A 500-Ωresistor, an uncharged 1.50-μF capacitor, and a 6.16-V emf are connected in series. (a) W

step 1 So in this problem, we connect the battery with the resistor and in addition with the capacitor

A 500-Ωresistor, an uncharged 1.50-μF capacitor, and a...

A $500-\Omega$ resistor, an uncharged $1.50-\mu \mathrm{F}$ capacitor, and a $6.16-\mathrm{V}$ emf are connected in series. (a) What is the initial current? (b) What is the $R C$ time constant? (c) What is the current after one time constant? (d) What is the voltage on the capacitor after one time constant?

Key Concepts

RC Circuit

An RC circuit is an electrical circuit composed of resistors and capacitors, connected in series or parallel, that exhibits time-dependent behavior when a voltage is applied. The charging and discharging of the capacitor through the resistor follow exponential relationships, which are essential in understanding transient circuit responses.

Initial Current Calculation

At the moment a circuit involving a capacitor is closed, the uncharged capacitor acts like a short circuit. This means that the full battery voltage appears across the resistor, and the initial current is determined using Ohm's law, given by I = V/R. This concept is crucial for analyzing the immediate behavior of RC circuits.

RC Time Constant (?)

The time constant, denoted as ? and defined by the product of the resistance and capacitance (? = RC), characterizes the rate at which the capacitor charges or discharges. After a time equal to the time constant, a charging capacitor reaches about 63.2% of the steady-state voltage, making ? a fundamental timing measure in transient analysis.

Exponential Decay of Current in a Charging Circuit

Once the capacitor begins charging, the current in an RC circuit decays exponentially over time. The mathematical expression I(t) = I? exp(-t/?) describes how the initial current decreases as the voltage across the capacitor builds up, reflecting the exponential behavior inherent in first-order circuits.

Exponential Charging of a Capacitor

The voltage across a charging capacitor increases according to an exponential function, given by V(t) = ? (1 - exp(-t/?)). This formula shows that the capacitor voltage asymptotically approaches the battery voltage, providing insight into the gradual energy storage in the capacitor over time.

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