An air filled parallel plate capacitor has plates of area 0.0406 m^2 and seperation of 2.5*10^-6 m.
a)What is the capacitence for the setup?
b)Our capacitor is connected to a 550 v battery, how much energy is stored in the capacitor?
c) If -1 mJ of work is performed by the capapcitor's plates but it is never disconnected from the 550 V battery, what is the new seperation of the plates?
a) The capacitance of a parallel plate capacitor can be calculated using the formula:
C = ε₀ * A / d
Where:
C = capacitance (in farads)
ε₀ = permittivity of free space (ε₀ = 8.85 x 10⁻¹² F/m)
A = area of the plates (in square meters)
d = separation between the plates (in meters)
Plugging in the given values:
A = 0.0406 m²
d = 2.5 x 10⁻⁶ m
C = (8.85 x 10⁻¹² F/m) * (0.0406 m²) / (2.5 x 10⁻⁶ m)
C = 0.1426 F
Therefore, the capacitance of the setup is 0.1426 Farads.
b) The energy stored in a capacitor can be calculated using the formula:
E = 0.5 * C * V²
Where:
E = energy stored in the capacitor (in joules)
C = capacitance (in farads)
V = voltage across the capacitor (in volts)
Plugging in the given values:
C = 0.1426 F
V = 550 V
E = 0.5 * (0.1426 F) * (550 V)²
E = 20.959 J
Therefore, the energy stored in the capacitor is 20.959 joules.
c) To find the new separation of the plates, we can use the work-energy principle. The work performed by the capacitor's plates can be calculated using the formula:
W = 0.5 * C * (V_f² - V_i²)
Where:
W = work performed by the capacitor (in joules)
C = capacitance (in farads)
V_f = final voltage across the capacitor (in volts)
V_i = initial voltage across the capacitor (in volts)
Plugging in the given values:
W = -1 mJ = -0.001 J
C = 0.1426 F
V_i = 550 V
-0.001 J = 0.5 * (0.1426 F) * (V_f² - (550 V)²)
Now, let's rearrange the equation to solve for V_f:
V_f² - (550 V)² = (2 * -0.001 J) / (0.1426 F)
V_f² = ((2 * -0.001 J) / (0.1426 F)) + (550 V)²
V_f² = -0.013999 + (550 V)²
V_f = √(-0.013999 + (550 V)²)
Once you find the value of V_f, you can use it to calculate the new separation (d') using the capacitance formula:
d' = ε₀ * A / C
Where:
d' = new separation between the plates (in meters)
A = area of the plates (in square meters)
C = capacitance (in farads)
Plug in the values you've given (A = 0.0406 m² and C = 0.1426 F) to find the new separation (d').