Understanding the Impact of Diameter Reduction on Copper Wire Resistance

How to Calculate the New Resistance

When a copper wire is stretched to reduce its diameter to half of its previous value, its resistance changes. To understand this, we can use the fundamental relationship of electrical resistance with resistivity, length, and cross-sectional area.

The resistance R of a wire is given by the formula:

R frac{rho L}{A}

R - Resistance (Ω) (rho) - Resistivity of the material, constant for copper (Ω·m) (L) - Length of the wire (m) (A) - Cross-sectional area of the wire ((m^2))

Step 1: Calculate the Original Area

The cross-sectional area (A) of a wire with diameter (d) is given by:

A frac{pi d^2}{4}

Step 2: Calculate the New Area After Diameter Reduction

If the diameter is reduced to half, the new diameter (d) is:

d' frac{d}{2}

The new cross-sectional area (A') becomes:

A' frac{pi d'^2}{4} frac{pi left(frac{d}{2}right)^2}{4} frac{pi d^2}{16}

Step 3: Relate the New Area to the Original Area

The original area (A) is:

A frac{pi d^2}{4}

Now we can express the new area in terms of the original area:

A' frac{A}{4}

Step 4: Consider the Length Change

When the wire is stretched, its length increases. The volume of the wire remains constant during stretching, so the relationship between areas and lengths before and after stretching can be expressed as:

A L A' L'

Given (A' frac{A}{4}), we can substitute:

A L frac{A}{4} L'

This simplifies to:

L' 4L

Step 5: Calculate the New Resistance

Now we can find the new resistance (R'):

R' frac{rho L'}{A'} frac{rho (4L)}{frac{A}{4}} frac{4 rho L}{frac{A}{4}} frac{4 times 4 rho L}{A} 16 frac{rho L}{A} 16 R

Conclusion

Thus, the new resistance (R') after the wire's diameter is reduced to half is:

R' 16 R

This means the resistance increases by a factor of 16.

To summarize, the relationship between resistance, resistivity, length, and cross-sectional area is crucial in determining how changes in wire diameter affect its electrical resistance. Reducing the diameter increases the resistance significantly.

Keywords: wire resistance, resistivity, cross-sectional area