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 with the given dielectric, we can use the formula for the capacitance of a capacitor with a dielectric material:
C = (κ * ε₀ * A) / d
where C is the capacitance, κ is the dielectric constant, ε₀ is the electric constant (also known as the permittivity of free space), A is the area of the dielectric between the plates, and d is the separation between the plates.
In this case, since capacitor B is a thick-walled cylinder, the area A can be determined by subtracting the area of the inner cylinder from the area of the outer cylinder:
A = π * (R1^2 - R2^2)
We know the values of R1, R2, d, and κ, so we can substitute them into the formula to find the capacitance.
Let's calculate the capacitance step by step:
1. Calculate the area A:
A = π * (R1^2 - R2^2)
A = π * ((5.15 cm)^2 - (3.65 cm)^2)
2. Convert the units to meters to maintain consistency:
A = π * ((0.0515 m)^2 - (0.0365 m)^2)
3. Calculate the capacitance:
C = (κ * ε₀ * A) / d
C = (3.87 * 8.85 x 10^-12 F/m * π * ((0.0515 m)^2 - (0.0365 m)^2)) / 0.00395 m
Now you can calculate the capacitance of capacitor B by substituting the values into the equation and using a calculator.