Question

What must be the diameter of a cylindrical 120-m long metal wire if its resistance is to be 6.0 ω? the resistivity of this metal is 1.68 × 10-8 ω ∙ m?

Answer 1

To find the diameter of a cylindrical wire with a given resistance and resistivity, we can use the formulas for resistance and cross-sectional area. Using these formulas, we can determine the diameter of the wire.

Step-by-step explanation:

To find the diameter of a cylindrical metal wire, we can use the formula for resistance:

R = (ρ * L) / A

where R is the resistance, ρ is the resistivity, L is the length of the wire, and A is the cross-sectional area of the wire. Rearranging the formula to solve for A:

A = (ρ * L) / R

Substituting the given values into the formula:

A = (1.68 × 10-8 ω ∙ m * 120 m) / 6.0 ω = 2.52 × 10-7 m^2

Finally, we can find the diameter of the wire using the formula for the area of a circle:

A = π * r^2

Substituting the value of A into the formula:

π * r^2 = 2.52 × 10-7 m^2

where r is the radius of the wire. Solving for r:

r^2 = (2.52 × 10-7 m^2) / π

r ≈ √(8.04 × 10-8) m ≈ 2.83 × 10-4 m

Finally, the diameter of the wire is twice the radius:

diameter ≈ 2 * (2.83 × 10-4 m) ≈ 5.67 × 10-4 m

Answer 2

The resistance of the cylindrical wire is
`R=(\rho l)/(A)`.

Here
`R` is the resistance,
`l` is the length of the wire and
`A` is the area of cross section. Since the wire is cylindrical
`A=(\pi d^2)/(4)`. Rearranging the above equation,

`image`

Here
`l=120, R=6, \rho=1.68(10^(-8))`.

Substituting numerical values,

`image`

Te diameter of the wire is
`0.6541 mm`