Question

A cylindrical capacitor is filled with two cylindrical layers of dielectric with permittivities E₁ and E₂ The inside radii of the layers are equal to R₁ and R₂ > R₁.

The maximum permissible values of electric field strength are equal to E₁ₘ and E₂ₘ for these dielectrics. At what relationship between
E,R and Eₘ will the voltage increase result in the field strength reaching the breakdown value for both dielectrics simultaneously?

Answer 1

The relationship between E, R, and Eₘ for reaching the breakdown value in a cylindrical capacitor with two dielectric layers can be expressed as V = (R₂ - R₁) * E₁ / E₂.

Step-by-step explanation:

To find the relationship between E, R, and Eₘ for the voltage increase to result in the field strength reaching the breakdown value for both dielectrics simultaneously, we need to consider the electric field between the cylinders in the cylindrical capacitor.

The electric field can be calculated using the formula E = V / d, where E is the electric field strength, V is the voltage, and d is the distance between the cylinders.

1. First, calculate the electric field for each dielectric using the maximum permissible field strengths: E₁ = E₁ₘ and E₂ = E₂ₘ.
2. Next, equate the electric fields for both dielectrics: E₁ = E₂.
3. Using the relationship E = V / d, we can rewrite the equation as V / d₁ = V / d₂, where d₁ and d₂ are the distances between the cylinders and the radii: d₁ = R₂ - R₁ and d₂ = R₂.
4. Solving for V, we get V = (d₁ / d₂) * d = (R₂ - R₁) * E₁ / E₂.

In conclusion, the relationship between E, R, and Eₘ for the voltage increase to result in the field strength reaching the breakdown value for both dielectrics simultaneously is given by V = (R₂ - R₁) * E₁ / E₂.