To determine the angular frequency of oscillations for a small particle of mass \( m \) and charge \( -q \) placed at point \( P \) inside a sphere of radius \( r \), when another charge \( Q \) is introduced, we can use concepts from electrostatics and harmonic motion. Given that \( r \gg x \), we can simplify our analysis by assuming that the electric field created by charge \( Q \) is uniform across the region where the particle oscillates.
Understanding the Forces at Play
When the particle is released, it experiences a force due to the electric field generated by charge \( Q \). The force \( F \) acting on the particle can be expressed using Coulomb's law:
F = k \cdot \frac{|qQ|}{x^2}
Here, \( k \) is Coulomb's constant, and \( x \) is the distance from charge \( Q \) to the particle. Since the particle has a negative charge, it will be attracted towards the positive charge \( Q \).
Establishing the Potential Energy
The potential energy \( U \) associated with the particle in the electric field is given by:
U = -\frac{k \cdot qQ}{x}
As the particle moves closer to charge \( Q \), the potential energy decreases, indicating that the particle is moving towards a more stable configuration.
Finding the Effective Force and Oscillation
When the particle is displaced slightly from its equilibrium position, it will experience a restoring force that tends to bring it back to that position. This restoring force can be approximated as:
F_{\text{restoring}} = -k_{\text{eff}} \cdot x
Where \( k_{\text{eff}} \) is the effective spring constant. For small displacements, we can relate the force to the angular frequency \( \omega \) of oscillation:
F = m \cdot a = m \cdot \frac{d^2x}{dt^2}
By equating the restoring force to the mass times acceleration, we have:
m \cdot \frac{d^2x}{dt^2} = -k_{\text{eff}} \cdot x
Relating Angular Frequency to Spring Constant
The angular frequency \( \omega \) of the oscillation can be expressed as:
\(\omega = \sqrt{\frac{k_{\text{eff}}}{m}}\)
To find \( k_{\text{eff}} \), we can differentiate the potential energy with respect to \( x \) and evaluate it at the equilibrium position:
k_{\text{eff}} = -\frac{d^2U}{dx^2}
After performing the differentiation and substituting back into the equation for \( \omega \), we find that:
\(\omega = \sqrt{\frac{k \cdot |Q|}{m \cdot r^3}}\)
Final Expression for Angular Frequency
Thus, the angular frequency of the oscillations of the particle in the electric field created by charge \( Q \) is given by:
\(\omega = \sqrt{\frac{k \cdot |Q|}{m \cdot r^3}}\)
This formula indicates that the angular frequency depends on the charge \( Q \), the mass \( m \) of the particle, and the radius \( r \) of the sphere. As you can see, the larger the radius, the smaller the angular frequency, which aligns with our intuition that a larger radius leads to a weaker restoring force.
