Consider an R C circuit with ℰ=12.0 V, R=175 Ω, and C=55.7 μF . Find (a) the time constant for t

step 1 So in this problem we have an RC circuit with an EMF or a voltage of 12 volts, a resistance of 1

Consider an R C circuit with ℰ=12.0 V, R=175 Ω, and C=55.7...

Consider an $R C$ circuit with $\mathcal{E}=12.0 \mathrm{V}, R=175 \Omega$, and $C=55.7 \mu F .$ Find $(a)$ the time constant for the circuit, $(b)$ the maximum charge on the capacitor, and (c) the initial current in the circuit.

Key Concepts

Initial Current Calculation

At the moment when the circuit is first closed (t = 0), the capacitor behaves like a short circuit (because it is initially uncharged), so the entire voltage appears across the resistor. The initial current can then be calculated using Ohm’s law by dividing the voltage by the resistance, illustrating an instantaneous peak in current that decays over time as the capacitor charges.

Maximum Charge on Capacitor

The maximum charge that a capacitor can accumulate in a circuit is determined by the product of its capacitance (C) and the applied voltage (V). This concept reflects the capacitor’s ability to store energy, and in a charging RC circuit, once the capacitor is fully charged, its voltage equals that of the battery, meaning the charge reaches its maximum value.

RC Circuit

An RC circuit is an electrical circuit that consists of a resistor (R) and a capacitor (C) connected in series or parallel with a voltage source. It is often used to study transient responses because the voltage across the capacitor changes over time as it charges or discharges, following an exponential behavior determined by the circuit's components.

Time Constant

The time constant, denoted as ? and defined by the product of the resistance (R) and the capacitance (C), characterizes how quickly the capacitor charges or discharges in an RC circuit. A larger time constant means the capacitor charges more slowly, as it takes more time to reach approximately 63% of the maximum charge.

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