Dipole oscillations To understand the motion of a diatomic molecule in a uniform

Question

Dipole oscillations

To understand the motion of a diatomic molecule in a uniform electric field, let us consider a dumbbell model where the molecule is made of two point particles with identical mass m = 9 × 10-27 kg and charges +Q and -Q (Q = 2 µC). These particles are separated a distance d = 4 µm from each other. The molecule is in a region of space with a uniform electric field of magnitude 300 N/C. Initially, the electric moment of the molecule makes an angle of 5° with the electric field. The molecule is released from rest.

(a) Find the magnitude of the initial torque on the molecule.

t = ??? N m

(b) Find the kinetic energy of the system as it passes through the equilibrium position.

KE = ??? J

(c) Since the answer to (b) is not zero, the molecule will keep moving past the equilibrium position, reach an angle of 5° on the opposite side and rotate in the opposite direction. Therefore the resulting motion is an oscillation about the equilibrium position. Find the frequency this oscillation.

f = ??? Hz

Explanation / Answer

m = 9 × 10^-27 kg, Q = 2 µC, d = 4 µm, E = 300 N/C, angle A = 5°, (a) Find the magnitude of the initial torque on the molecule. t = F*d*sinA = QE*d*sinA = 2.09*10^-10 Nm (b) Find the kinetic energy of the system as it passes through the equilibrium position. KE = integral of F*d*sinA (-dA) = F*d*cosA at equilibrium, A = 0, KE = F*d = QE*d = 2.40*10^-9 J (c) Since the answer to (b) is not zero, the molecule will keep moving past the equilibrium position, reach an angle of 5° on the opposite side and rotate in the opposite direction. Therefore the resulting motion is an oscillation about the equilibrium position. Find the frequency this oscillation. restoring torque = -F*d*sinA if A is small, then restoring torque = -F*d*A = moment of inertia * angular acceleration -F*d*A = [m*(d/2)^2 + m*(d/2)^2] * d^2 A /dt^2 (md^2/2) * d^2 A /dt^2 + QE*d*A = 0 angular frequency = w = sqrt[QE*d/(md^2/2)] = sqrt[2QE/(md)] = 1.83*10^14 frequency f = w/(2*pi) = 2.91*10^13 Hz

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