A comon flashlight bulb is rated 0.30A and 2.7V (the values of the current and voltage under operating conditions.) If the resistance of the tungsten bulb filament at room temperature `20^(@)C` is `1.0Omega` and its temperature coefficient of resistivity is `4.0 xx 10^(-3)C^(-1)`, then find the temperature in centigrade of the filament when the bulb is on. (Consider the variationi of resistance to be linear with temperature.)

Answer

To find the temperature of the filament when the bulb is on, we will follow these steps:

### Step 1: Calculate the resistance of the bulb when it is on.
According to Ohm's Law, the resistance \( R \) can be calculated using the formula:
\[ R = \frac{V}{I} \]
Where:
- \( V = 2.7 \, \text{V} \) (voltage)
- \( I = 0.30 \, \text{A} \) (current)

Substituting the values:
\[ R = \frac{2.7 \, \text{V}}{0.30 \, \text{A}} = 9 \, \Omega \]

### Step 2: Use the resistance-temperature relationship.
The resistance of the filament at a temperature \( T \) can be expressed as:
\[ R_T = R_0 (1 + \alpha (T - T_0)) \]
Where:
- \( R_T = 9 \, \Omega \) (resistance at operating temperature)
- \( R_0 = 1 \, \Omega \) (resistance at room temperature, \( T_0 = 20^\circ C \))
- \( \alpha = 4.0 \times 10^{-3} \, \text{C}^{-1} \) (temperature coefficient of resistivity)

### Step 3: Substitute known values into the equation.
Substituting the known values into the equation:
\[ 9 = 1 (1 + 4.0 \times 10^{-3} (T - 20)) \]

### Step 4: Simplify the equation.
This simplifies to:
\[ 9 = 1 + 4.0 \times 10^{-3} (T - 20) \]
Subtracting 1 from both sides gives:
\[ 8 = 4.0 \times 10^{-3} (T - 20) \]

### Step 5: Solve for \( T - 20 \).
Dividing both sides by \( 4.0 \times 10^{-3} \):
\[ T - 20 = \frac{8}{4.0 \times 10^{-3}} = 2000 \]

### Step 6: Solve for \( T \).
Adding 20 to both sides gives:
\[ T = 2000 + 20 = 2020^\circ C \]

### Final Answer:
The temperature of the filament when the bulb is on is \( 2020^\circ C \).

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