If the magnitude of the electric field in air exceeds roughly 3 X10^6 N/C, the air breaks down and a spark forms. For a two-disk capacitor of radius 52 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 X10^-12 C2/(N·m2).
Calculate the capacitance using the diameter and gap. Assume the gap is filled with air with the breakdown field strength you were given.
Plate area A = 0.849 m^2
Gap d = 0.001 m
C = epsilon0*A/d = 8.85*10^-12*0.849/0.001 = 7.51*10^-13 farads
Maximum allowed E = 3*10^6 V/m
Maximum allowed voltage:
Vmax = E*d = 3000 V
Max charge Q = C*Vmax = 2.25*10^-9 C
E is approximately (Q/A)/e(o)
so rearranging the formula we get
AEe(o)=Q
hence Q= 2.26e-5
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 between the disks can withstand. We can then use this electric field to determine the maximum charge.
Step 1: Calculate the electric field between the disks.
The electric field between the disks of a capacitor is given by the formula:
E = V/d
where E is the electric field, V is the voltage across the capacitor, and d is the distance between the disks.
In this case, the distance between the disks (d) is given as 1 mm, which is equal to 0.001 m.
Step 2: Calculate the voltage across the capacitor.
The voltage across the capacitor can be calculated using the formula:
V = Ed
where E is the electric field and d is the distance between the disks.
Given the maximum electric field of 3 X 10^6 N/C and the distance of 0.001 m, we can calculate the voltage:
V = (3 X 10^6 N/C) * (0.001 m) = 3000 V
Step 3: Calculate the maximum charge.
The charge (Q) on a capacitor can be calculated using the formula:
Q = C * V
where Q is the charge, C is the capacitance, and V is the voltage across the capacitor.
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, A is the area of the plates, and d is the distance between the plates.
In this case, the radius of the disks (r) is given as 52 cm, which is equal to 0.52 m. Thus, the area (A) of each disk is given by:
A = π * r^2 = 3.14 * (0.52 m)^2 = 0.848 m^2
Now we can calculate the capacitance:
C = (8.85 X 10^-12 C^2/(Nm^2)) * (0.848 m^2) / (0.001 m) = 7.538 X 10^-9 F
Finally, we can calculate the maximum charge:
Q = (7.538 X 10^-9 F) * (3000 V) = 0.0226 C (approximately)
Therefore, the maximum charge (plus and minus) that can be placed on the disks without a spark forming is approximately 0.0226 C.
C = epsilon0*A/d = 8.85*10^-12*0.849/0.001 = 7.51*10^-13 farads
Your steps are correct but your answer is miscalculated.