Study Notes on Current Density Calculation for Wire 3

Finding Current Density of Wire 3

1. Definition of Current Density

• The current density (denoted as J) is defined as the total electric current (I) flowing through a unit cross-sectional area (A) of a material.

• Mathematically, it can be expressed as: $J = \frac{I}{A}$ where:

  • J is the current density in amperes per square meter (A/m²) or other relevant units.

  • I is the current in amperes (A).

  • A is the cross-sectional area in square meters (m²).

2. Calculating the Area of Wire 3

• The cross-section of a wire is typically circular. To find the area (A) of a circular wire:

  • Use the formula:
    $A = \pi r^2$

• However, it is essential to express this in terms of the diameter provided.

• Given:

  • Diameter of wire 3: 1.5 mm

  • The radius (r) is half of the diameter:
    $r = \frac{\text{Diameter}}{2} = \frac{1.5 \, \text{mm}}{2} = 0.75 \, \text{mm}$

• Convert radius from millimeters to meters for standard SI unit consistency:
  $r = 0.75 \, \text{mm} = 0.75 \times 10^{-3} \, \text{m}$

• Now apply the radius in the area formula:

  • Substituting the radius into the area formula gives:
    $A = \pi (0.75 \times 10^{-3})^2$

  • Solve for A:
    $A = \pi (0.75^2 \times 10^{-6}) \approx 1.7671 \times 10^{-6} \, \text{m}^2$

• To convert to square millimeters:

  • 1 m² = $(10^3)^2$ mm² = 1,000,000 mm²

  • Therefore:
    $A = 1.7671 \times 10^{-6} \, \text{m}^2 \times 10^6 \, \text{mm}^2/\text{m}^2 \approx 1.7671 \, \text{mm}^2$

3. Express Current Density in Amperes per Square Millimeter

• Now that we have the area, we can find the current density (J3) in amperes per square millimeter.

• If the total current (I) flowing through the wire is known, substitute it into the current density formula:

  • For example, if the total current (I3) in wire 3 is known, the equation becomes:
    $J3 = \frac{I3}{A}$

• Express the final answer to two significant figures in units of amperes per square millimeter (A/mm²).