The capacitor in FIGURE P28.73 begins to charge after the switch closes at t = 0 s. Find an expression for the current I at time t. Graph I from t = 0 to t = 5τ.

Verified step by step guidance

1. Step 1: Recall the relationship for the charging current in an RC circuit. The current I(t) as a function of time t is given by the equation: $I(t)\; =\; \backslash frac\{\backslash varepsilon\}\{R\}\; e^\{-t/\backslash tau\}$, where $\backslash varepsilon$ is the emf of the battery, $R$ is the resistance, and $\backslash tau\; =\; RC$ is the time constant of the circuit.
2. Step 2: Substitute the time constant $\backslash tau\; =\; RC$ into the equation. The expression becomes: $I(t)\; =\; \backslash frac\{\backslash varepsilon\}\{R\}\; e^\{-t/(RC)\}$. This is the general expression for the current in the circuit as a function of time.
3. Step 3: To graph the current from $t\; =\; 0$ to $t\; =\; 5\backslash tau$, note that the current starts at its maximum value $I(0)\; =\; \backslash frac\{\backslash varepsilon\}\{R\}$ when $t\; =\; 0$, and it decreases exponentially toward zero as time increases.
4. Step 4: Plot the graph of $I(t)$ versus time. The x-axis represents time $t$, and the y-axis represents the current $I(t)$. The curve will start at $\backslash frac\{\backslash varepsilon\}\{R\}$ at $t\; =\; 0$ and approach zero asymptotically as $t$ increases. Mark key points such as $t\; =\; \backslash tau$, $t\; =\; 2\backslash tau$, and so on, where the current decreases significantly.
5. Step 5: Interpret the graph. The exponential decay of the current reflects the charging process of the capacitor. At $t\; =\; \backslash tau$, the current has decreased to approximately 37% of its initial value, and by $t\; =\; 5\backslash tau$, the current is nearly zero, indicating the capacitor is almost fully charged.

Key Concepts

Capacitance

Capacitance is the ability of a system to store electric charge per unit voltage. It is measured in farads (F) and is defined by the formula C = Q/V, where Q is the charge stored and V is the voltage across the capacitor. Understanding capacitance is crucial for analyzing how capacitors behave in circuits, especially during charging and discharging phases.

RC Time Constant

The RC time constant, denoted as τ (tau), is a measure of the time it takes for the voltage across a capacitor to charge to approximately 63.2% of its maximum value when connected to a voltage source. It is calculated as τ = R × C, where R is the resistance in the circuit and C is the capacitance. This concept is essential for determining the charging and discharging rates of capacitors in circuits.

Current in a Charging Capacitor

The current I in a charging capacitor can be described by the equation I(t) = (V/R) * e^(-t/τ), where V is the voltage of the power source, R is the resistance, and τ is the time constant. This equation shows that the current decreases exponentially over time as the capacitor charges. Understanding this relationship is key to graphing the current over time and analyzing the behavior of the circuit.