When a dielectric material is introduced between the plates of a charged parallel plate capacitor, it significantly affects the charge on the plates, the potential difference (p.d.) between them, and the energy stored in the capacitor. Let's break down how each of these parameters changes when a dielectric is inserted.
Charge on the Plates
The charge on the plates of a capacitor is influenced by the dielectric constant (κ) of the material inserted. The expression for the charge (Q) on the plates can be modified as follows:
Q = C × V
Where:
- C is the capacitance of the capacitor.
- V is the potential difference across the plates.
When a dielectric is introduced, the capacitance increases by a factor of κ:
C' = κ × C
Thus, the new charge becomes:
Q' = C' × V' = κ × C × V'
Since the potential difference decreases when a dielectric is added, we can express the new charge as:
Q' = κ × Q
Potential Difference Between the Plates
The potential difference (V) across the plates of a capacitor is inversely related to the capacitance. When a dielectric is inserted, the potential difference changes as follows:
V' = V / κ
This indicates that the potential difference decreases by a factor of κ when a dielectric is present, assuming the charge remains constant.
Stored Energy in the Capacitor
The energy (U) stored in a capacitor is given by the formula:
U = 1/2 × C × V²
With the introduction of a dielectric, the energy stored changes due to the increased capacitance and decreased potential difference:
U' = 1/2 × C' × V'²
Substituting the expressions for C' and V', we get:
U' = 1/2 × (κ × C) × (V/κ)² = 1/2 × (κ × C) × (V²/κ²) = (1/2) × (C × V²) / κ
This shows that the energy stored in the capacitor decreases by a factor of κ when a dielectric is introduced.
Summary of Variations
- Charge on the plates: Q' = κ × Q
- Potential difference: V' = V / κ
- Stored energy: U' = (1/2) × (C × V²) / κ
In essence, adding a dielectric increases the charge on the plates while decreasing both the potential difference and the energy stored in the capacitor. This interplay of factors is crucial for understanding how capacitors function in various electrical applications.
