Nine identical wires , each of diameter d and length L, are connected in parallel. The combination has the resistance as a single similar wire of length L but whose diameter is ______?

wire resistance is inversely proportional to cross sectional area (radius^2)

9* r^2 = (3r)^2

To find the equivalent resistance of the parallel combination of identical wires, we can use the formula:

1/Req = (1/R1) + (1/R2) + (1/R3) + ... + (1/Rn)

where Req is the equivalent resistance, R1, R2, R3, ..., Rn are the resistances of each individual wire.

Since all the wires are identical, their resistances are the same. Let's denote the resistance of each wire as R.

Therefore, the formula can be simplified as:

1/Req = n/R

Rearranging the equation:

Req = R/n

We are given that the length and diameter of each wire are the same. Let's denote the diameter as d and the length as L.

The resistance of a wire is given by the formula:

R = ρ * (L/A)

where ρ is the resistivity of the material, L is the length, and A is the cross-sectional area.

Since all the wires have the same diameter, their cross-sectional areas are the same.

The cross-sectional area of a wire can be calculated using the formula:

A = π * (d/2)^2

Now, we need to find the equivalent diameter of a wire that has the same resistance as the parallel combination.

Substituting the values into the formula:

Req = R/n

Req = ρ * (L/A) / n

The cross-sectional area of the equivalent wire can be calculated using the formula:

A_eq = π * (d_eq/2)^2

To find the diameter of the equivalent wire, we solve for d_eq:

A_eq = π * (d_eq/2)^2

A_eq = π * (d_eq^2 / 4)

d_eq^2 = 4 * (A_eq / π)

d_eq = √(4 * (A_eq / π))

Substituting the value of A_eq:

d_eq = √(4 * (π * (d/2)^2) / π)

Simplifying the equation:

d_eq = √(4 * (d/2)^2)

d_eq = √(4 * (d^2/4))

d_eq = √(d^2)

d_eq = d

Therefore, the diameter of the equivalent wire is the same as the diameter of the individual wires, which is d.

To find the equivalent resistance of a combination of parallel wires, we need to consider the formula for resistance in a wire, which is given by:

R = (ρL) / A

where R is the resistance, ρ 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 nine identical wires connected in parallel. Let's assume that the diameter of each wire is d.

1. Resistance of a single wire:
The resistance of a single wire can be calculated using the formula mentioned above. However, since we need to compare it to a similar wire but of a different diameter, we can represent it as:

R₁ = (ρL) / A₁

2. Resistance of the combination of wires:
Since the wires are connected in parallel, the total resistance of the combination (R_combination) can be found using the formula:

1 / R_combination = 1 / R₁ + 1 / R₁ + ... + 1 / R₁ (for each wire)

Substituting the values of resistivity (ρ), length (L), and cross-sectional area (A₁) into the equation, we get:

1 / R_combination = 1 / [(ρL) / A₁] + 1 / [(ρL) / A₁] + ... + 1 / [(ρL) / A₁]

Simplifying this equation, we get:

1 / R_combination = 1 / R₁ + 1 / R₁ + ... + 1 / R₁ (summing up for each wire)

1 / R_combination = 9 / R₁

Since the resistances of all the wires are the same (identical wires), we can write:

1 / R_combination = (9 * A₁) / (ρL)

Finally, rearranging the equation, we find:

R_combination = (ρL) / (9 * A₁)

Now, we need to find the equivalent resistance of a single similar wire of length L but with a different diameter.

Let's assume the diameter of the similar wire is D. We can calculate its cross-sectional area (A₂) using the formula:

A₂ = π * (D / 2)^2 = (π * D^2) / 4

Substituting this value into the previous equation, we get:

R_combination = (ρL) / (9 * A₁) = (ρL) / (9 * (π * d^2) / 4)

Simplifying further, we find:

R_combination = (4ρL) / (9 * π * d^2)

Therefore, the diameter of the similar wire would be d' such that:

d' = sqrt((4ρL) / (9 * π * R_combination))

You can substitute the known values for resistivity (ρ), length (L), and the resistance of the combination (R_combination) into this equation to find the diameter (d') of the similar wire.