When a capacitor charges and discharges during each time constant, by what percentage does the voltage change toward the fully-charged state?

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Multiple Choice

When a capacitor charges and discharges during each time constant, by what percentage does the voltage change toward the fully-charged state?

When a capacitor charges through a resistor, the voltage across the capacitor increases according to an exponential function. Specifically, after one time constant (denoted as τ, or tau), the voltage across the capacitor reaches approximately 63.2% of the difference between its initial voltage (usually 0 volts for a fully discharged capacitor) and the final voltage it is charging towards (the supply voltage). This percentage comes from the mathematical relationship that describes charging and discharging in RC (resistor-capacitor) circuits. The formula for the voltage across a charging capacitor is V(t) = V_final * (1 - e^(-t/τ)), where V_final is the final voltage, t is time, and e is Euler's number. At t = τ, this simplifies to V(τ) = V_final * (1 - e^(-1)), which results in approximately 63.2% of the final voltage. This characteristic of charging and discharging is a fundamental property of capacitors and is crucial for understanding the timing and behavior of circuits involving them. Thus, understanding that the voltage changes by about 63.2% toward the fully charged state after each time constant is essential when working with capacitors in various applications.

When a capacitor charges through a resistor, the voltage across the capacitor increases according to an exponential function. Specifically, after one time constant (denoted as τ, or tau), the voltage across the capacitor reaches approximately 63.2% of the difference between its initial voltage (usually 0 volts for a fully discharged capacitor) and the final voltage it is charging towards (the supply voltage).

This percentage comes from the mathematical relationship that describes charging and discharging in RC (resistor-capacitor) circuits. The formula for the voltage across a charging capacitor is V(t) = V_final * (1 - e^(-t/τ)), where V_final is the final voltage, t is time, and e is Euler's number. At t = τ, this simplifies to V(τ) = V_final * (1 - e^(-1)), which results in approximately 63.2% of the final voltage.

This characteristic of charging and discharging is a fundamental property of capacitors and is crucial for understanding the timing and behavior of circuits involving them. Thus, understanding that the voltage changes by about 63.2% toward the fully charged state after each time constant is essential when working with capacitors in various applications.

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