To determine the diameter of the 2 mm tungsten filament given its resistance, we need to follow these steps:
1. Understand the given parameters:
- Resistance (R): [tex]\( 0.05 \Omega \)[/tex]
- Length (L): [tex]\( 2 \)[/tex] mm (which is [tex]\( 2 \times 10^{-3} \)[/tex] meters)
- Resistivity ([tex]\(\rho\)[/tex]): [tex]\( 5.6 \times 10^{-8} \,\Omega \cdot \text{meters} \)[/tex]
2. Use the formula for resistance of a wire:
[tex]\[ R = \rho \frac{L}{A} \][/tex]
Here, [tex]\( A \)[/tex] is the cross-sectional area of the filament.
3. Rearrange the formula to solve for the cross-sectional area [tex]\( A \)[/tex]:
[tex]\[ A = \rho \frac{L}{R} \][/tex]
Plugging in the given values:
[tex]\[ A = 5.6 \times 10^{-8} \cdot \frac{2 \times 10^{-3}}{0.05} \][/tex]
[tex]\[ A = 5.6 \times 10^{-8} \cdot 40 \times 10^{-3} \][/tex]
[tex]\[ A = 2.24 \times 10^{-9} \, \text{m}^2 \][/tex]
4. Relate the cross-sectional area to the diameter:
The cross-sectional area of a circular wire is given by:
[tex]\[ A = \pi \left(\frac{d}{2}\right)^2 \][/tex]
Rearrange to solve for the diameter [tex]\( d \)[/tex]:
[tex]\[ d = 2 \sqrt{\frac{A}{\pi}} \][/tex]
Plugging in the value for [tex]\( A \)[/tex]:
[tex]\[ d = 2 \sqrt{\frac{2.24 \times 10^{-9}}{\pi}} \][/tex]
5. Calculate the diameter:
Carry out the calculation step-by-step:
[tex]\[ \frac{2.24 \times 10^{-9}}{\pi} \approx 7.13 \times 10^{-10} \][/tex]
[tex]\[ \sqrt{7.13 \times 10^{-10}} \approx 2.67 \times 10^{-5} \][/tex]
[tex]\[ d = 2 \times 2.67 \times 10^{-5} \approx 5.34 \times 10^{-5} \][/tex]
6. Conclusion:
The diameter of the tungsten filament is approximately [tex]\( 5.34 \times 10^{-5} \)[/tex] meters or [tex]\( 53.4 \)[/tex] micrometers.
