Two capacitors are identical, except that one is empty and the other is filled with a dielectric (k = 4.5). The empty capacitor is connected to a 15 -V battery. What must be the potential difference across the plates of the capacitor filled with a dielectric so that it stores the same amount of electrical energy as the empty capacitor?
energy=1/2 C v^2
so v=sqrt(1/k)
check that.
To determine the potential difference across the plates of the capacitor filled with a dielectric, we need to use the concept of capacitance.
The capacitance (C) of a capacitor can be calculated using the formula:
C = (k * ε₀ * A) / d
Where:
- k is the relative permittivity of the dielectric material (given as 4.5).
- ε₀ is the vacuum permittivity (approximately 8.85 x 10^-12 F/m).
- A is the area of the capacitor plates.
- d is the distance between the capacitor plates.
Since the two capacitors are identical, their area (A) and distance (d) are the same.
Now, let's consider the electrical energy stored in a capacitor. The energy (U) stored in a capacitor can be calculated using the formula:
U = (1/2) * C * V²
Where:
- U is the electrical energy stored in the capacitor.
- C is the capacitance.
- V is the potential difference across the capacitor plates.
Given that the empty capacitor is connected to a 15-V battery, its potential difference (V₁) is 15 V.
To find the potential difference (V₂) across the plates of the capacitor filled with a dielectric that stores the same amount of electrical energy as the empty capacitor, we equate the energies:
(1/2) * C₁ * V₁² = (1/2) * C₂ * V₂²
Since the capacitors have the same area and distance, their capacitances are inversely proportional to the relative permittivity:
C₁ / C₂ = k₂ / k₁
Substituting the value of k₁ = 1 and k₂ = 4.5, we get:
C₁ / C₂ = 4.5 / 1
Simplifying further, we have:
C₁ = 4.5 * C₂
Substituting this expression in the energy equation, we get:
(1/2) * (4.5 * C₂) * V₁² = (1/2) * C₂ * V₂²
Canceling out the common factors, we have:
4.5 * V₁² = V₂²
Taking the square root of both sides, we get:
V₂ = √(4.5 * V₁²)
Substituting the given value of V₁ = 15V, we can calculate V₂ as:
V₂ = √(4.5 * 15²)
V₂ ≈ 28.87 V
Therefore, the potential difference across the plates of the capacitor filled with a dielectric should be approximately 28.87 V to store the same amount of electrical energy as the empty capacitor.