A capacitor consists of a solid metal sphere of radius 3 cm placed at the center of a thin metal spherical shell of radius 12 cm. The space between is empty. What is the capacitance?
For the solution to this problem (and the formula to use for C), see
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capsph.html
You will need the value of epsilon-zero, which I presume you have learned.
Thank you so much for your help! I got the answer.
To find the capacitance of this system, we need to use the formula for the capacitance of a spherical capacitor:
C = 4πε₀(ab) / (b - a)
where C is the capacitance, ε₀ is the permittivity of free space, a is the radius of the inner sphere, and b is the radius of the outer shell.
In this case, we have a solid metal sphere with radius a = 3 cm at the center, and a thin metal spherical shell with radius b = 12 cm surrounding it. The space between them is empty, so there is no dielectric material present.
Plugging these values into the formula, we get:
C = 4πε₀(3 cm * 12 cm) / (12 cm - 3 cm)
Now, we need to determine the value of ε₀, which is the permittivity of free space. It has a value of approximately 8.85 x 10⁻¹² F/m.
Substituting this value and simplifying the expression, we have:
C = 4π * 8.85 x 10⁻¹² F/m * (3 cm * 12 cm) / (12 cm - 3 cm)
Next, we need to convert the centimeter measurements to meters:
C = 4π * 8.85 x 10⁻¹² F/m * (0.03 m * 0.12 m) / (0.12 m - 0.03 m)
Simplifying further:
C = 4π * 8.85 x 10⁻¹² F/m * 0.0036 m² / 0.09 m
C = (4π * 8.85 x 10⁻¹² F/m * 0.0036 m²) / 0.09 m
Finally, we can evaluate this expression:
C = 4 * 3.14 * 8.85 x 10⁻¹² F/m * 0.0036 m² / 0.09 m
C ≈ 4.98 x 10⁻¹³ F
Therefore, the capacitance of the system is approximately 4.98 x 10⁻¹³ Farads.
To find the capacitance of this system, we need to use the formula for the capacitance of a parallel plate capacitor:
C = (ε₀ * A) / d
Where C is the capacitance, ε₀ is the permittivity of free space, A is the area of one of the capacitor plates, and d is the distance between the plates.
In the case of a spherical capacitor, we can approximate it as a parallel plate capacitor with the plates being two concentric spheres. The inner sphere will act as one plate, and the outer shell will act as the other plate.
To find the capacitance, we need to calculate the area of one of the plates and the distance between them.
First, let's find the area of one of the plates:
A = 4π * r^2
Where A is the area of one of the plates, and r is the radius of the inner sphere.
Given that the radius of the inner sphere is 3 cm, we can calculate the area:
A = 4π * (0.03 m)^2
Next, we need to find the distance between the plates. In this case, the distance is simply the difference between the radii of the two spheres:
d = R₂ - R₁
Where d is the distance between the plates, R₂ is the radius of the outer sphere, and R₁ is the radius of the inner sphere.
Given that the radius of the outer shell is 12 cm and the radius of the inner sphere is 3 cm, we can calculate the distance:
d = (0.12 m) - (0.03 m)
Now, we have all the necessary parameters to calculate the capacitance:
C = (ε₀ * A) / d
Substitute the values into the equation:
C = (8.85 x 10^-12 F/m * 4π * (0.03 m)^2) / ((0.12 m) - (0.03 m))
After performing the calculation, the capacitance of the system should be obtained in Farads (F).