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A 100 W, 200 V filament lamp has operating temperature of 2000ºC. The filament material has resistance temperature coefficient of 0.005 at OºC per ºC. The current taken by the lamp at the instant of switching with 200 V supply with filament temperature of 20ºC will be
- a)1A
- b)3A
- c)5A
- d)10A
Correct answer is option 'C'. Can you explain this answer?
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A 100 W, 200 V filament lamp has operating temperature of 2000ºC....
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A 100 W, 200 V filament lamp has operating temperature of 2000ºC....
Given:
- Power of the filament lamp, P = 100 W
- Voltage across the filament lamp, V = 200 V
- Operating temperature of the filament, T = 2000 °C
- Resistance temperature coefficient of the filament material, α = 0.005 Ω/°C
- Filament temperature at the instant of switching, T0 = 20 °C
To find:
The current taken by the lamp at the instant of switching with a 200 V supply.
Explanation:
1. The resistance of the filament lamp can be calculated using the formula:
R = V^2 / P
where R is the resistance.
2. Substituting the given values, we get:
R = (200 V)^2 / 100 W
= 40000 V^2 / 100 W
= 400 Ω
3. The resistance-temperature relationship can be given as:
R = R0(1 + α(T - T0))
where R0 is the resistance at temperature T0.
4. Rearranging the equation, we get:
(1 + α(T - T0)) = R / R0
5. At the instant of switching, the filament temperature is T0 = 20 °C and the resistance is R = 400 Ω.
6. Substituting the values, we get:
(1 + 0.005(T - 20)) = 400 / R0
7. Solving for R0, we get:
R0 = 400 / (1 + 0.005(T - 20))
= 400 / (1 + 0.005(2000 - 20))
= 400 / (1 + 0.005(1980))
= 400 / (1 + 9.9)
= 400 / 10.9
≈ 36.7 Ω
8. The current at the instant of switching can be calculated using Ohm's Law:
I = V / R0
= 200 V / 36.7 Ω
≈ 5.45 A
Conclusion:
The current taken by the lamp at the instant of switching with a 200 V supply and a filament temperature of 20 °C is approximately 5.45 A. Therefore, the correct answer is option (c) 5 A.
- Power of the filament lamp, P = 100 W
- Voltage across the filament lamp, V = 200 V
- Operating temperature of the filament, T = 2000 °C
- Resistance temperature coefficient of the filament material, α = 0.005 Ω/°C
- Filament temperature at the instant of switching, T0 = 20 °C
To find:
The current taken by the lamp at the instant of switching with a 200 V supply.
Explanation:
1. The resistance of the filament lamp can be calculated using the formula:
R = V^2 / P
where R is the resistance.
2. Substituting the given values, we get:
R = (200 V)^2 / 100 W
= 40000 V^2 / 100 W
= 400 Ω
3. The resistance-temperature relationship can be given as:
R = R0(1 + α(T - T0))
where R0 is the resistance at temperature T0.
4. Rearranging the equation, we get:
(1 + α(T - T0)) = R / R0
5. At the instant of switching, the filament temperature is T0 = 20 °C and the resistance is R = 400 Ω.
6. Substituting the values, we get:
(1 + 0.005(T - 20)) = 400 / R0
7. Solving for R0, we get:
R0 = 400 / (1 + 0.005(T - 20))
= 400 / (1 + 0.005(2000 - 20))
= 400 / (1 + 0.005(1980))
= 400 / (1 + 9.9)
= 400 / 10.9
≈ 36.7 Ω
8. The current at the instant of switching can be calculated using Ohm's Law:
I = V / R0
= 200 V / 36.7 Ω
≈ 5.45 A
Conclusion:
The current taken by the lamp at the instant of switching with a 200 V supply and a filament temperature of 20 °C is approximately 5.45 A. Therefore, the correct answer is option (c) 5 A.
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