Question

A metal cylinder of radius 2 mm is concentric with another metal cylinder of radius 5 mm. If the space between the cylinders is filled with air, what is the capacitance per unit length of the arrangement?

a. 6.1×10 −¹¹
b. F/m 4.4×10 −¹¹
c. F/m 2.2×10−¹¹
d. F/m 3.3×10 −¹¹
d. F/m 1.2×10 −¹⁰

Answer 1

The capacitance per unit length of the arrangement formed by two concentric metal cylinders is approximately 2.2 x 10^-11 F/m, using the permittivity of free space and the radii of the cylinders.

Step-by-step explanation:

The question refers to the capacitance per unit length of the arrangement formed by two concentric cylinders with air in between. To calculate this, we can use the formula for the capacitance of a cylindrical capacitor:

C = 2πε0L / ln(b/a)

where

• C is the capacitance,
• ε0 is the permittivity of free space, 8.85 × 10-12 F/m,
• L is the length of the cylinders, which is considered per unit length for this example (so L = 1 m),
• ln denotes the natural logarithm,
• a is the radius of the inner cylinder (2 mm = 2 × 10-3 m), and
• b is the radius of the outer cylinder (5 mm = 5 × 10-3 m).

Plugging in the values, we get:

C = 2π×8.85 × 10-12 / ln(5 × 10-3 /2 × 10-3)

C ≈ 2π×8.85 × 10-12 / ln(2.5)

C ≈ 2.2 × 10-11 F/m

Therefore, the capacitance per unit length of the cylindrical capacitor arrangement is approximately 2.2 × 10-11 F/m.