Question

A wire of resistivity ρ must be replaced in a circuit by a wire of the same material but four times as long. If, however, the total resistance is to remain as before, the diameter of the new wire mustA- be four times the original diameter.B- be one-fourth the original diameter.C- be the same as the original diameter.D- be one-half the original diameter.E- be two times the original diameter.

Answer 1

Answer:

E. two times the original diameter

Explanation:

Resistance of a wire is:

R = ρ L/A

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

For a round wire with diameter d:

R = ρ L / (¼ π d²)

The two wires must have the same resistance, so:

ρ₁ L₁ / (¼ π d₁²) = ρ₂ L₂ / (¼ π d₂²)

The wires are made of the same material, so ρ₁ = ρ₂:

L₁ / (¼ π d₁²) = L₂ / (¼ π d₂²)

The new length is four times the old, so 4 L₁ = L₂:

L₁ / (¼ π d₁²) = 4 L₁ / (¼ π d₂²)

1 / (¼ π d₁²) = 4 / (¼ π d₂²)

Solving:

1 / (d₁²) = 4 / (d₂²)

(d₂²) / (d₁²) = 4

(d₂ / d₁)² = 4

d₂ / d₁ = 2

So the new wire must have a diameter twice as large as the old wire.