A 2.0-m-long, 1.0-mm-diameter wire has a variable resistivity given by where x is measured from one end of the wire. What is the current if this wire is connected to the terminals of a 9.0 V battery?

Verified step by step guidance

1. Understand the problem: The wire has a variable resistivity, meaning the resistivity changes along its length. To find the current, we need to calculate the total resistance of the wire and then use Ohm's Law, \( I = \frac{V}{R} \), where \( V \) is the voltage and \( R \) is the total resistance.
2. Express the resistance of a small segment of the wire: The resistance of a small segment \( dx \) of the wire is given by \( dR = \frac{\rho(x) dx}{A} \), where \( \rho(x) \) is the resistivity as a function of \( x \), \( A \) is the cross-sectional area of the wire, and \( dx \) is the infinitesimal length of the segment.
3. Integrate to find the total resistance: The total resistance \( R \) is the integral of \( dR \) over the length of the wire, \( R = \int_0^L \frac{\rho(x)}{A} dx \), where \( L \) is the total length of the wire. Substitute the expression for \( \rho(x) \) and calculate the integral.
4. Calculate the cross-sectional area: The cross-sectional area \( A \) of the wire is given by \( A = \pi r^2 \), where \( r \) is the radius of the wire. Since the diameter is given as 1.0 mm, the radius is \( r = 0.5 \times 10^{-3} \) m. Substitute this value into the formula for \( A \).
5. Use Ohm's Law to find the current: Once the total resistance \( R \) is determined from the integration, use Ohm's Law, \( I = \frac{V}{R} \), where \( V = 9.0 \) V, to calculate the current through the wire.

Key Concepts

Resistivity

Resistivity is a material property that quantifies how strongly a given material opposes the flow of electric current. It is denoted by the symbol ρ (rho) and is measured in ohm-meters (Ω·m). In this question, the resistivity varies along the length of the wire, which means that the resistance will also change depending on the position along the wire.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. It is mathematically expressed as I = V/R. This law is fundamental for analyzing circuits and will be essential for calculating the current in the wire when connected to the battery.

Resistance of a Wire

The resistance (R) of a wire can be calculated using the formula R = ρ(L/A), where ρ is the resistivity, L is the length of the wire, and A is the cross-sectional area. For a cylindrical wire, the cross-sectional area can be determined using the diameter. Since the resistivity varies along the length of the wire in this problem, the total resistance will need to be calculated by integrating the resistivity function over the length of the wire.