Question

A \(12.0 \mathrm{~V}\) battery with an internal resistance \(R_{\mathrm{j}}=4.00 \Omega\) is attached across an external resistor of resistance \(R\). Find the maximum power that can be delivered to the resistor.

Short answer

Answer: The maximum power that can be delivered to the external resistor is 9.00 W.

Step-by-step solution

Write down given values and formula for maximum power transfer theorem

Given values: Voltage of the battery, V = 12.0 V Internal resistance of the battery, \(R_j\) = 4.00 Ω Maximum power transfer theorem states that the maximum power is transferred to the load when the load resistance (R) is equal to the internal resistance of the source (\(R_j\)). Therefore, \(R = R_j\).

Calculate the equivalent resistance

Now, we will find the equivalent resistance of the circuit, which is the sum of the internal resistance and the external resistance (\(R_{eq} = R_j + R\)). Since \(R = R_j\), we can write: \(R_{eq} = R_j + R_j = 2R_j\) Plugging in the given value for \(R_j\): \(R_{eq} = 2(4.00 \Omega) = 8.00 \Omega\)

Calculate the current through the circuit

Now we can use Ohm's law to calculate the current (I) through the circuit: \(I = \frac{V}{R_{eq}}\) Substitute the values for V and \(R_{eq}\): \(I = \frac{12.0 V}{8.00 \Omega} = 1.50 \,\mathrm{A}\)

Calculate the maximum power delivered to the resistor

Finally, we can calculate the maximum power (P) that can be delivered to the resistor using the formula: \(P = I^2 R\) We know that \(R = R_j\), so: \(P = I^2 R_j\) Now, substitute the given values: \(P = (1.50 \,\mathrm{A})^2 (4.00 \Omega) = 2.25 \,\mathrm{A^2} (4.00 \Omega) = 9.00 \,\mathrm{W}\) The maximum power that can be delivered to the resistor is 9.00 W.

Key concepts

Ohm's Law

Ohm's Law is a fundamental principle in the field of electrical circuits. It describes the relationship between voltage (V), current (I), and resistance (R) in an electrical circuit. Mathematically, it is expressed as: \[ V = IR \]This means that the voltage across a resistor is equal to the product of the current flowing through it and its resistance. It's a simple and yet powerful tool for understanding how electric circuits operate.

In the context of our problem with the battery and resistors, Ohm’s Law helps us determine the current flowing through the circuit. By knowing the equivalent resistance and the voltage of the battery, we can easily calculate the current using this law.

• Voltage (V) is the electric potential difference between two points in a circuit.
• Current (I) is the flow of electric charge through a conductor.
• Resistance (R) is a measure of how much a material opposes the flow of current.

Understanding Ohm's Law is crucial for analyzing and designing electrical circuits, making it a foundational concept in electronics.

Electrical Circuits

Electrical circuits consist of various components that facilitate the flow of electric current. Typical components include resistors, batteries, capacitors, and inductors. They are connected in different configurations to perform various functions, such as powering devices, amplifying signals, or storing energy.

In our exercise, we are dealing with a simple circuit made up of:

• A battery that provides voltage and acts as the power source.
• Resistors, including both the internal resistance of the battery and the external resistor connected to it.

Understanding the configuration and interaction between these components is key to calculating parameters like current and power.
The equivalent resistance in a circuit, like the one in our problem, determines the total opposition that the current faces. By knowing the equivalent resistance, we can determine the circuit's current and how power is distributed across the circuit’s components. Key concepts in analyzing electrical circuits include:

• Series and parallel connections, which affect how resistances add up.
• Current conservation, where the total current entering a junction equals the current leaving.
• Voltage conservation, where the total voltage around any closed loop must equal zero.

Power Calculation

Power calculation is essential for understanding how much energy is transferred or converted in an electrical circuit. The power (P) in a circuit is the rate at which energy is consumed or transformed, and it can be calculated using several formulas depending on the known values.
In our exercise, the power delivered to the resistor is calculated using the formula:\[ P = I^2 R \] Here, I is the current through the resistor, and R is the resistance of the resistor. This formula shows that power is directly proportional to the square of the current and the resistance. Hence, even small changes in current can lead to significant changes in power.

When applying the Maximum Power Transfer Theorem:

• The load resistance R should be equal to the internal resistance R_j for maximum power to be transferred to the load.
• This ensures that energy losses are minimized, and the circuit operates most efficiently.

Besides, understanding power calculations helps in evaluating the performance of electrical devices and ensuring that components within an electrical circuit can handle the expected loads without overheating or failing.