A 0.65-mm-diameter copper wire carries a tiny current of 3.2 μA. Estimate the electric field in the wire.

Verified step by step guidance

1. Determine the cross-sectional area of the wire. The diameter of the wire is given as 0.65 mm, so the radius is half of that. Use the formula for the area of a circle: \( A = \pi r^2 \), where \( r \) is the radius of the wire.
2. Convert the given current \( I = 3.2 \mu A \) into amperes by using the conversion \( 1 \mu A = 10^{-6} A \).
3. Use Ohm's Law, \( E = \rho J \), to find the electric field \( E \) in the wire. Here, \( \rho \) is the resistivity of copper (a known material property), and \( J \) is the current density. The resistivity of copper is approximately \( \rho = 1.68 \times 10^{-8} \ \Omega \cdot m \).
4. Calculate the current density \( J \) using the formula \( J = \frac{I}{A} \), where \( I \) is the current and \( A \) is the cross-sectional area of the wire.
5. Substitute the values of \( \rho \) and \( J \) into the formula \( E = \rho J \) to estimate the electric field in the wire.

Key Concepts

Electric Field

The electric field is a vector field that represents the force per unit charge experienced by a charged particle in a region of space. It is defined as the gradient of the electric potential and is measured in volts per meter (V/m). In conductive materials like copper, the electric field drives the flow of charge carriers, which constitutes an electric current.

Current Density

Current density is a measure of the electric current flowing per unit area of a cross-section of a conductor. It is expressed in amperes per square meter (A/m²) and is crucial for understanding how current distributes within a wire. The relationship between current density, electric field, and conductivity is given by J = σE, where J is the current density, σ is the conductivity, and E is the electric field.

Ohm's Law

Ohm's Law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance of the conductor. Mathematically, it is expressed as V = IR, where V is voltage, I is current, and R is resistance. This law is fundamental in calculating the electric field in a wire, as it relates the electric field to the current and resistance.