An air-filled spherical capacitor is constructed

with inner and outer shell radii of 6.4 cm and
16.1 cm, respectively.
Calculate the capacitance of the device.
The value of the Coulomb constant is
8.98755 × 10
9 N·m^2/C^2
Answer in units of pF

a is inner , b is outer

a has +q
b has -q inside, +q outside

Voltage of a (integrate E to 0 at infinity
Va = q/(4 pi eo ra)

Voltage of b (same way with q)
Vb = q/(4 pi eo rb)

V = Va-Vb = (q/4 pi eo) (1/ra - 1/rb)
= (q/4 pi eo) (rb-ra)/(ra rb)

C = q/V = 4 pi eo ra rb / (rb-ra)

To calculate the capacitance of the air-filled spherical capacitor, we can use the formula:

C = (4πε₀r₁r₂) / (r₂ - r₁)

Where:
C - capacitance of the capacitor
ε₀ - Coulomb constant (8.98755 × 10⁹ N·m²/C²)
r₁ - inner shell radius (6.4 cm)
r₂ - outer shell radius (16.1 cm)
π - pi (approximately 3.14159)

Let's substitute the values into the formula and calculate the capacitance:

C = (4πε₀r₁r₂) / (r₂ - r₁)
= (4π * 8.98755 × 10⁹ * 6.4 * 16.1) / (16.1 - 6.4)
= (4 * 3.14159 * 8.98755 × 10⁹ * 6.4 * 16.1) / 9.7

Calculating this expression gives us the capacitance of the device.