Air isn't a perfect electric insulator, but it has a very high resistivity. Dry air has a resistivity of approximately 3 x 10^13 Ù x m. A capacitor has square plates 10 cm on a side separated by 1.2 mm of dry air.
If the capacitor is charged to 250 V, what fraction of the charge will flow across the air gap in 1 minute? Make the approximation that the potential difference doesn't change as the charge flows.
Calculate C, the capacitance
Q = charge = C V
R = resistance = 3*10^13 *.0012/(.1*.1)
i =V/R
change in Q = i t = (V/R)*1*60
fraction = change in Q / Q
Can you clarify on this part:
Calculate C, the capacitance
Q = charge = C V
To determine the fraction of charge that flows across the air gap in 1 minute, we can use Ohm's Law to calculate the current flowing through the air gap. Then we can use the relationship between current and charge to find the fraction of charge that flows.
1. Calculate the resistance of the air gap:
- The resistivity of dry air is given as 3 x 10^13 Ω x m.
- The distance between the plates is given as 1.2 mm, which is 0.0012 m.
- The area of one plate is 0.1 m x 0.1 m = 0.01 m^2 (since the plates are square).
The resistance (R) of the air gap is then given by:
R = (resistivity x distance) / area
R = (3 x 10^13 Ω x m x 0.0012 m) / 0.01 m^2
R ≈ 3.6 x 10^11 Ω
2. Calculate the current (I) flowing through the air gap:
- The potential difference (V) across the capacitor is given as 250 V.
- We can use Ohm's Law: V = I x R, where R is the resistance.
- Rearranging the equation, we can solve for I: I = V / R
I = 250 V / 3.6 x 10^11 Ω
I ≈ 6.94 x 10^-10 A
3. Calculate the charge (Q) that flows through the air gap in 1 minute:
- The current is given in amperes, so we need to convert 1 minute to seconds (60 seconds).
- The charge flowing through a circuit can be calculated by multiplying the current by the time (Q = I x t).
Q = (6.94 x 10^-10 A) x (60 s)
Q ≈ 4.17 x 10^-8 C
4. Calculate the fraction of charge that flows across the air gap:
- The total charge on the capacitor is given as Q_total = C x V, where C is the capacitance.
- The fractional charge flow is given by: Q_fractional = Q / Q_total.
Since the capacitance is not given, we cannot calculate the exact fraction of charge that flows without this information.
Therefore, without knowing the capacitance, we cannot determine the specific fraction of charge that will flow across the air gap in 1 minute.
To determine the fraction of charge that will flow across the air gap in 1 minute, we need to calculate the current flowing through the air and then use time and current equation to find the charge.
1. Calculate the resistance of the air gap:
The resistivity of air is given as 3 x 10^13 Ù x m. Since the plates of the capacitor are square, and the separation between them is 1.2 mm (or 0.0012 m), the resistance of the air gap can be calculated using the formula:
R = resistivity x (length / area)
Here, the length is the separation, and the area is the square of the plate side length.
R = (3 x 10^13 Ù x m) x (0.0012 m / (0.1 m)^2)
R = 0.036 Ù
2. Calculate the current flowing through the air gap:
Using Ohm's Law, we can find the current flowing through the air gap given that the potential difference (voltage) across the capacitor is 250 V and the resistance of the air gap is 0.036 Ù.
I = V / R
I = 250 V / 0.036 Ù
I ≈ 6944 A (rounded to four significant figures)
3. Calculate the charge flowed across the air gap in 1 minute:
To find the charge, we can use the formula Q = I x t, where Q is the charge, I is the current, and t is the time.
Q = (6944 A) x (60 s)
Q ≈ 416,640 C
4. Calculate the fraction of charge:
The fraction of charge that flowed across the air gap can be calculated by dividing the charge across the air gap by the total charge on the capacitor.
Fraction of charge = (Charge across air gap) / (Total charge on capacitor)
Fraction of charge = 416,640 C / [ ( Capacitance ) x ( Voltage across capacitor ) ]
Since the area of the capacitor plates is (0.1 m)^2 and the separation is 0.0012 m, the capacitance can be calculated using the formula:
C = (ε₀ x A ) / (d)
where ε₀ is the permittivity of vacuum and is approximately 8.85 x 10^-12 F/m.
Fraction of charge = 416,640 C / [ ( (8.85 x 10^-12 F/m) x ( (0.1 m)^2 ) ) / (0.0012 m)]
Fraction of charge ≈ 0.002362 (rounded to six decimal places)
Therefore, approximately 0.002362 (or 0.2362%) of the charge will flow across the air gap in 1 minute.