A cylindrical capacitor (Example 24–2) has Rₐ = 3.5 mm and R₆.= 0.50 mm. The two conductors have a potential difference of 625 V, with the inner conductor at the higher potential. Calculate the energy stored in a 1.0-m length of the capacitor.

Verified step by step guidance

1. Understand the problem: A cylindrical capacitor consists of two concentric cylindrical conductors. The inner conductor has radius R₆ = 0.50 mm, and the outer conductor has radius Rₐ = 3.5 mm. The potential difference between the conductors is 625 V. We are tasked with calculating the energy stored in a 1.0-m length of the capacitor.
2. Step 1: Write the formula for the capacitance per unit length of a cylindrical capacitor. The capacitance per unit length (C') is given by: $C\text{'}=\frac{2\pi \epsilon ₀}{ln\left(\frac{R}{a}\right)}$, where ε₀ is the permittivity of free space (8.85 × 10⁻¹² F/m), Rₐ is the radius of the outer conductor, and R₆ is the radius of the inner conductor.
3. Step 2: Substitute the given values into the formula for C'. Convert the radii to meters: Rₐ = 3.5 mm = 3.5 × 10⁻³ m and R₆ = 0.50 mm = 0.50 × 10⁻³ m. Then calculate the natural logarithm term: $ln\left(\frac{3.5\times {10}^{-}}{0.50\times {10}^{-}}\right)$.
4. Step 3: Use the formula for the energy stored in a capacitor. The energy stored (U) is given by: $U=\frac{1}{2}C\text{'}V{2}^{}\times l$, where V is the potential difference (625 V), l is the length of the capacitor (1.0 m), and C' is the capacitance per unit length calculated in Step 2.
5. Step 4: Substitute the values of C', V, and l into the energy formula. Perform the necessary calculations to find the energy stored in the capacitor. Ensure all units are consistent (e.g., meters, volts, farads).

Key Concepts

Capacitance

Capacitance is the ability of a system to store electric charge per unit voltage. For cylindrical capacitors, the capacitance can be calculated using the formula C = (2πε₀L) / ln(Rₐ/R₆), where ε₀ is the permittivity of free space, L is the length of the capacitor, and Rₐ and R₆ are the radii of the outer and inner conductors, respectively.

Energy Stored in a Capacitor

The energy (U) stored in a capacitor can be calculated using the formula U = 1/2 C V², where C is the capacitance and V is the potential difference across the capacitor. This relationship shows how energy is directly proportional to the square of the voltage and the capacitance, highlighting the importance of both factors in energy storage.

Potential Difference

Potential difference, or voltage, is the difference in electric potential between two points in an electric field. In this context, it is the voltage applied across the cylindrical capacitor, which influences both the charge stored and the energy contained within the capacitor. A higher potential difference results in greater energy storage.