3 }J$