To determine the ratio of capacitance for the two configurations of a parallel plate capacitor, we first need to understand how capacitance is calculated. The capacitance \( C \) of a parallel plate capacitor is given by the formula:
Capacitance Formula
The formula for capacitance is:
C = \frac{\varepsilon_0 \cdot A}{d}
Where:
- C is the capacitance.
- \(\varepsilon_0\) is the permittivity of free space (a constant).
- A is the area of one of the plates.
- d is the distance between the plates.
Initial Configuration
In the initial setup, let’s denote the area of each plate as \( A \) and the distance between the plates as \( d \). The capacitance \( C_1 \) can be expressed as:
C_1 = \frac{\varepsilon_0 \cdot A}{d}
Modified Configuration
Now, according to the problem, the area of each plate is doubled, so the new area \( A' \) becomes:
A' = 2A
Additionally, the distance between the plates is halved, so the new distance \( d' \) is:
d' = \frac{d}{2}
Now, we can calculate the new capacitance \( C_2 \) using the modified values:
C_2 = \frac{\varepsilon_0 \cdot A'}{d'} = \frac{\varepsilon_0 \cdot (2A)}{\frac{d}{2}}
This simplifies to:
C_2 = \frac{\varepsilon_0 \cdot 2A \cdot 2}{d} = \frac{4 \varepsilon_0 \cdot A}{d}
Calculating the Ratio
Now, we can find the ratio of the capacitances \( \frac{C_2}{C_1} \):
\(\frac{C_2}{C_1} = \frac{\frac{4 \varepsilon_0 \cdot A}{d}}{\frac{\varepsilon_0 \cdot A}{d}} = \frac{4}{1} = 4\)
Final Thoughts
The ratio of the capacitance in the two cases is 4. This means that by doubling the area of the plates and halving the distance between them, the capacitance increases by a factor of four. This illustrates how both the area and the distance between the plates significantly affect the capacitance of a parallel plate capacitor.
