If the magnitude of the electric field in air exceeds roughly 3 106 N/C, the air breaks down and a spark forms. For a two-disk capacitor of radius 49 cm with a gap of 1 mm, what is the maximum charge (plus and minus) that can be placed on the disks without a spark forming (which would permit charge to flow from one disk to the other)? The constant å0 = 8.85 10-12 C2/(N·m2).
To find the maximum charge that can be placed on the disks without a spark forming, we need to calculate the maximum electric field that air can withstand before it breaks down.
Given:
Radius of the disks (r) = 49 cm = 0.49 m
Gap between the disks (d) = 1 mm = 0.001 m
Electric constant (ε0) = 8.85 x 10^(-12) C^2/(N·m^2)
Formula to calculate the electric field between the disks of a parallel-plate capacitor:
E = σ / (ε0)
where E is the electric field, σ is the surface charge density, and ε0 is the electric constant.
To find the maximum charge that can be placed on the disks, we need to calculate the maximum surface charge density.
The formula for surface charge density on a disk is:
σ = Q / A
where σ is the surface charge density, Q is the charge, and A is the area of the disk.
The area of a disk is given by:
A = πr^2
Therefore, the electric field between the disks can be rewritten as:
E = Q / (ε0 * A)
Since we want to find the maximum charge, the surface charge density (σ) will be the highest we can have without a spark forming, which is determined by the maximum electric field before air breaks down.
Therefore, the maximum charge can be calculated by rearranging the formula:
Q = E * ε0 * A
Substituting the values, we have:
A = π * (0.49 m)^2
E = 3 x 10^6 N/C
Calculating the area:
A = 3.14 * (0.49 m)^2
A = 0.7535 m^2
Finally, we can calculate the maximum charge:
Q = (3 x 10^6 N/C) * (8.85 x 10^(-12) C^2/(N·m^2)) * (0.7535 m^2)
Calculating the maximum charge:
Q = 2.110 x 10^(-5) C
Therefore, the maximum charge that can be placed on the disks without a spark forming is approximately 2.110 x 10^(-5) C.
To find the maximum charge that can be placed on the disks without a spark forming, we need to calculate the maximum electric field that the air can withstand before it breaks down. Once we have the maximum electric field, we can use it to calculate the maximum charge.
Step 1: Calculate the maximum electric field:
The maximum electric field is given as 3*10^6 N/C.
Step 2: Use the formula for electric field between two parallel plates to find the maximum voltage:
The electric field between two parallel plates is given by:
E = V / d
where E is the electric field, V is the voltage, and d is the distance between the plates.
Rearranging the formula, we have:
V = E * d
Given:
E = 3*10^6 N/C
d = 1 mm = 0.001 m
Substituting the values into the formula, we find:
V = (3*10^6 N/C) * (0.001 m)
V = 3000 N·m/C = 3000 V
Step 3: Calculate the maximum charge that can be placed on the disks:
The formula for the capacitance of a parallel plate capacitor is given by:
C = (ε0 * A) / d
where C is the capacitance, ε0 is the vacuum permittivity (8.85*10^-12 C^2/(N·m^2)), A is the area of the plates, and d is the distance between the plates.
Given:
ε0 = 8.85*10^-12 C^2/(N·m^2)
A = π * (0.49 m)^2 (since the radius is given as 49 cm)
d = 0.001 m
Substituting the values into the formula, we find:
C = (8.85*10^-12 C^2/(N·m^2)) * (π * (0.49 m)^2) / (0.001 m)
C ≈ 0.678 * 10^-9 F
Now that we have the capacitance, we can calculate the maximum charge using the formula:
Q = C * V
where Q is the charge and V is the voltage.
Substituting the values into the formula, we find:
Q = (0.678 * 10^-9 F) * (3000 V)
Q ≈ 0.00203 C
Therefore, the maximum charge that can be placed on the disks without a spark forming is approximately 0.00203 C (plus and minus).