An electric dipole consists of 2.0 g spheres charged to 5.0 nC (positive and negative) at the ends of a 12 cm long massless rod. The dipole rotates on a frictionless pivot at its center. The dipole is held perpendicular to a uniform electric field with field strength 1400V, then released. What is the dipole’s angular velocity at the instant it is aligned with the electric field?
To find the dipole's angular velocity at the instant it is aligned with the electric field, we can use the principle of conservation of energy. The initial potential energy of the dipole is converted into its final kinetic energy when it aligns with the electric field.
First, let's calculate the initial potential energy of the dipole. The potential energy of a dipole in an electric field is given by:
U = - pE cosθ
Where:
U = potential energy
p = dipole moment
E = electric field strength
θ = angle between dipole moment and electric field direction
In this case, the dipole is initially perpendicular to the electric field, so θ = 90 degrees. The dipole moment (p) is the product of the charge (q) and the distance between the charges (d):
p = qd
Given:
q = 5.0 nC = 5.0 x 10^-9 C
d = 12 cm = 0.12 m
E = 1400 V/m (electric field strength)
Calculating the dipole moment:
p = (5.0 x 10^-9 C) x (0.12 m)
p = 6.0 x 10^-10 C·m
Now, substituting the values of p, E, and θ into the potential energy equation, when the dipole is perpendicular to the electric field:
U = - pE cosθ
U = - (6.0 x 10^-10 C·m) x (1400 V/m) x cos(90 degrees)
U = - 6.0 x 10^-10 C·m x 1400 V/m x 0
U = 0 Joules
Since the initial potential energy is zero, all the potential energy is converted into kinetic energy at the final position when the dipole is aligned with the electric field.
The kinetic energy (K) of the dipole can be calculated using:
K = (1/2) I ω^2
Where:
K = kinetic energy
I = moment of inertia
ω = angular velocity
The moment of inertia (I) for a rigid body rotating about an axis passing through its center and perpendicular to the plane of rotation is given by:
I = m r^2
Where:
m = mass of each sphere
r = distance from the center to each sphere
Given:
mass of each sphere (m) = 2.0 g = 2.0 x 10^-3 kg
distance from the center to each sphere (r) = 0.12 m / 2 = 0.06 m
Calculating the moment of inertia:
I = (2.0 x 10^-3 kg) x (0.06 m)^2
I = 2.4 x 10^-5 kg·m^2
Since all the potential energy is converted into kinetic energy, we have:
K = U = 0 Joules
So, substituting the values of I and K into the kinetic energy equation, we can solve for the angular velocity (ω):
0 = (1/2) (2.4 x 10^-5 kg·m^2) ω^2
0 = 1.2 x 10^-5 kg·m^2 ω^2
Solving for ω:
ω^2 = 0 / (1.2 x 10^-5 kg·m^2)
ω^2 = 0
ω = 0 radians/second
Therefore, the dipole's angular velocity at the instant it is aligned with the electric field is 0 radians/second.