(a) Find the capacitance of a parallel-plate capacitor made of two 5 cm×5 cm square plates separated by 1 mm.
(b) If 50 V is applied to this capacitor, how much energy is stored?
(a) C=εₒ•A/d=8.85•10^-12•25•10^-4/0.001=
=2.21•10^-11F,
(b) E=CU²/2=...
(a) To find the capacitance of a parallel-plate capacitor, we can use the formula:
C = ε₀ * (A / d)
Where:
C is the capacitance
ε₀ is the permittivity of free space (8.85 x 10⁻¹² F/m)
A is the area of one of the plates
d is the separation distance between the plates
Given:
A = (5 cm) * (5 cm) = 25 cm² = 0.0025 m²
d = 1 mm = 0.001 m
Plugging the values into the formula, we have:
C = (8.85 x 10⁻¹² F/m) * (0.0025 m² / 0.001 m)
C = 8.85 x 10⁻¹² F * 2.5
C = 22.125 x 10⁻¹² F
Therefore, the capacitance of the parallel-plate capacitor is approximately 22.125 picofarads (pF).
(b) The energy stored in a capacitor can be calculated using the formula:
E = 0.5 * C * V²
Where:
E is the energy stored in the capacitor
C is the capacitance
V is the voltage applied across the capacitor
Given:
C = 22.125 x 10⁻¹² F
V = 50 V
Plugging the values into the formula, we have:
E = 0.5 * (22.125 x 10⁻¹² F) * (50 V)²
E = 0.5 * (22.125 x 10⁻¹² F) * (2500 V²)
E = 27.65625 x 10⁻⁷ J
Therefore, the energy stored in the capacitor when 50 V is applied is approximately 27.65625 nanojoules (nJ).
To find the capacitance of a parallel-plate capacitor, we can use the formula:
C = ε₀ * (A / d),
where C is the capacitance, ε₀ is the permittivity of free space, A is the area of one of the plates, and d is the separation distance between the plates.
(a) To find the capacitance, we need to calculate the area and the separation distance:
- The area of one of the plates is A = (5 cm) * (5 cm).
- The separation distance between the plates is d = 1 mm.
Now, we can substitute these values into the formula:
C = ε₀ * (A / d).
The permittivity of free space, ε₀, is a constant value approximately equal to 8.854 x 10^-12 F/m.
So, substituting the given values, we have:
C = (8.854 x 10^-12 F/m) * [(5 cm) * (5 cm) / (1 mm)].
To solve the equation, we need to convert the centimeters to meters and the millimeters to meters:
1 cm = 0.01 m, and 1 mm = 0.001 m.
Therefore:
C = (8.854 x 10^-12 F/m) * [(0.05 m) * (0.05 m) / (0.001 m)].
Calculating this expression will give us the capacitance value in Farads (F).
(b) To calculate the energy stored in the capacitor when 50 V is applied, we can use the formula:
E = (1/2) * C * V^2,
where E is the energy stored, C is the capacitance, and V is the applied voltage.
Using the previously calculated capacitance value and the given voltage of 50 V, substitute these values into the formula:
E = (1/2) * C * (50 V)^2.
Calculating this expression will give us the energy stored in joules (J).