Question

A copper wire has radius \(r=0.0250 \mathrm{~cm},\) is \(3.00 \mathrm{~m}\) long, has resistivity \(\rho=1.72 \cdot 10^{-5} \Omega \mathrm{m},\) and carries a current of \(0.400 \mathrm{~A}\). The wire has a charge-carrier density of \(8.50 \cdot 10^{24}\) electrons/m \(^{3}\) a) What is the resistance, \(R,\) of the wire? b) What is the clectric potential difference, \(\Delta V\), across the wire? c) What is the electric field. F. in the wire?

Short answer

Answer: The resistance of the copper wire is 2.6223 Ω, the electric potential difference is 1.0489 V, and the electric field is 0.3496 V/m.

Step-by-step solution

Find the Resistance

To find the resistance, we will use the formula for resistance in terms of resistivity, length, and cross-sectional area of the wire: $$ R = \frac{\rho L}{A} $$ With the given resistivity \(\rho\), length L, and the cross-sectional area A, which can be calculated using the wire radius r as: $$ A = \pi r^2 $$ Substitute the given values into the formula and find the resistance.

Calculate the cross-sectional area

Calculate the cross-sectional area using the given radius: $$ A = \pi r^2 = \pi(0.0250 \times 10^{-2})^2 $$ $$ A = 1.9635 \times 10^{-6} m^2 $$

Calculate the resistance

Now, we can substitute the values of resistivity, length and cross-sectional area in the resistance formula: $$ R = \frac{1.72 \times 10^{-5} \Omega \mathrm{m} \times 3.00 \mathrm{~m}}{1.9635 \times 10^{-6} m^2} $$ $$ R = 2.6223 \Omega $$ The resistance of the wire is \(2.6223 \Omega\).

Find the electric potential difference

To find the electric potential difference, we will use Ohm's law, which relates the potential difference to the current and resistance of the wire: $$ \Delta V = IR $$ Substitute the given current and calculated resistance into the formula to find the potential difference.

Calculate the electric potential difference

Now, plug in the given current and calculated resistance values: $$ \Delta V = (0.400 A) (2.6223 \Omega) $$ $$ \Delta V = 1.0489 V $$ The electric potential difference across the wire is \(1.0489 V\).

Find the electric field

To find the electric field, we will use the formula that relates electric field to potential difference and length of the wire: $$ E = \frac{\Delta V}{L} $$ Substitute the potential difference and length values into the formula to find the electric field.

Calculate the electric field

Now, substitute the calculated potential difference and given length values into the formula: $$ E = \frac{1.0489 V}{3.00 \mathrm{~m}} $$ $$ E = 0.3496\, V/m $$ The electric field in the wire is \(0.3496\, V/m\).