a)A dielectric in the shape of a thick-walled cylinder of outer radius R1 = 5.15 cm, inner radius R2 = 3.65 cm, thickness d = 3.95 mm, and dielectric constant κ = 3.87 is placed between the plates, coaxial with the plates, as shown in the figure. Calculate the capacitance of capacitor B, with this dielectric.
To calculate the capacitance of capacitor B, we first need to determine the electric field between the plates of the capacitor.
1. Start by finding the electric field inside the dielectric material.
- The electric field inside a dielectric is given by the equation: E = V / d.
- In this case, the potential difference (V) between the plates is the same as the potential difference between the outer and inner radii of the dielectric.
- Since the electric field (E) is uniform between the plates, we can calculate it using the potential difference and the thickness of the dielectric.
2. Calculate the potential difference between the outer and inner radii of the dielectric.
- The potential difference (V) can be calculated using the equation: V = Ed.
- Here, E is the electric field and d is the thickness of the dielectric.
3. Calculate the electric field inside the dielectric.
- Use the equation E = V / d, where V is the potential difference and d is the thickness of the dielectric.
- Plug in the values obtained in step 2 to calculate the electric field E.
4. Calculate the electric field inside the capacitor.
- Due to the presence of the dielectric material, the electric field between the plates of the capacitor is reduced.
- The electric field inside the capacitor is related to the electric field inside the dielectric by the equation E_capacitor = E / κ, where κ is the dielectric constant.
- Plug in the values obtained in step 3 to calculate the electric field inside the capacitor.
5. Calculate the capacitance of capacitor B.
- The capacitance (C) of a capacitor is given by the equation C = ε * A / d, where ε is the permittivity of free space, A is the area of the capacitor plates, and d is the separation between the plates.
- The area of the capacitor plates can be calculated using the equation A = π * (R1^2 - R2^2).
- Plug in the values obtained in step 4 to calculate the capacitance C of capacitor B.