Chapter 4: Q6E (page 842)

You want to produce three 1.00-mm-diameter cylindrical wires,
each with a resistance of 1.00 Ω at room temperature. One wire is gold, one
is copper, and one is aluminum. Refer to Table 25.1 for the resistivity
values. (a) What will be the length of each wire? (b) Gold has a density of $1.93 \times 10^{- 4} \frac{kg}{m^{3}}$ .
What will be the mass of the gold wire? If you consider the current price of gold, is
this wire very expensive?

Short Answer

Expert verified

(a) The length of each wire is,

Length of copper wire – 45.66m

Length of gold wire – 32.19

Length of aluminum wire – 28.35m

(b)The mass of the gold wire is 0.488 kg and the wire is very expensive.

Step by step solution

Formula of resistance and resistivity of wires.

Consider the formula for the resistance as:

$R = ρ \frac{L}{A} L = \frac{RA}{ρ}$

Here, $ρ$ is the resistivity, L is the length and A is the area of cross section.

Consider the resistivity values- as:

Copper - $1.72 \times 10^{- 8} Ω . m$ .

Gold - $2.44 \times 10^{- 8} Ω . m$ .

Aluminum - $2.77 \times 10^{- 8} Ω . m$ .

Consider the formula for the density as:

$M = Vρ M = (AL) ρ$

Calculate the length of each wire.

(a)

Solve for the length of the copper wire.

$L = \frac{(1 Ω) (\frac{π}{4} {(1 mm)}^{2})}{(1.72 \times 10^{- 8}) Ω . m} = 45.66 m$

Solve for the length of the gold wire.

$L = \frac{(1 Ω) (\frac{π}{4} {(1 mm)}^{2})}{(2.44 \times 10^{- 8}) Ω . m} = 32.19 m$

Solve for the length of the aluminum wire.

$L = \frac{(1 Ω) (\frac{π}{4} {(1 mm)}^{2})}{(2.77 \times 10^{- 8}) Ω . m} = 28.35 m$

Calculate the mass of the gold wire.

(b)

The density of gold is given as $1.93 \times 10^{4} \frac{kg}{m^{3}}$

Therefore mass of gold wire is,

$M = (1.93 \times 10^{4} \frac{kg}{m^{3}}) (7.854 \times 10^{- 7} m^{2}) (32.19 m) = 0.488 kg$

Therefore, the mass of the gold wire is 0.488 kg and the wire is very expensive.

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Short Answer

Step by step solution

Formula of resistance and resistivity of wires.

Calculate the length of each wire.

Calculate the mass of the gold wire.

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