26. Total Heat Dissipated
Step 1: Why is this a Geometric Progression?
Concept: The current decays exponentially ($I \propto e^{-t}$). Power is proportional to $I^2$, so Power also decays exponentially ($P \propto e^{-2t}$).
A unique property of exponential decay is that for equal time intervals (here $\Delta t = 0.1s$), the energy (area under the curve) decreases by a constant ratio.
Therefore, $H_1, H_2, H_3…$ always form a G.P.
A unique property of exponential decay is that for equal time intervals (here $\Delta t = 0.1s$), the energy (area under the curve) decreases by a constant ratio.
Therefore, $H_1, H_2, H_3…$ always form a G.P.
Step 2: Identify the Series
We are given the heat dissipated in the first two intervals:
- Heat in interval 1 ($H_1$): $0.01$ J
- Heat in interval 2 ($H_2$): $0.006$ J
We find the Common Ratio ($r$) of this decay:
$$ r = \frac{H_2}{H_1} = \frac{0.006}{0.01} = 0.6 $$
Step 3: Sum of Infinite Series
The total heat dissipated until the current vanishes ($t \to \infty$) is the sum of the infinite Geometric Progression series ($S_{\infty} = \frac{a}{1-r}$).
$$ H_{\text{total}} = \frac{H_1}{1 – r} $$
$$ H_{\text{total}} = \frac{0.01}{1 – 0.6} = \frac{0.01}{0.4} $$
Total Heat = 0.025 J