Calculating the Resistivity of a Wire

Problem:

When 93.6 V is applied across a wire that is 9.6 m long and has a 0.23 mm radius, the magnitude of the current density is 1.6×10^4 A/m². Find the resistivity of the wire.

Final answer:

We determine the resistivity of the wire by first calculating the cross-sectional area and electric field, then using these in the formula for resistivity.

Explanation:

The student asked for the resistivity of a wire given its voltage, length, radius, and current density. To find this, we need to utilize the formula for resistivity (ρ), which is expressed as R = ρA/L, where R is the resistance, L is the length, and A is the cross-sectional area of the wire.

Given the definition of current density (J = I/A), we can identify that current density J = 1.6x10^4 A/m². We first calculate the cross-sectional area (A) of the wire using the formula, A = πr², where r = 0.23mm. The radius needs to be converted into meters so the units are consistent, giving us r = 0.23 x 10^-3 meters.

Now, using the formula for the electric field (E), which is E = V/L, where V is the voltage and L is the length of the wire, we can determine E. Rewriting Ohm's law (V = IR), we express this in terms of the electric field and current density, J = σE, where σ is the electrical conductivity. Then we calculate the wire's resistivity (ρ), which is reciprocal of electrical conductivity, ρ = 1/σ.