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When we think of a wire, its cross-sectional area is one of the most important factors determining how much electric current it can handle. Cross-sectional area refers to the size of the face you would see if you cut the wire perpendicular to its length. For a circular wire, this area is essential in determining current capacity.

The formula to calculate the cross-sectional area of a circular wire is \[ A = \frac{\pi}{4} d^2 \]where \( A \) is the area and \( d \) is the diameter. In simple terms, this means the larger the diameter, the larger the cross-sectional area, which in turn means more room for current to flow.

• For instance, a diameter increase from 0.4 mm to 0.6 mm greatly increases the cross-sectional area because the area grows with the square of the diameter.
• This increased area allows the wire to carry more current without overheating, as the resistance to current flow is reduced.

Current density is a measure that describes how much electric current flows through a unit area of cross-section within the wire. It's an important concept because it helps us understand how electricity is distributed over an area and it affects how much current can safely pass through a wire.

We express current density \( J \) in amperes per square meter (A/m²) using the formula:
\[ J = \frac{I}{A} \]where \( I \) is the current flowing through the wire, and \( A \) is the cross-sectional area. This formula tells us how concentrated the current is over a particular area.

• In a small-diameter wire, the current density can become very high because a large current has less area over which it can spread, leading to a potential risk of overheating.
• By increasing the diameter of the wire (thus increasing the cross-sectional area), we spread the same current over a larger area, reducing the current density and making the wire safer for higher currents.

Calculating the electric current involves understanding the relationship between current and cross-sectional area, particularly in the context of fuse wires which are designed to protect electrical circuits by breaking when the current exceeds a safe level.

The exercise demonstrates how increasing the diameter (and hence, cross-sectional area) allows a wire to conduct more current. Let's calculate using our knowledge from earlier:

• The original wire with a diameter of 0.4 mm has a cross-sectional area of \(0.04\pi \).
• The new wire with a diameter of 0.6 mm has a cross-sectional area of \(0.09\pi \).

The ratio of their areas \( \left( \frac{0.09\pi}{0.04\pi} \right) = \left( \frac{3}{2} \right)^2 \) indicates that the new wire can handle a proportionately larger current.

Thus, if the original wire carried 3 A, the new wire can carry:
\[ I_2 = I_1 \times \left( \frac{3}{2} \right)^2 = 3 \times \frac{9}{4} \]Simplifying gives \( \frac{27}{4} = 6.75 \) A, meaning the larger wire can safely carry more current without overheating. This illustrates how adjusting the dimensions of wire affects its capacity.