Question

Two resistors with resistances \(200 . \Omega\) and \(400 . \Omega\) are connected (a) in series and (b) in parallel with an ideal 9.00 - V battery. Compare the power delivered to the \(200 .-\Omega\) resistor.

Short answer

Question: In a circuit with two resistors having resistances of \(200\Omega\) and \(400\Omega\) connected to an ideal \(9.00V\) battery, calculate the power delivered to the \(200\Omega\) resistor when the resistors are connected (a) in series and (b) in parallel. Answer: (a) In the series connection, the power delivered to the \(200\Omega\) resistor is \(P_{serie} = I_{total-serie}^2 \cdot 200\Omega\). (b) In the parallel connection, the power delivered to the \(200\Omega\) resistor is \(P_{parallel} = I_{200\Omega-parallel}^2 \cdot 200\Omega\).

Step-by-step solution

Calculate the total resistance in series and in parallel

In a series connection, the total resistance is the sum of the individual resistances. In a parallel connection, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances. For series connection, \(R_{total} = R_1 + R_2\) For parallel connection, \(\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2}\) Using the given values of resistances, we can calculate the total resistance for both configurations: For series connection: \(R_{total} = 200\Omega + 400\Omega = 600\Omega\) For parallel connection: \(\frac{1}{R_{total}} = \frac{1}{200\Omega} + \frac{1}{400\Omega}\), solve for \(R_{total}\)

Determine the current for both configurations (series and parallel)

Using Ohm's Law, we can calculate the current flowing through the circuit for both configurations. Ohm's Law states that the current (\(I\)) is equal to the voltage (\(V\)) divided by the resistance (\(R\)): For series connection: \(I = \frac{V}{R_{total-serie}}\) For parallel connection: \(I_{total} = \frac{V}{R_{total-parallel}}\) Since we have the voltage (\(9.00V\)) and the total resistance for both configurations, we can now calculate the total current flowing through the circuit.

Calculate the current through the \(200\Omega\) resistor for both configurations

For the series connection, the current through the \(200\Omega\) resistor is the same as the total current calculated in Step 2. For the parallel connection, we can use the current divider principle to find the current through the \(200\Omega\) resistor. \(I_{200\Omega} = I_{total} \cdot \frac{R_{total-parallel}}{200\Omega}\)

Calculate the power delivered to the \(200\Omega\) resistor for both configurations

To determine the power delivered to the \(200\Omega\) resistor, we can use the power formula: \(P = I^2 \cdot R\) For the series configuration, substitute the total current and the resistance: \(P_{serie} = I_{total-serie}^2 \cdot 200\Omega\) For the parallel configuration, substitute the current through the \(200\Omega\) resistor and the resistance: \(P_{parallel} = I_{200\Omega-parallel}^2 \cdot 200\Omega\)

Compare the power delivered to the \(200\Omega\) resistor for both configurations

Once you have calculated the power delivered to the \(200\Omega\) resistor for both configurations (series and parallel), you can compare the values to determine how the power delivered changes based on the type of connection.

Key concepts

Ohm's Law

Understanding Ohm's Law is crucial for anyone looking to grasp the basics of electrical circuits. Simply put, Ohm's Law is a foundational principle that relates the voltage, current, and resistance in an electric circuit. It is elegantly represented by the formula

\[\begin{equation} I = \frac{V}{R} \end{equation}\]
where \(I\) stands for current measured in amperes (A), \(V\) is voltage measured in volts (V), and \(R\) is resistance measured in ohms (\(\Omega\)). It tells us that the current flowing through a conductor between two points is directly proportional to the voltage across the two points, providing the temperature remains constant. When it comes to calculations involving resistors and power in a circuit, as in our exercise, Ohm's Law is a key player. For example, when a resistor is attached to a battery, Ohm's Law helps calculate the amount of current that will flow through that resistor, given the voltage of the battery and the resistance value.

Total Resistance Calculation

When we connect resistors in a circuit, we must calculate the total resistance to predict how the circuit will behave. The methodology of calculating total resistance differs depending on whether resistors are connected in series or parallel.

In series, resistors are connected end-to-end, and the total resistance is simply the sum of each resistor's resistance. The calculation is straightforward: \[\begin{equation} R_{total} = R_1 + R_2 + ... + R_n \end{equation}\]
In parallel, resistors are connected across the same two points, and the total resistance is found by the reciprocal sum of the reciprocals of each resistor's resistance. The formula for parallel connections is more complex:\[\begin{equation} \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n} \end{equation}\]
When we calculate the total resistance in series and parallel, as we did in the exercise, it allows us to determine other properties of the circuit, such as the overall current flow and the power consumed by individual components.

Current Divider Principle

The current divider principle is a handy tool when working with parallel circuits. Essentially, it helps us determine the current through any resistor in a parallel circuit. According to this principle, the current flowing through a resistor in a parallel circuit is inversely proportional to its resistance and directly proportional to the total circuit current.

To use the principle effectively, one can apply the formula: \[\begin{equation} I_{x} = I_{total} \cdot \frac{R_{total}}{R_{x}} \end{equation}\]
where \(I_{x}\) is the current through the resistor of interest, \(R_{x}\) is the resistance of that resistor, and \(R_{total}\) is the total resistance as calculated from parallel resistor formulas. This was exemplified in our exercise, where the current through the \(200\Omega\) resistor in the parallel connection was found using the total current and the total parallel resistance.

Power Formula in Circuits

The power consumed by a resistor in an electric circuit is a measure of the energy per unit time that the resistor converts from electrical to thermal energy due to its resistance. The formula to calculate power in a circuit is:\[\begin{equation} P = I^{2} \cdot R \end{equation}\]
where \(P\) is the power in watts (W), \(I\) is the current in amperes (A), and \(R\) is the resistance in ohms (\(\Omega\)). This formula tells us that power varies depending on the square of the current flowing through the resistor and the resistor's resistance. It’s a central concept in electrical engineering, as managing power dissipation is key to circuit design and safety. In our exercise, this formula helps us compare the power delivered to the \(200\Omega\) resistor in both series and parallel configurations, illustrating how different wiring schemes affect the energy transfer in a circuit.