Let's solve the problem step by step. ### **Step 1: Understanding the Problem** We have a structure made of two dielectric layers: - **Layer 1:** Thickness \( a_1 = 1 \) cm, dielectric constant \( \varepsilon_1 = 2 \), dielectric strength \( E_{d1} = 20 \) kV/cm. - **Layer 2:** Thickness \( a_2 = 2 \) cm, dielectric constant \( \varepsilon_2 = 1 \), dielectric strength \( E_{d2} = 5 \) kV/cm. - **Total applied voltage \( U = 50 \) kV.** We are to check if this structure can safely operate under this voltage or if breakdown will occur. --- ### **Step 2: Electric Field Distribution** For **series capacitors** (or dielectric layers in series), the electric displacement \( D \) is the same in both layers: \[ D = \varepsilon_0 \varepsilon_1 E_1 = \varepsilon_0 \varepsilon_2 E_2 \] \[ \Rightarrow \varepsilon_1 E_1 = \varepsilon_2 E_2 \] \[ \Rightarrow E_1 = \frac{\varepsilon_2}{\varepsilon_1} E_2 \] \[ \Rightarrow E_1 = \frac{1}{2} E_2 \] --- ### **Step 3: Voltage Across Each Layer** The total voltage is the sum of voltages across each layer: \[ U = U_1 + U_2 = E_1 a_1 + E_2 a_2 \] But since \( E_1 = \frac{1}{2}E_2 \): \[ U = \left(\frac{1}{2}E_2\right)a_1 + E_2 a_2 \] \[ U = \frac{a_1}{2}E_2 + a_2 E_2 \] \[ U = E_2 \left( \frac{a_1}{2} + a_2 \right) \] Substitute values: - \( a_1 = 1 \) cm - \( a_2 = 2 \) cm - \( U = 50 \) kV \[ 50 = E_2 \left( \frac{1}{2} + 2 \right) = E_2 \times 2.5 \] \[ E_2 = \frac{50}{2.5} = 20 \text{ kV/cm} \] Now, \( E_1 = \frac{1}{2}E_2 = \frac{1}{2} \times 20 = 10 \text{ kV/cm} \) --- ### **Step 4: Compare with Breakdown Strength** - **Layer 1:** \( E_1 = 10 \) kV/cm < \( E_{d1} = 20 \) kV/cm (Safe) - **Layer 2:** \( E_2 = 20 \) kV/cm > \( E_{d2} = 5 \) kV/cm (NOT SAFE) --- ### **Step 5: Conclusion** - **Layer 2** will **break down** because the electric field exceeds its dielectric strength. - **The material cannot safely operate at 50 kV.** --- ## **Final Answer** **This material cannot safely operate at 50 kV. Layer 2 will break down because the electric field (20 kV/cm) exceeds its dielectric strength (5 kV/cm).** --- **Summary Table:** | Layer | Max Field (kV/cm) | Actual Field (kV/cm) | Safe? | |---------|-------------------|----------------------|-------| | Layer 1 | 20 | 10 | Yes | | Layer 2 | 5 | 20 | No |