Problem 3 - Applied Physics

Question

The emf of a battery is \(12.0 \mathrm{~V}\). If the internal resistance is \(0.300 \Omega\) and the voltage applied to the circuit is \(11.6 \mathrm{~V}\), what is the current through the battery?

Step-by-Step Solution

Answer

The current through the battery is approximately 1.33 A.

Step 1: Understanding the Problem

We need to find the current flowing through a battery. We know the electromotive force (emf) of the battery is 12.0 V, the internal resistance is 0.300 Ω, and the voltage across the circuit is 11.6 V.

Step 2: Identify the Formula

We will use the formula for the voltage across a battery considering its internal resistance, which is given by: \[ V = ext{emf} - I imes r \]Where:- \(V\) is the voltage across the circuit,- \(I\) is the current,- \(r\) is the internal resistance, and- \(\text{emf}\) is the electromotive force of the battery.

Step 3: Rearranging the Formula

Rearrange the formula to solve for the current \(I\):\[ I = \frac{\text{emf} - V}{r} \]

Step 4: Substitute the Values

Substitute the given values into the rearranged formula:\[ I = \frac{12.0 \, V - 11.6 \, V}{0.300 \, \Omega} \]

Step 5: Perform the Calculation

Calculate the difference in voltage and then divide by the internal resistance:\[ I = \frac{0.4 \, V}{0.300 \, \Omega} \]\[ I \approx 1.33 \, A \]

Step 6: Conclusion

The current flowing through the battery, considering the given parameters, is approximately 1.33 A.

Key Concepts

Electromotive Force

Electromotive force (emf) is a critical concept in understanding how batteries work. It represents the maximum potential difference between two terminals of a battery when no current is flowing through the circuit. This is often thought of as the 'push' or 'driving force' that drives electrons to move through an electrical circuit.

A battery has a specific emf that determines its initial capability to cause charge to flow. However, in real-world applications, the actual voltage across the terminals can be less than the emf due to the battery's internal resistance. In the exercise provided, the emf is 12.0 V, which is the theoretical upper limit of voltage the battery can supply when it's not connected to any load.

• Understanding emf helps in identifying the capability and efficiency of a power source.
• Emf is measured in Volts (V), just like any other form of voltage.

In summary, electromotive force is like having a car with plenty of horsepower. It tells us what is possible if everything works perfectly, but real-world variables like internal resistance joint in to influence the actual performance.

Internal Resistance

Internal resistance is another important concept to grasp, especially when dealing with practical circuits. It refers to the inherent resistance to current flow within the battery or power source itself. This resistance is not ideal because it reduces the efficiency of the battery.

In the exercise, the internal resistance is given as 0.300 Ω. This resistance slightly reduces the voltage available to the connected circuit as some of the energy is used against the battery's internal friction. As a result, although the emf of the battery is 12.0 V, the actual voltage experienced by the external circuit is 11.6 V:

• This voltage drop is due to the internal resistance working against the flow of electrons.
• Reducing internal resistance can improve the overall efficiency of power sources.

Understanding internal resistance helps in appreciating why voltage delivered is not always the same as the emf, and it is crucial in designing efficient electrical systems.

Ohm's Law

Ohm's Law is foundational in the field of electronics and physics. It states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance between them. Expressed mathematically, it tells us:

\[ V = I \times R \]
This formula can be rearranged to solve for different variables depending on what is provided or required:

• If you know the voltage and the resistance, you can find the current: \( I = \frac{V}{R} \)
• If you need to find voltage: \( V = I \times R \)
• If resistance is required: \( R = \frac{V}{I} \)

In the step-by-step solution, we used a form of Ohm's Law to calculate the current flowing through the battery by adjusting it to consider internal resistance. With \( V = \text{emf} - I \times r \), it accommodated the real-life case where resistance within the battery itself affects the outcome. This illustrates Ohm's Law’s importance in helping to analyze and solve problems in electric circuits quickly and accurately.