To determine the time it takes for a [tex]$22-\mu F$[/tex] capacitor to charge when connected in series with a [tex]$90-k\Omega$[/tex] resistor, we need to use the concept of the time constant ([tex]$\tau$[/tex]) in an RC circuit. Here's a step-by-step solution:
1. Understand the Time Constant (τ):
The time constant [tex]$\tau$[/tex] of an RC circuit is a measure of how quickly the capacitor charges or discharges. It is given by:
[tex]\[
\tau = R \times C
\][/tex]
where [tex]\(R\)[/tex] is the resistance and [tex]\(C\)[/tex] is the capacitance.
2. Substitute the Given Values:
- Capacitance, [tex]\(C = 22 \mu F = 22 \times 10^{-6} F\)[/tex]
- Resistance, [tex]\(R = 90 k\Omega = 90 \times 10^3 \Omega\)[/tex]
Now, calculate the time constant:
[tex]\[
\tau = (90 \times 10^3 \Omega) \times (22 \times 10^{-6} F)
\][/tex]
[tex]\[
\tau = 1.98 \, \text{seconds}
\][/tex]
3. Determine the Charging Time:
The capacitor in an RC circuit does not charge instantaneously. It takes a certain amount of time to charge fully. A commonly used approximation is that it takes about 5 times the time constant ([tex]$5\tau$[/tex]) for the capacitor to charge to over 99% of its maximum voltage.
Therefore, the charging time:
[tex]\[
\text{Time to charge} = 5 \times \tau
\][/tex]
[tex]\[
\text{Time to charge} = 5 \times 1.98 \, \text{seconds}
\][/tex]
[tex]\[
\text{Time to charge} = 9.9 \, \text{seconds}
\][/tex]
Thus, it will take approximately 9.9 seconds for the [tex]$22-\mu F$[/tex] capacitor to charge when connected in series with a [tex]$90-k\Omega$[/tex] resistor.
