To determine the maximum voltage of the spherical capacitor you've described, we need to consider both the dielectric material and the vacuum region between the electrodes. The maximum voltage is influenced by the dielectric strengths of both the dielectric material and the vacuum, as well as the geometry of the capacitor. Let's break this down step by step.
Understanding the Configuration
In your setup, we have a spherical capacitor with the following parameters:
- Inner electrode radius (a) = 1 cm
- Outer electrode inner radius (b) = 1.67 cm
- Dielectric thickness (d) = 0.5 cm
- Relative permittivity of the dielectric (Ɛ) = 5Ɛ₀
- Dielectric strength of vacuum (E_sv) = 30 kV/cm
- Dielectric strength of the dielectric (E_s) = 60 kV/cm
Calculating the Maximum Voltage
The maximum voltage that the capacitor can handle before breakdown occurs is determined by the weaker dielectric strength of the materials involved. In this case, we have two regions: the dielectric and the vacuum. The breakdown voltage for each region can be calculated using the formula:
Breakdown Voltage Formula
The breakdown voltage (U) can be calculated as:
U = E × d
Where:
- E is the dielectric strength (in kV/cm)
- d is the thickness of the dielectric or the distance across the vacuum (in cm)
Calculating for the Dielectric Region
For the dielectric region:
U_d = E_s × d = 60 kV/cm × 0.5 cm = 30 kV
Calculating for the Vacuum Region
For the vacuum region, the distance is the space between the outer radius of the dielectric and the inner radius of the outer electrode:
Distance in vacuum = b - (a + d) = 1.67 cm - (1 cm + 0.5 cm) = 0.17 cm
Now, we can calculate the breakdown voltage for the vacuum:
U_v = E_sv × distance in vacuum = 30 kV/cm × 0.17 cm = 5.1 kV
Determining the Maximum Voltage of the Capacitor
The maximum voltage the capacitor can withstand without experiencing breakdown is the lower of the two calculated breakdown voltages:
U_max = min(U_d, U_v) = min(30 kV, 5.1 kV) = 5.1 kV
Considering the Result
However, you mentioned a result of 9 kV. This discrepancy suggests that there may be additional factors or constraints in your specific scenario that were not accounted for in the basic calculations. It’s possible that the effective voltage rating of the capacitor is influenced by other design considerations or safety factors.
Final Thoughts
In summary, the maximum voltage of the capacitor is primarily determined by the dielectric strength of the materials involved and the geometry of the capacitor. In this case, the vacuum region limits the maximum voltage to 5.1 kV, which is below the dielectric strength of the dielectric material. If you have any further details or specific conditions that might affect this calculation, feel free to share!
