91Ó°ÊÓ

To calculate the current in a wire, you need to understand the relationship between current density and the wire's cross-sectional area. Current density, denoted as \( J \), represents the amount of current \( I \) flowing per unit area \( A \). The formula is given by:

• \( J = \frac{I}{A} \)

By rearranging this formula, you can solve for the current:

• \( I = J \times A \)

The cross-sectional area of the wire is essential here, and since the wire in our exercise is circular, you find the area using: \[ A = \pi \times \left(\frac{d}{2}\right)^2 \]where \( d \) is the diameter of the wire.
In our problem, the diameter is given as 1.02 mm, which needs to be converted to meters for standard unit consistency:

• \( 1.02\, \text{mm} = 1.02 \times 10^{-3} \text{ m} \)

Substitute the diameter into the area formula to get the cross-sectional area. Then multiply it by the current density \( J = 1.50 \times 10^6 \text{ A/m}^2 \) to calculate the current \( I \).
Understanding each step helps ensure that the calculated current is accurate based on the wire's characteristics.

Drift velocity is a measure of the average velocity that charge carriers, typically electrons, achieve due to an electric field. To find drift velocity \( v_d \), use the relationship between current, charge carrier density, and area:

• \( I = n \times A \times e \times v_d \)

Here:

• \( I \) is the current through the wire,
• \( n \) is the charge carrier density,
• \( e \) is the elementary charge (approximately \( 1.60 \times 10^{-19} \text{ C} \)), and
• \( v_d \) is what we're solving for, the drift velocity.

Re-arrange the formula to solve for drift velocity:

• \( v_d = \frac{I}{n \times A \times e} \)

Given our problem, assuming the charge carrier density \( n \) for copper is \( 8.5 \times 10^{28} \text{ electrons/m}^3 \), we first need to know \( I \) and \( A \) from the earlier calculations.
Plug these values into the equation to calculate \( v_d \). It's essential to note that the drift velocity is generally very small compared to the thermal velocities of the electrons, so don't be surprised by a small value.

Charge carrier density \( n \) refers to the number of charge carriers, such as electrons, per unit volume in a material. It significantly influences how current flows through a conductor. For most metals, charge carrier density is relatively high, which allows these materials to conduct electricity efficiently.
Cu material is commonly assumed as our conductor, with a typical charge carrier density of \( 8.5 \times 10^{28} \text{ electrons/m}^3 \). This high density means there are plenty of electrons available in a cubic meter for carrying charge, making copper an excellent conductor.Understanding charge carrier density helps clarify why certain materials conduct electricity better than others. A higher \( n \) usually equates to lower resistance and higher conductivity. By considering both \( n \) and the physical dimensions of the wire, when conducting experiments or solving problems, you can reliably estimate the current flow and drift velocity.