In the circuit, the initial energy stored in the capacitor...

In the circuit, the initial energy stored in the capacitor is 25 J. At $t=0$ the switch is closed. (a) Sketch a graph of the voltage across the resistor $\left(V_{\mathrm{R}}\right)$ as a function of $t .$ Label the vertical axis with key numerical value(s) and units. (b) At what time is the energy stored in the capacitor $1.25 \mathrm{J} ?$

In the circuit, the initial energy stored in the capacitor is 25 J. At $t=0$ the
switch is closed.
(a) Sketch a graph of the voltage
across the resistor $\left(V_{\mathrm{R}}\right)$ as a function of $t .$ Label the vertical axis with key numerical value(s) and units.
(b) At what time is the energy stored in the capacitor $1.25 \mathrm{J} ?$

Key Concepts

RC Circuit

An RC circuit is a basic electrical circuit composed of a resistor (R) and a capacitor (C), which is used to study charging and discharging behaviors of the capacitor. The circuit dynamics are determined by these two components interacting, and such circuits are fundamental in understanding transient responses in electronics.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. In the context of RC circuits, both the voltage across the capacitor and the current in the circuit decay exponentially with time, reflecting the gradual release of stored energy.

Time Constant (?)

The time constant, defined as ? = RC, characterizes the speed of the charging or discharging process in an RC circuit. It represents the time required for a quantity to either build up or decay to approximately 63% of its final value, making it a critical parameter in transient analysis.

Capacitor Energy

The energy stored in a capacitor is given by the formula E = 0.5 * C * V^2, linking the electrical potential difference to the energy content. This relationship is essential for understanding how the energy in a circuit evolves over time during charging or discharging processes.

Voltage Across Resistor

In an RC circuit, the voltage across the resistor reflects the current flowing as the capacitor discharges, and it also follows an exponential decay. This voltage is initially equal to the capacitor voltage at t = 0 and then decreases over time, providing insight into the transient response of the circuit.

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