Circuit Current: Calculate Current Through A 4Ω Resistor
Hey guys! Let's break down this physics problem step-by-step. We're dealing with a circuit that has two batteries and a resistor, and our goal is to figure out the current flowing through one of the resistors. This involves understanding Ohm's Law, Kirchhoff's Laws, and how batteries behave in a circuit. So, buckle up and let's dive in!
The Problem: A Circuit with Two Batteries and a Resistor
Here's the scenario: We have a circuit with two batteries connected in a way that their voltages oppose each other. One battery has an electromotive force (EMF) of 24V and an internal resistance of 2Ω. The other battery has an EMF of 6V and the same internal resistance of 2Ω. These batteries are connected in series with a 4Ω resistor (R2). Our mission, should we choose to accept it, is to calculate the current flowing through this 4Ω resistor.
To get started, let's visualize the circuit. Imagine a loop where the current flows. The 24V battery is trying to push current in one direction, while the 6V battery is trying to push it in the opposite direction. This difference in voltage is what ultimately drives the current through the circuit. The resistors, both the internal resistances of the batteries and the external 4Ω resistor, will impede the flow of current.
Now, let’s dive deeper into the concepts we need to solve this problem:
Key Concepts: Electromotive Force (EMF), Internal Resistance, and Ohm's Law
- Electromotive Force (EMF): Think of EMF as the "push" or voltage provided by a battery. It's the energy per unit charge that the battery can supply. In our case, we have two EMFs: 24V and 6V.
- Internal Resistance: Real-world batteries aren't perfect. They have an internal resistance that opposes the flow of current. This internal resistance acts like a resistor inside the battery itself. We have a 2Ω internal resistance for each battery.
- Ohm's Law: This is the bread and butter of circuit analysis! It states that the voltage (V) across a resistor is directly proportional to the current (I) flowing through it, and the constant of proportionality is the resistance (R). Mathematically, V = IR. This law will be crucial in finding the current.
Applying Kirchhoff's Laws: The Foundation of Circuit Analysis
Kirchhoff's Laws are fundamental principles that govern how current and voltage behave in electrical circuits. They're like the rules of the road for electrons! We'll primarily use Kirchhoff's Voltage Law (KVL) to solve this problem.
- Kirchhoff's Voltage Law (KVL): KVL states that the sum of the voltage drops and voltage rises around any closed loop in a circuit must equal zero. In simpler terms, if you start at a point in a circuit and travel around a loop, adding up all the voltage changes (both positive and negative), you should end up back where you started with no net change in voltage.
Solving the Circuit: Step-by-Step
Now, let’s put these concepts together to solve the problem. Here's how we can approach it:
- Determine the Net Voltage: Since the batteries are opposing each other, we subtract the smaller voltage from the larger voltage to find the net voltage driving the current. So, 24V - 6V = 18V. This 18V is the effective voltage pushing current through the circuit.
- Calculate the Total Resistance: We need to consider all the resistances in the circuit. We have the two internal resistances (2Ω each) and the external resistor (4Ω). Since they are in series, we simply add them up: 2Ω + 2Ω + 4Ω = 8Ω. This is the total resistance the current
