A resistor is formed in the shape of a hollow ,quarter cylinder from a material of resistivity p.The length of cylinder is L,inner and outer radii are R1 and R2 respectively .the resistance between the end faces will be

To find the resistance of a hollow quarter-cylinder formed by a resistor, we can break down the problem step by step, focusing on the geometric properties and the resistivity of the material. The resistance between the end faces of the quarter-cylinder can be derived using the principles of electrical resistance and geometry.

Understanding the Geometry

The hollow quarter-cylinder has an inner radius (R1) and an outer radius (R2), with a length (L) along the axis. This shape is crucial for calculating the resistance, as the resistance formula is influenced by the cross-sectional area through which the current flows.

Resistance Formula

The resistance (R) of a material can be expressed using the formula:

• R = ρ * (L / A)

Where:

• R is the resistance.
• ρ (rho) is the resistivity of the material.
• L is the length of the conductor.
• A is the cross-sectional area through which the current flows.

Calculating Cross-Sectional Area

For a hollow quarter-cylinder, the cross-sectional area (A) can be determined by the area of the outer circle minus the area of the inner circle. The formula for the area of a circle is πr², so for our quarter-cylinder, we have:

• A_outer = (1/4) * π * R2²
• A_inner = (1/4) * π * R1²

Thus, the effective cross-sectional area through which the current flows is:

• A = A_outer - A_inner = (1/4) * π * (R2² - R1²)

Substituting Values into the Resistance Formula

Now that we have the cross-sectional area, we can substitute it into the resistance formula:

• R = ρ * (L / A)

Substituting for A, we get:

• R = ρ * (L / ((1/4) * π * (R2² - R1²)))

This simplifies to:

• R = (4ρL) / (π * (R2² - R1²))

Final Expression

Therefore, the resistance between the end faces of the hollow quarter-cylinder is given by:

• R = (4ρL) / (π * (R2² - R1²))

This formula allows you to calculate the resistance using the known values of resistivity (ρ), length (L), and the inner and outer radii (R1 and R2). It's a practical application of basic electrical principles combined with geometry.