To find the time constant of a parallel plate capacitor filled with a dielectric liquid that is decreasing in height, we need to analyze how the capacitance changes as the liquid level drops. Let's break this down step by step.
Understanding the Capacitance of a Parallel Plate Capacitor
The capacitance \( C \) of a parallel plate capacitor can be expressed using the formula:
C = \frac{\varepsilon A}{d}
Where:
- \( \varepsilon \) is the permittivity of the dielectric material.
- A is the area of the plates (which is 1 unit area in this case).
- d is the separation between the plates.
Dielectric Constant and Effective Capacitance
When the capacitor is filled with a dielectric liquid, the permittivity \( \varepsilon \) can be expressed as:
\( \varepsilon = \varepsilon_0 \cdot \kappa \)
Where:
- \( \varepsilon_0 \) is the permittivity of free space.
- \( \kappa \) is the dielectric constant of the liquid, which is given as 2.
Thus, the effective capacitance when the liquid level is at height \( h \) can be written as:
C(h) = \frac{2 \varepsilon_0 \cdot 1}{d - h}
Capacitance as the Liquid Level Decreases
Initially, the liquid level is at \( h = \frac{d}{3} \). As the liquid level decreases at a constant speed \( V \), we can express the height of the liquid as:
h(t) = \frac{d}{3} - Vt
Substituting this into the capacitance formula gives:
C(t) = \frac{2 \varepsilon_0 \cdot 1}{d - \left(\frac{d}{3} - Vt\right)} = \frac{2 \varepsilon_0}{\frac{2d}{3} + Vt}
Time Constant Calculation
The time constant \( \tau \) in an RC circuit is defined as:
\( \tau = R \cdot C \)
Where \( R \) is the resistance and \( C \) is the capacitance. In this case, we are interested in how the capacitance changes over time due to the decreasing height of the liquid. The time constant will depend on the rate of change of capacitance with respect to time.
Finding the Time Constant as a Function of Time
To find the time constant as a function of time, we need to differentiate the capacitance with respect to time:
\( \frac{dC}{dt} = -\frac{2 \varepsilon_0 V}{\left(\frac{2d}{3} + Vt\right)^2} \)
Now, substituting this into the expression for the time constant, we get:
\( \tau(t) = R \cdot C(t) \)
Since \( R \) is constant, the time constant will vary with the changing capacitance. Thus, we can express the time constant as:
\( \tau(t) = R \cdot \frac{2 \varepsilon_0}{\frac{2d}{3} + Vt} \)
Final Expression
In summary, the time constant as a function of time \( t \) for the parallel plate capacitor with a decreasing liquid level is:
\( \tau(t) = R \cdot \frac{2 \varepsilon_0}{\frac{2d}{3} + Vt} \)
This equation shows how the time constant decreases as the liquid level drops, affecting the overall behavior of the capacitor over time.
