Problem 35 - College Physics

Question

How much charge flows through a 12-V battery when a \(2.0-\mu \mathrm{F}\) capacitor is connected across its terminals?

Step-by-Step Solution

Answer

The charge is 2.4 × 10⁻⁵ C.

Step 1: Understanding the Problem

We are asked to determine the amount of charge that flows through a 12-V battery when a 2.0-microfarad (μF) capacitor is connected across it.

Step 2: Key Formula

The amount of charge (Q) stored in a capacitor is given by the formula: \[ Q = C imes V \]where \( C \) is the capacitance in Farads and \( V \) is the voltage in Volts.

Step 3: Substitute the Given Values

We have \( C = 2.0 \, \mu F = 2.0 \, \times 10^{-6} \, F \) and \( V = 12 \, V \). Substitute these values into the equation:\[ Q = (2.0 \, \times 10^{-6}) \, F \, \times \, 12 \, V \]

Step 4: Calculate the Charge

Now, perform the multiplication to find the charge:\[ Q = 2.0 \, \times \, 10^{-6} \, \times \, 12 = 2.4 \, \times \, 10^{-5} \, C \]Thus, the charge that flows through the battery is \(2.4 \, \times \, 10^{-5} \, C \).

Key Concepts

Capacitance

Capacitance is the ability of a system to store electric charge. It is represented by the symbol "C" and is measured in units called Farads (F). A capacitor is a common component in electrical circuits that utilizes this property to hold and release electric charge when needed. The larger the capacitance, the more charge a capacitor can store at a given voltage. Think of capacitance like a bucket that holds water. The larger the bucket, the more water (or charge) you can store. Similarly, a larger capacitance means more charge can be held by the capacitor at the same voltage. In practical terms, circuit designers select capacitors based on the needed capacitance to store sufficient charge for circuit operations. This ensures that devices work correctly, whether it's smoothing out voltage in power supplies or storing energy for later use.

Voltage

Voltage, often described as electric potential difference, is the force that pushes electrical current through a circuit. It is represented by the symbol "V" and is measured in volts (V). Consider voltage like a pressure that drives water through a hose. Higher voltage means stronger pressure, allowing more electric current to flow. In the exercise, we have a 12-V battery. This means the battery provides a consistent potential difference of 12 volts across the capacitor, driving charge into it. In simple circuits like this one, the voltage dictates how much electrical energy is available to do work.

Charge Calculation

To calculate the charge stored in a capacitor, we use a simple formula: \[ Q = C \times V \]Here, \( Q \) is the charge in coulombs (C), \( C \) is the capacitance in farads (F), and \( V \) is the voltage in volts (V). This formula tells us the total charge stored in the capacitor is directly proportional to both the capacitance and the voltage. To find the charge, multiply the capacitance value by the voltage. For example, substituting the exercise's capacitance of \(2.0 \, \mu F\) (which is \(2.0 \, \times \, 10^{-6} \, F\)) and a 12-V battery, the calculation becomes: \[ Q = (2.0 \, \times \, 10^{-6}) \, F \, \times \, 12 \, V = 2.4 \, \times \, 10^{-5} \, C \]Thus, the stored charge is \(2.4 \, \times \, 10^{-5} \, C\). This process showcases how simple multiplication can determine how much electrical charge is involved.

Electrical Circuits

Electrical circuits are pathways that allow electric charge to flow. They consist of various components such as resistors, capacitors, and batteries, each performing distinct functions. In our exercise, a battery and capacitor form a basic circuit. The battery provides the circuit with electrical energy by maintaining a consistent voltage, while the capacitor stores the charge. When these components are connected in a circuit, electrons flow from the battery into the capacitor, charging it. Understanding circuits involves noting:

• How components like capacitors and batteries interact.
• The role of voltage in driving charge movement.
• The effects of varying capacitance and voltage on the stored charge.

This foundational knowledge of circuits helps you solve problems like calculating charge and understanding how devices function in real-world applications.