A bob of a mass m with a charge -Q is suspended from a string of length l inside a parallel plate capacitor whose electric intensity vertically upwords. The period of oscillation is what

To determine the period of oscillation of a charged bob suspended in a parallel plate capacitor, we need to consider the forces acting on the bob and how they interact with the electric field. The bob, with mass m and charge -Q, experiences both gravitational and electric forces. Let's break this down step by step.

Understanding the Forces at Play

When the bob is suspended in the electric field of the capacitor, two main forces act on it:

• Gravitational Force (Fg): This force acts downward and is given by the equation Fg = mg, where g is the acceleration due to gravity.
• Electric Force (Fe): Since the bob has a charge -Q and is in an electric field (E) directed upwards, the electric force acting on the bob is Fe = -QE. The negative sign indicates that the force acts in the opposite direction of the electric field because the charge is negative.

Net Force and Equilibrium

In equilibrium, the forces acting on the bob balance each other. Therefore, we can set up the following equation:

mg = QE

From this, we can express the electric field strength as:

E = mg/Q

Small Angle Approximation for Oscillations

When the bob is displaced slightly from its equilibrium position, it will experience a restoring force that causes it to oscillate. For small angles (θ), we can use the small angle approximation, where sin(θ) ≈ θ (in radians). The restoring force due to gravity can be expressed as:

F_restoring = -mg sin(θ) ≈ -mgθ

According to Hooke's law, this restoring force is analogous to that of a spring, leading to simple harmonic motion (SHM). The angular frequency (ω) of SHM is given by:

ω = √(g/l)

Incorporating the Electric Field

However, we must also consider the effect of the electric field. The effective acceleration due to gravity in this scenario is modified by the electric field. The new effective gravitational acceleration (g_eff) can be expressed as:

g_eff = g - (QE/m)

Substituting the expression for E we derived earlier, we get:

g_eff = g - (mg/Q)(Q/m) = g - g = 0

This indicates that the electric field can completely counteract the gravitational force, leading to a situation where the bob can theoretically float. However, for oscillations, we need to consider the balance of forces when the bob is displaced.

Calculating the Period of Oscillation

The period (T) of oscillation for a simple harmonic oscillator is given by:

T = 2π√(l/g_eff)

In our case, since the electric field modifies the effective gravitational force, we can express the period as:

T = 2π√(l/(g - (QE/m)))

Final Thoughts

In summary, the period of oscillation of the charged bob in the electric field of the capacitor is influenced by both gravitational and electric forces. The final formula for the period takes into account the effective gravitational acceleration, which is modified by the electric field. This interplay of forces is a fascinating example of how electric fields can affect mechanical systems.