Question

Two conducting wires of the same material are to have the same resistance. One wire is 17.0 m long and 0.44 mm in diameter. If the other wire is 0.30 mm in diameter, how long should it be?

Answer 1

Answer:

L = 7.90  m

Explanation:

• The general expression for the resistance of a resistor is as follows:

        $R = \frac{\rho*L}{A}$

• When we are talking about a wire, of a given diameter, the above formula becomes as follows:

       $R = \frac{\rho*L}{\frac{\pi*d^{2}}{4}}$

• where ρ = resistivity of the material, L= length of the wire and d= diameter of the wire.
• If both wires are from the same material, and have the same resistance, we can say the following, simplifying common terms:

       $R_{1} = R_{2} = \frac{L_{1} }{d_{1}^{2} } =\frac{L_{2} }{d_{2}^{2} }$

• where L₁ = 17.0 m, d₁= 0.44 mm = 4.4*10⁻⁴ m, d₂= 3*10⁻⁴ m.
• Replacing these values, we can solve for L₂, as follows:

       $L_{2} = \frac{L_{1} *d_{2} ^{2}}{d_{1}^{2}} = \frac{17m*(3e-4m)^{2} }{(4.4e-4m)^{2}} \ = 7.90 m$

• The wire should have 7.90 m length.