A capacitor is made from two hollow, coaxial, iron cylinders, one inside the other. The inner cylinder is negatively charged and the outer is positively charged; the magnitude of the charge on each is $10.0$ pC. The inner cylinder has radius $0.50$ mm, the outer one has radius $5.00$ mm, and the length of each cylinder is $18.0$ cm.
(a) What is the capacitance?
(b) What applied potential difference is necessary to produce these charges on the cylinders?

Verified step by step guidance

1. Step 1: Understand the geometry of the capacitor. We have two coaxial cylinders, one inside the other, forming a cylindrical capacitor. The inner cylinder has a radius of 0.50 mm, and the outer cylinder has a radius of 5.00 mm. Both cylinders have a length of 18.0 cm.
2. Step 2: Use the formula for the capacitance of a cylindrical capacitor. The capacitance C is given by the formula: $C=\frac{2\pi \epsilon l}{\ln \left(\frac{b}{a}\right)}$, where $\epsilon$ is the permittivity of free space, $l$ is the length of the cylinders, $a$ is the radius of the inner cylinder, and $b$ is the radius of the outer cylinder.
3. Step 3: Convert all units to meters for consistency. The inner radius is 0.50 mm = 0.0005 m, the outer radius is 5.00 mm = 0.005 m, and the length is 18.0 cm = 0.18 m.
4. Step 4: Calculate the capacitance using the formula from Step 2. Substitute the values: $a$ = 0.0005 m, $b$ = 0.005 m, $l$ = 0.18 m, and $\epsilon$ = 8.85 x 10^-12 F/m.
5. Step 5: To find the applied potential difference, use the formula $V=\frac{Q}{C}$, where $Q$ is the charge on the cylinders (10.0 pC = 10.0 x 10^-12 C) and $C$ is the capacitance calculated in Step 4.

Key Concepts

Capacitance of a Coaxial Capacitor

Capacitance is a measure of a capacitor's ability to store charge per unit voltage. For coaxial cylinders, the capacitance can be calculated using the formula C = (2πε₀L) / ln(b/a), where ε₀ is the permittivity of free space, L is the length of the cylinders, and a and b are the radii of the inner and outer cylinders, respectively. This formula accounts for the cylindrical geometry and the electric field distribution between the cylinders.

Electric Potential Difference

The electric potential difference, or voltage, between two points is the work done per unit charge to move a charge between those points. For a coaxial capacitor, the potential difference V can be found using the relationship V = Q/C, where Q is the charge on the cylinders and C is the capacitance. This concept is crucial for determining the voltage required to achieve a specific charge distribution on the capacitor.

Permittivity of Free Space

Permittivity of free space, denoted as ε₀, is a fundamental physical constant that characterizes the ability of a vacuum to permit electric field lines. It appears in the equations for capacitance and electric fields, influencing how electric fields interact with materials. In the context of a coaxial capacitor, ε₀ is used to calculate the capacitance, reflecting how the electric field is distributed between the charged cylinders.