A 1.50 μF capacitor is charged through a 125 Ωresistor and then discharged through the same resi

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A 1.50 μF capacitor is charged through a 125 Ωresistor and...

A $1.50 \mu \mathrm{F}$ capacitor is charged through a $125 \Omega$ resistor and then discharged through the same resistor by short-circuiting the battery. While the capacitor is being charged, find (a) the time for the charge on its plates to reach $1-1 / e$ of its maximum value and $(b)$ the current in the circuit at that time. (c) During the discharge of the capacitor, find the time for the charge on its plates to decrease to $1 / e$ of its initial value. Also, find the time for the current in the circuit to decrease to $1 / e$ of its initial value.

Key Concepts

RC Circuit

An RC circuit consists of a resistor and a capacitor connected in series or parallel, and it’s a fundamental configuration for studying transient phenomena in electrical circuits. This type of circuit demonstrates how capacitors charge and discharge over time when connected to a voltage source, making it a practical example of first-order linear systems in electronics.

Time Constant (?)

The time constant, denoted by ? and defined as the product of resistance (R) and capacitance (C), is a crucial parameter in RC circuits. It characterizes the rate at which the capacitor charges or discharges, with the voltage or charge changing significantly (by a factor of about 63.2% during charging or decaying to 36.8% during discharging) in one time constant.

Exponential Charging Equation

In an RC charging circuit, the voltage or charge on the capacitor increases exponentially toward its maximum value. This behavior is described by the equation Q(t) = Q_max * (1 - e^(-t/?)), where Q_max represents the maximum charge the capacitor can hold. The term (1 - 1/e) arises naturally in this equation, indicating the time at which the capacitor reaches approximately 63.2% of its full charge.

Exponential Discharging Equation

For an RC discharging circuit, the capacitor’s charge or voltage decays exponentially from its initial value. This decay is captured by the formula Q(t) = Q_0 * e^(-t/?), where Q_0 is the initial charge. The concept of the '1/e' value is used to describe the time it takes for the charge to fall to about 36.8% of its original value, a key metric in analyzing transient behavior during discharge.

Current Behavior in RC Circuits

In both charging and discharging scenarios, the current in the circuit also exhibits an exponential behavior. During charging, the current starts at its maximum value and decays as I(t) = (V/R) * e^(-t/?), while during discharging, it similarly follows an exponential decay from its initial value. Understanding this relationship is vital for analyzing how quickly the energy stored in the capacitor is delivered or dissipated in the circuit.

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