After two time constants, to what percentage of the supply voltage is a capacitor in an RC circuit charged?

After two time constants, to what percentage of the supply voltage is a capacitor in an RC circuit charged?

• A. 63.2%
• B. 86.5%
• C. 50%
• D. 100%

Answer: (B)

In an RC (resistor-capacitor) circuit, the charging of the capacitor is governed by an exponential function, characterized by the time constant (τ), which is the product of the resistance (R) and capacitance (C) in the circuit. After a period equal to one time constant, a capacitor will charge to about 63.2% of the supply voltage. As time progresses, the charging continues to approach the supply voltage, and at two time constants (2τ), the capacitor charges to about 86.5% of the supply voltage. This is derived from the formula for the voltage across the capacitor at any time (t), which is: \[ V(t) = V_{supply} \times (1 - e^{-\frac{t}{\tau}}) \] Where \( e \) is the base of the natural logarithm, and after substituting 2τ for t, the expression yields approximately 86.5% of the supply voltage. This percentage illustrates how the charging process slows down as the capacitor gets closer to full charge, following the asymptotic nature of exponential growth. Thus, choosing 86.5% is correct as it reflects the theoretical understanding of capacitor charging

In an RC (resistor-capacitor) circuit, the charging of the capacitor is governed by an exponential function, characterized by the time constant (τ), which is the product of the resistance (R) and capacitance (C) in the circuit. After a period equal to one time constant, a capacitor will charge to about 63.2% of the supply voltage.

As time progresses, the charging continues to approach the supply voltage, and at two time constants (2τ), the capacitor charges to about 86.5% of the supply voltage. This is derived from the formula for the voltage across the capacitor at any time (t), which is:

[ V(t) = V_{supply} \times (1 - e^{-\frac{t}{\tau}}) ]

Where ( e ) is the base of the natural logarithm, and after substituting 2τ for t, the expression yields approximately 86.5% of the supply voltage. This percentage illustrates how the charging process slows down as the capacitor gets closer to full charge, following the asymptotic nature of exponential growth.

Thus, choosing 86.5% is correct as it reflects the theoretical understanding of capacitor charging