Difference between revisions of "Exercise 100320"
From 105/106 Lecture Notes by OBM
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==Problem== | ==Problem== | ||
| − | Find an expression for the oscillation frequency of an electric dipole of dipole moment p | + | Find an expression for the oscillation frequency of an electric dipole of dipole moment <math>\vec{p}</math> and rotational inertia <math>I</math> for small amplitudes of oscillation about its equilibrium position in a uniform electric field of magnitude <math>E</math>. |
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==Solution== | ==Solution== | ||
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small angle approximation: | small angle approximation: | ||
| − | <math>\sin\theta\rightarrow\theta<math>. | + | <math>\sin\theta\rightarrow\theta</math>. |
| − | + | Thus, | |
<math>\tau\approx-pE\theta</math>. | <math>\tau\approx-pE\theta</math>. | ||
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Since this exhibits a simple negative proportionality to the angle of rotation, the dipole oscillates in simple harmonic motion, like a torsional pendulum with torsion constant <math>\kappa=pE</math>. The angular frequency <math>\omega</math> is given by | Since this exhibits a simple negative proportionality to the angle of rotation, the dipole oscillates in simple harmonic motion, like a torsional pendulum with torsion constant <math>\kappa=pE</math>. The angular frequency <math>\omega</math> is given by | ||
| − | <math>\omega^2=\frac{\kappa}{I}=\frac{ | + | <math>\omega^2=\frac{\kappa}{I}=\frac{pE}{I}</math> |
where I is the rotational inertia of the dipole. The frequency of oscillation is | where I is the rotational inertia of the dipole. The frequency of oscillation is | ||
| − | <math>\mathcal{f}=\frac{\omega}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{ | + | <math>\mathcal{f}=\frac{\omega}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{pE}{I}}</math> |
Latest revision as of 10:42, 10 March 2020
Problem
Find an expression for the oscillation frequency of an electric dipole of dipole moment and rotational inertia for small amplitudes of oscillation about its equilibrium position in a uniform electric field of magnitude .
Solution
is a restoring torque, trying to bring the tilted dipole back to its aligned equilibrium position.
small angle approximation:
.
Thus,
.
Since this exhibits a simple negative proportionality to the angle of rotation, the dipole oscillates in simple harmonic motion, like a torsional pendulum with torsion constant . The angular frequency is given by
where I is the rotational inertia of the dipole. The frequency of oscillation is
