A parallel plate capacitor with plates of area 320 cm.^2 is charged to a potential difference V and is then disconnected from the voltage source. When the plates are moved 0.3 cm. farther apart, the voltage between the plates is increased by 152 V. What is the charge on the positive plate of the capacitor?
Having trouble setting up a ratio to solve for charge
To solve this problem, let's first establish the relationships involved. The capacitance of a parallel plate capacitor is given by the formula:
C = ε₀ * A / d
where C is the capacitance, ε₀ is the permittivity of free space (approximately 8.85 x 10^-12 F/m), A is the area of the plates, and d is the distance between the plates.
Now, let's denote the initial distance between the plates as d₁ and the final distance as d₂. We are given that the initial area of the plates (A) is 320 cm^2, which is equivalent to 320 x 10^-4 m^2.
From the problem, we learn that when the plates are moved 0.3 cm (which is 0.3 x 10^-2 m) farther apart, the voltage between the plates increases by 152 V.
Since the capacitance remains constant for a specific arrangement of plates and dielectric, the initial and final voltages (V₁ and V₂) are related by:
V₁ / V₂ = C₂ / C₁
where C₁ and C₂ are the initial and final capacitance values.
By rearranging this equation, we can write:
V₂ = (C₂ / C₁) * V₁
Now, substituting the respective formulas for capacitance and rearranging, we get:
V₂ = (ε₀ * A / d₂) / (ε₀ * A / d₁) * V₁
V₂ = (d₁ / d₂) * V₁
Given that the voltage increases by 152 V, we can write:
V₂ - V₁ = 152 V
Substituting the expression for V₂ from above, we have:
(d₁ / d₂) * V₁ - V₁ = 152 V
Simplifying, we get:
V₁ * (d₁ / d₂ - 1) = 152 V
Finally, solving for V₁, we have:
V₁ = 152 V / (d₁ / d₂ - 1)
Now that we have determined the initial voltage, we can use the formula for capacitance and the known values to find the charge (Q) on the positive plate:
Q = C₁ * V₁
Substituting the values, we get:
Q = (ε₀ * A / d₁) * V₁
Now, you can plug in the values for the constants (permittivity of free space, area of plates, initial and final distances) and solve for charge.