The temperature of the filament after a minute (t) is approximately 803°C.
To solve for the temperature of the filament (t) after a minute, we can follow these steps:
**1. Relate initial current (I0) to resistance (R0):**
- I0 = 1.97 A (given)
- V = constant voltage (given)
- R0 = V / I0 ≈ 5.08 Ω (calculated using Ohm's Law)
**2. Consider the change in resistance due to temperature increase:**
- ΔR = α * R0 * ΔT (where α is the resistivity temperature coefficient and ΔT is the temperature change)
- We know R0 from step 1 and α is given (4.5 * 10^(-3) K^(-1)), but ΔT is unknown.
**3. Apply conservation of energy to relate current change to temperature increase:**
- Power dissipated by the filament at initial and final states:
- P0 = I0^2 * R0
- P1 = I1^2 * R1 (where I1 is the current after 1 minute, 0.64 A)
- Since the voltage is constant, the power input remains the same (P0 = P1).
- Substitute R1 = R0 + ΔR and solve for ΔT:
- ΔT ≈ (P0 - P1) / (α * R0^2)
**4. Calculate the final temperature (t):**
- t = T0 + ΔT (where T0 is the initial temperature, 23 °C)
**Calculation:**
1. P0 = (1.97 A)^2 * 5.08 Ω ≈ 19.68 W
2. P1 = (0.64 A)^2 * 5.08 Ω ≈ 2.05 W
3. ΔT ≈ (19.68 W - 2.05 W) / (4.5 * 10^(-3) K^(-1) * 5.08 Ω^2) ≈ 780 K
4. t = 23 °C + 780 K ≈ 803 °C
**Therefore, the temperature of the filament after a minute (t) is approximately 803°C.**