Question

Two identical light bulbs are connected to a battery. Will the light bulbs be brighter if they are connected in series or in parallel?

Short answer

Answer: The light bulbs will be brighter when connected in parallel.

Step-by-step solution

Calculate Current and Voltage in Series Connection

When the light bulbs are connected in series, the total resistance (R_total) is the sum of their individual resistances (R), i.e., R_total = 2R. Since the battery provides a constant voltage (V), we can use Ohm's law (V = I * R) to find the total current (I_total) in the circuit. In this case, I_total = V / (2R).

Calculate Current and Voltage in Parallel Connection

When the light bulbs are connected in parallel, the total resistance (R_total) in the circuit is given by the formula (1/R_total) = (1/R) + (1/R). Solving for R_total, we find that R_total = R/2. Again, using Ohm's law, we can find the total current (I_total) in the circuit. In this case, I_total = V / (R/2) = 2V / R.

Calculate Power (Brightness) in Series Connection

The power (P) of a light bulb is directly related to its brightness and can be calculated using the formula P = I^2 * R. Since the bulbs are identical, they have the same resistance and current through them in series connection. Therefore, the power of each light bulb in series connection is P_series = I_total^2 * R = (V^2 / (4R^2)) * R = V^2 / (4R).

Calculate Power (Brightness) in Parallel Connection

For the parallel connection, we'll find the power of each bulb again using P = I^2 * R. However, in a parallel connection, the voltage across each bulb is the same as the battery voltage, and the currents through each bulb are equal (since they have the same resistance) and sum up to the total current (I_total). That means the current through each bulb is I_parallel = I_total / 2 = (2V / R) / 2 = V / R. The power of each light bulb in parallel connection is P_parallel = I_parallel^2 * R = (V^2 / R^2) * R = V^2 / R.

Compare Brightness in Series and Parallel Connection

Now, let's compare the brightness (power) of the bulbs in series and parallel connection. We have P_series = V^2 / (4R) and P_parallel = V^2 / R. Since P_parallel > P_series, the light bulbs will be brighter when connected in parallel compared to when they are connected in series.

Key concepts

Ohm's Law

Ohm's Law is a fundamental principle in electrical engineering that relates voltage (\( V \)), current (\( I \)), and resistance (\( R \) in a circuit. It's presented as the simple equation \( V = I \times R \). This equation tells us that the current flowing through a conductor between two points is directly proportional to the voltage across the two points, assuming the temperature remains constant. The resistance in the circuit acts as the proportionality constant.

When applying Ohm's Law to solve for the brightness of bulbs in a circuit, as in the textbook exercise, it's crucial to understand how altering the resistance—whether by arranging bulbs in series or parallel—affects the current. In a series connection, resistances add up which increases the total resistance, hence reducing the current. Conversely, in a parallel connection, resistances

Electrical Resistance

Electrical resistance is a measure of the opposition to the flow of electric current. It's what determines how much current will flow through the circuit for a given voltage, following Ohm's Law. Resistance is measured in ohms (\( \(\Omega \) \).

In the context of the textbook exercise, when bulbs are placed in series, the total resistance increases as each bulb adds its own resistance to the circuit. Therefore, \( R_{\text{total}} = R_1 + R_2 + ... + R_n \), where \( R_i \) is the resistance of the \( i^{\text{th}} \) bulb. However, when bulbs are arranged in parallel, the total resistance is less than any individual resistance and can be calculated using the formula \( 1/R_{\text{total}} = 1/R_1 + 1/R_2 + ... + 1/R_n \). This decreased resistance leads to a higher current for a given voltage, based on Ohm's Law.

Circuit Brightness

Circuit brightness is a term often used to describe the luminous efficacy of a circuit component, like a light bulb. It is largely dependent on the electrical power (\( P \) consumed by the bulb, where \( P = I^2 \times R \) or \( P = V \times I \), depending on which values are known. Brightness, in this sense, is a practical way of visualizing power. The greater the power dissipated by a bulb, the brighter it will shine.

From the step-by-step solution provided, we understand that the bulbs in a parallel circuit are brighter than those in a series circuit. This conclusion aligns with the calculation of power in each configuration: in a series circuit, the power is divided among the components, while in a parallel circuit, each bulb can draw power equivalent to being directly connected to the voltage source. Therefore, due to the higher power drawn in parallel, each light bulb exhibits greater brightness.

Power Calculation in Circuits

Power calculation in circuits is crucial for understanding the workload, or electrical energy consumption, of circuit components per unit time. Measured in watts (\( W \), power in an electrical circuit is given by \( P = V \times I \), where \( V \) is voltage and \( I \) is current. Another useful form, especially for resistive loads like incandescent light bulbs, derives from Ohm's Law: \( P = I^2 \times R \) or \( P = V^2 / R \.

Applying these formulas to our exercise, for bulbs connected in series, power is dissipated over a greater resistance, which results in less power per bulb and correspondingly, less brightness. For bulbs in parallel, each has the full circuit voltage across it and operates at a power level as if alone in the circuit. Consequently, light bulbs in parallel shine brighter, which is evident when comparing \( P_{\text{parallel}} \) to \( P_{\text{series}} \). This is an important consideration for real-world applications where the distribution of power affects the performance and functionality of electrical devices.