To tackle the question about Dumet wire and its role in lightbulb technology, we need to delve into the thermal expansion properties of the materials involved. The goal is to ensure that the wire expands at a rate similar to that of the glass, preventing any stress or breakage during temperature changes. Let's break this down step by step.

Understanding Thermal Expansion

When materials are heated, they tend to expand. This property is quantified by the coefficient of thermal expansion (CTE), which indicates how much a material will expand per degree of temperature increase. For Dumet wire, the core is made of nickel-steel (invar), while the outer sheath is copper. The key is to match the expansion of the wire with that of the glass used in lightbulbs.

Setting Up the Ratio

Let’s denote:

• r_n = radius of the nickel-steel core
• r_c = radius of the copper sheath

The ratio we want to find is r_n / r_c. To ensure that the wire expands similarly to the glass, we need to consider the thermal expansion coefficients of both materials:

• CTE of glass (approximately 3.2 x 10^-6 /°C)
• CTE of nickel-steel (invar, approximately 1.2 x 10^-6 /°C)
• CTE of copper (approximately 16 x 10^-6 /°C)

Calculating the Ratio

To find the ratio of the radii, we can use the relationship between the thermal expansions of the core and the sheath. The effective expansion of the wire should match that of the glass:

Let’s assume the expansion of the wire is a weighted average based on the volume fractions of the core and sheath:

Using the formula for linear expansion:

ΔL = L0 * α * ΔT

Where ΔL is the change in length, L0 is the original length, α is the coefficient of thermal expansion, and ΔT is the change in temperature. For the wire to match the glass, we can set up the equation:

(αn * Vn + αc * Vc) = αglass * (Vn + Vc)

Here, V_n and V_c are the volumes of the nickel-steel core and copper sheath, respectively. The volumes can be expressed in terms of their radii:

Vn = (4/3)πrn3,  Vc = (4/3)πrc3

Substituting these into the equation and simplifying will lead us to the desired ratio.

Typical Value for the Ratio

In practice, the ratio of the nickel-steel radius to the copper sheath radius is typically around 0.5. This means that the radius of the nickel-steel core is about half that of the copper sheath. This ratio is crucial because it ensures that the Dumet wire can effectively expand and contract with the glass without causing any mechanical failure.

Conclusion

In summary, the Dumet wire is designed with careful consideration of the thermal expansion properties of its materials. By maintaining a specific ratio between the nickel-steel core and the copper sheath, we can ensure that the wire performs reliably in lightbulb applications. This careful engineering allows for the longevity and efficiency of lightbulbs, making them a staple in our daily lives.