Question

The filament of a 100 W (120 V) light bulb is a tungsten wire 0.035 mm in diameter. At the filament's operating temperature, the resistivity is 5.0×10⁻⁷ Ω ⋅ m. What is the resistance of the filament?

1) 0.00035 Ω
2) 0.0035 Ω
3) 0.035 Ω
4) 0.35 Ω

Answer 1

The resistance of the filament in the light bulb can be calculated using the formula R = (resistivity * length) / (cross-sectional area). By substituting the given values for resistivity, diameter, and voltage, and solving for the length of the filament, we find that it should be approximately 2.8 cm long.

Step-by-step explanation:

The resistance of the filament can be calculated using the formula for the resistance of a wire: R = (resistivity * length) / (cross-sectional area).

First, we need to find the cross-sectional area of the filament. The diameter is given as 0.035 mm, so the radius is half of that, which is 0.0175 mm. Converting to meters, the radius is 1.75 x 10^(-5) m.

The cross-sectional area of a circle is given by the formula A = π * r^2, so the cross-sectional area of the filament is (3.14159)(1.75x10^-5)^2 = 9.62x10^-10 m^2.

Using the given resistivity of 5.0 x 10^(-7) Ω * m and the length of the filament, we can substitute into the resistance formula: R = (5.0 x 10^(-7) Ω * m)(length) / (9.62 x 10^(-10) m^2).

Given that the power is 100 W and the voltage is 120 V, we can use the formula P = V^2 / R to find the resistance: R = V^2 / P = (120^2) / 100 = 144 Ω.

Comparing the two expressions for resistance, we have (5.0 x 10^(-7) Ω * m)(length) / (9.62 x 10^(-10) m^2) = 144 Ω. Simplifying, we obtain length = (144 Ω)(9.62 x 10^(-10) m^2) / (5.0 x 10^(-7) Ω * m).

Calculating the numerical value, we find length ≈ 0.028 m = 2.8 cm.