A wire of uniform cross-section has a resistance of R Ω. What would be the resistance of a similar wire, made of the same material, but twice as long and of twice the diameter?

To calculate the resistance of a wire, we need to use the formula:

R = (ρ * L) / A

where:
R is the resistance of the wire,
ρ (rho) is the resistivity of the material,
L is the length of the wire, and
A is the cross-sectional area of the wire.

In this case, we have a wire of uniform cross-section with a resistance of R Ω. Let's assume the length of the wire is L and the diameter is d.

Now, we are given that the similar wire is twice as long and has twice the diameter. Therefore, the length of the new wire would be 2L, and the diameter would be 2d.

To find the resistance of the new wire, we need to calculate the new cross-sectional area (A') and the new length (L').

The cross-sectional area of a wire is given by the formula:

A = π * (d/2)^2

where π (pi) is a constant (approximately 3.1416).

Substituting the values, we have:

A = π * (d/2)^2
A' = π * [(2d)/2]^2
A' = π * d^2

Next, we need to find the new length:

L' = 2L

Now, let's substitute these values into the resistance formula:

R' = (ρ * L') / A'
R' = (ρ * (2L)) / (π * d^2)

However, we know that the material remains the same, so the resistivity (ρ) of the material is constant. Therefore, we can simplify the formula:

R' = 2 ( ρ * L ) / ( π * d^2 )
R' = 2R

So, the resistance of the new wire would be twice the resistance of the original wire.

To calculate the resistance of a wire, we can use 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 wire, and A is the cross-sectional area of the wire.

Given that the wire has a resistance of R Ω, let's assume that the resistivity of the material and the temperature remain constant.

Now, if we double the length of the wire and the diameter, we can determine the resistance of the new wire.

Let's break down the steps to find the resistance of the new wire:

Step 1: Determine the new length of the wire
Since the wire is twice as long as the original wire, the new length, L₂, would be 2L.

Step 2: Determine the new diameter of the wire
Since the wire has twice the diameter, the new diameter, d₂, would be 2d.

Step 3: Calculate the new cross-sectional area
The cross-sectional area is given by the formula A = π * (d/2)^2. Thus, the new cross-sectional area, A₂, can be calculated as A₂ = π * (d₂/2)^2.

Step 4: Calculate the resistance of the new wire
Using the formula R = (ρ * L₂) / A₂, we can substitute the values we obtained in Steps 1 and 3, respectively.

R₂ = (ρ * 2L) / A₂

R₂ = (ρ * 2L) / (π * (d₂/2)^2)

Now, substitute the value of d₂ in terms of d:

R₂ = (ρ * 2L) / (π * ((2d)/2)^2)

R₂ = (ρ * 2L) / (π * (d^2))

Finally, simplify the equation:

R₂ = (ρ * 2L) / (π * d^2)

So, the resistance of the wire that is twice as long and has twice the diameter will be (ρ * 2L) / (π * d^2) Ω.

R = pL/A

A = pi*r^2 = pi*(2r)^2 = pi*4r^2.

R2 = 2pL/(pi*4r^2) = pL/Pi*2r^2.