Question
How many time constants will elapse before the current in a charging RC circuit drops to half of its initial value? Answer the same question for a discharging RC circuit.
✓
Answer
$\frac{\text{Q}}{2}=\text{Q}\Big(1-\text{e}^{\frac{-\text{t}}{\text{CR}}}\Big)$
$\Rightarrow\frac{1}{2}=\Big(1-\text{e}^{\frac{-\text{t}}{\text{CR}}}\Big)$
$\Rightarrow\text{e}^{\frac{-\text{t}}{\text{CR}}}=\frac{1}{2}$
$\Rightarrow\frac{\text{t}}{\text{RC}}=\log2\Rightarrow\text{n}=0.69.$
$\Rightarrow\frac{1}{2}=\Big(1-\text{e}^{\frac{-\text{t}}{\text{CR}}}\Big)$
$\Rightarrow\text{e}^{\frac{-\text{t}}{\text{CR}}}=\frac{1}{2}$
$\Rightarrow\frac{\text{t}}{\text{RC}}=\log2\Rightarrow\text{n}=0.69.$
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