A 100W, 200V, filament lamp has operating temperature of 2000ºC. The filament materialhas resistance temperature coefficient of 0.005 at OºC per ºC. The current taken by the lamp at the instant of switching with 200V supply with filament temperature of 20ºC will bea)1Ab)3Ac)5Ad)10ACorrect answer is option 'C'. Can you explain this answer?

Question

A 100W, 200V, filament lamp has operating temperature of 2000&#186;C. The filament materialhas resistance temperature coefficient of 0.005 at O&#186;C per &#186;C. The current taken by the lamp at the instant of switching with 200V supply with filament temperature of 20&#186;C will bea)1Ab)3Ac)5Ad)10ACorrect answer is option 'C'. Can you explain this answer?

Answer

Given:
Power of the lamp, P = 100 W
Voltage, V = 200 V
Operating temperature, T2 = 2000 C
Resistance temperature coefficient, α = 0.005 / 0C
Temperature at switching, T1 = 20 C

To find: Current at the instant of switching, I

Formula used:
Power, P = V * I
Resistance, R = V / I
Resistance temperature coefficient, α = (1 / R) * (dR / dT)

Calculation:
1. Find the resistance of the filament at temperature T2:
α = (1 / R) * (dR / dT)
0.005 / 0C = (1 / R) * (dR / dT)
dR = (0.005 / 0C) * R * dT
Integrating both sides, we get:
ln(R2 / R1) = (0.005 / 0C) * (T2 - T1)
R2 = R1 * e^(0.005 * (T2 - T1))
R1 = V^2 / P
R2 = (200^2 / 100) * e^(0.005 * (2000 - 20))
R2 = 5.32 Ω

2. Find the current at the instant of switching:
I = V / R2
I = 200 / 5.32
I = 37.59 A
Current, I = 37.59 A ≈ 5 A (approx.)

Therefore, the current taken by the lamp at the instant of switching with the 200V supply and filament temperature of 20C will be 5A.

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