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

Verified step by step guidance

1. Determine the cross-sectional area of the copper wire. The wire is cylindrical, so the cross-sectional area can be calculated using the formula for the area of a circle: $A=\pi r{2}^2$, where $r$ is the radius of the wire. Convert the diameter (0.65 mm) to meters and divide by 2 to find the radius.
2. Substitute the radius into the formula for the area to calculate the cross-sectional area. Ensure that the units are consistent (meters for the radius).
3. Recall the formula for current density: $J=\frac{I}{A}$, where $J$ is the current density, $I$ is the current, and $A$ is the cross-sectional area.
4. Substitute the given current (3.2 μA, converted to amperes) and the calculated cross-sectional area into the formula for current density.
5. Simplify the expression to find the current density. Ensure that the units are consistent, and express the result in standard units (A/m²).

Key Concepts

Current Density

Current density is defined as the amount of electric current flowing per unit area of a conductor. It is represented by the symbol 'J' and is calculated using the formula J = I/A, where 'I' is the current in amperes and 'A' is the cross-sectional area in square meters. Understanding current density is crucial for analyzing how current distributes within a conductor.

Cross-Sectional Area

The cross-sectional area of a wire is the area of a slice taken perpendicular to its length. For a cylindrical wire, this area can be calculated using the formula A = π(d/2)², where 'd' is the diameter of the wire. This concept is essential for determining how much current flows through a given area, which directly affects the current density.

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. It is expressed as V = IR, where 'V' is voltage, 'I' is current, and 'R' is resistance. This law helps in understanding the relationship between current, voltage, and resistance in electrical circuits.