### RC Circuit Problem**Problem Statement:**An RC circuit with a 1-Ω resistor and a 0.000001-F capacitor is driven by a voltage \( E(t) = \sin 100t \, V \). If the initial capacitor voltage is zero, determine the subsequent resistor and capacitor voltages and the current.In this problem, we have an RC (resistor-capacitor) circuit with specified components and a driving voltage. The components' values and voltage function are given as follows:- **Resistor (R):** 1 Ω- **Capacitor (C):** 0.000001 F (1 µF)- **Voltage Source:** \( E(t) = \sin 100t \, V \)- **Initial Capacitor Voltage:** 0 V**Objective:**Determine the subsequent:1. Resistor Voltage (\( V_R(t) \))2. Capacitor Voltage (\( V_C(t) \))3. Circuit Current (\( I(t) \))### Solution Approach:1. **Kirchhoff's Voltage Law (KVL):** Using Kirchhoff’s Voltage Law, the sum of the voltage drops across the resistor and the capacitor is equal to the applied voltage \( E(t) \). \[ E(t) = V_R(t) + V_C(t) \]2. **Ohm's Law for the Resistor:** The voltage across the resistor \( V_R(t) \) is related to the current \( I(t) \) by Ohm's law: \[ V_R(t) = I(t) \times R \]3. **Capacitor Voltage-Current Relationship:** The current through the capacitor can be related to the rate of change of the capacitor voltage: \[ I(t) = C \frac{dV_C(t)}{dt} \]4. **Differential Equation:** Combining the above equations, we can setup a differential equation to solve for the capacitor voltage: \[ E(t) = I(t) \times R + V_C(t) \] \[ \sin 100t = 1 \times I(t) + V_C(t) \] \[ I(t) = \frac{dV_C(t)}{dt} \] \[ \sin 100t = \frac{dV

According to kirchoff's law," Net voltage drop in a closed loop is equal to zero."

2. An RC circuit with a 1-0 resistor and a 0.000001-F capacitor is driven by a voltage E (t) = sin 100t V. If the initial capacitor voltage is zero, determine the subsequent resistor and capacitor voltages and the current.

2. An RC circuit with a 1-0 resistor and a 0.000001-F capacitor is driven by a voltage E (t) = sin 100t V. If the initial capacitor voltage is zero, determine the subsequent resistor and capacitor voltages and the current.

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Chapter1: Units, Trigonometry. And Vectors

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Question

### RC Circuit Problem

**Problem Statement:**

An RC circuit with a 1-Ω resistor and a 0.000001-F capacitor is driven by a voltage \( E(t) = \sin 100t \, V \). If the initial capacitor voltage is zero, determine the subsequent resistor and capacitor voltages and the current.

In this problem, we have an RC (resistor-capacitor) circuit with specified components and a driving voltage. The components' values and voltage function are given as follows:

- **Resistor (R):** 1 Ω
- **Capacitor (C):** 0.000001 F (1 µF)
- **Voltage Source:** \( E(t) = \sin 100t \, V \)
- **Initial Capacitor Voltage:** 0 V

**Objective:**

Determine the subsequent:
1. Resistor Voltage (\( V_R(t) \))
2. Capacitor Voltage (\( V_C(t) \))
3. Circuit Current (\( I(t) \))

### Solution Approach:

1. **Kirchhoff's Voltage Law (KVL):**

Using Kirchhoff’s Voltage Law, the sum of the voltage drops across the resistor and the capacitor is equal to the applied voltage \( E(t) \).

\[ E(t) = V_R(t) + V_C(t) \]

2. **Ohm's Law for the Resistor:**

The voltage across the resistor \( V_R(t) \) is related to the current \( I(t) \) by Ohm's law:

\[ V_R(t) = I(t) \times R \]

3. **Capacitor Voltage-Current Relationship:**

The current through the capacitor can be related to the rate of change of the capacitor voltage:

\[ I(t) = C \frac{dV_C(t)}{dt} \]

4. **Differential Equation:**

Combining the above equations, we can setup a differential equation to solve for the capacitor voltage:

\[ E(t) = I(t) \times R + V_C(t) \]
\[ \sin 100t = 1 \times I(t) + V_C(t) \]
\[ I(t) = \frac{dV_C(t)}{dt} \]
\[ \sin 100t = \frac{dV

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