A cylindrical capacitor has an inner conductor of radius 1.5 mm and an outer conductor of radius

step 1 Hi, everybody. So we need find what is the capsence per length here? And we have C over an induc

A cylindrical capacitor has an inner conductor of radius 1.5...

A cylindrical capacitor has an inner conductor of radius 1.5 $\mathrm{mm}$ and an outer conductor of radius 3.5 $\mathrm{mm}$ . The two conductors are separated by vacuum, and the entire capacitor is 2.8 $\mathrm{m}$ long. (a) What is the capacitance per unit length? (b) The potential of the inner conductor is 350 $\mathrm{mV}$ higher than that of the outer conductor. Find the charge (magnitude and sign) on both conductors.

Key Concepts

Relationship between Charge, Capacitance, and Voltage

The relationship Q = CV is central to understanding capacitors, where Q is the charge stored, C is the capacitance, and V is the potential difference between the conductors. This concept implies that for a given potential difference, the stored charge is directly proportional to the capacitance. This fundamental relation allows for the determination of either charge, capacitance, or voltage when the other two quantities are known.

Vacuum Permittivity

Vacuum permittivity, denoted as ??, is a fundamental physical constant that characterizes the ability of the vacuum to permit electric field lines. It appears in many electromagnetic formulas, including the capacitance of capacitors, and is essential for understanding how electric fields interact with different media. Its value determines the scaling between electric field and potential in free space.

Cylindrical Capacitor Geometry

A cylindrical capacitor consists of two coaxial cylinders – an inner conductor and an outer conductor. The geometry of the capacitor defines how the electric field is distributed between the conductors and plays a crucial role in determining its capacitance. The formula for a cylindrical capacitor typically involves the radii of the inner and outer cylinders and includes a logarithmic function due to the radial symmetry of the electric field.

Capacitance per Unit Length

For a cylindrical capacitor, the capacitance per unit length is derived considering the geometry of the system and the characteristics of the dielectric medium. The expression C' = (2???) / ln(b/a) is used, where 'a' is the radius of the inner conductor, 'b' is the radius of the outer conductor, and ?? is the vacuum permittivity. This concept is critical in problems where the capacitor's length is significant, and the total capacitance is obtained by multiplying this value by the length of the capacitor.

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