Two Wires, one of copper and other of magnin have equal lengths and equal resistances. Which is thinker?

To determine which wire is thicker between a copper wire and a manganin wire, given that they have equal lengths and equal resistances, we need to consider the properties of the materials involved, particularly their resistivities and how they relate to the dimensions of the wires.

Understanding Resistivity and Resistance

Resistance (\(R\)) of a wire can be expressed using the formula:

R = ρ (L/A)

Where:

• R = resistance
• ρ = resistivity of the material
• L = length of the wire
• A = cross-sectional area of the wire

Since both wires have equal lengths and resistances, we can set their resistance equations equal to each other:

ρ_Cu (L/A_Cu) = ρ_Mn (L/A_Mn)

Resistivity Values

Next, we need to look at the resistivity values of copper and manganin:

• Resistivity of copper (\(ρ_{Cu}\)) is approximately \(1.68 \times 10^{-8} \, \Omega \cdot m\).
• Resistivity of manganin (\(ρ_{Mn}\)) is about \(4.82 \times 10^{-7} \, \Omega \cdot m\).

Since manganin has a higher resistivity than copper, we can infer that for the resistances to be equal, the cross-sectional area of the manganin wire must be larger than that of the copper wire.

Cross-Sectional Area and Thickness

The cross-sectional area (\(A\)) of a wire is related to its thickness. For a circular wire, the area is given by:

A = π(d/2)²

Where \(d\) is the diameter of the wire. A larger cross-sectional area means a thicker wire. Since we established that \(A_{Mn} > A_{Cu}\) to maintain equal resistance, it follows that:

d_Mn > d_Cu

Conclusion

In summary, the manganin wire must be thicker than the copper wire to achieve the same resistance given their equal lengths. This is primarily due to the higher resistivity of manganin compared to copper, necessitating a larger cross-sectional area to balance the resistance equation. Thus, the thicker wire is the one made of manganin.