Question

Which of the arrangements of three identical light bulbs shown in the figure draws most current from the battery? a) \(A\) d) All three draw equal current. b) \(B\) e) \(\mathrm{A}\) and \(\mathrm{C}\) are tied for drawing the most current. c) \(C\)

Short answer

Answer: a) Three light bulbs connected in parallel.

Step-by-step solution

Identify type of connections in arrangements A, B, and C

In arrangement A, the three light bulbs are connected in parallel. In arrangement B, the light bulbs are connected in series. In arrangement C, one light bulb is connected in parallel to two others that are connected in series.

Calculate the total resistance of each arrangement

Let R be the resistance of a single lightbulb. For arrangement A, since all light bulbs are in parallel, the reciprocal of the total resistance is found by summing the reciprocals of individual resistances: \(\frac{1}{R_A} = \frac{1}{R} + \frac{1}{R} + \frac{1}{R}\). For arrangement B, since all light bulbs are in series, the total resistance is found by summing the individual resistances: \(R_B = R + R + R\). For arrangement C, note that two light bulbs are connected in series and their total resistance is \(R + R\), which is then connected in parallel to another lightbulb, thus we have: \(\frac{1}{R_C} = \frac{1}{R} + \frac{1}{R+R}\).

Use Ohm's Law to find the current drawn in each arrangement

Ohm's Law states that \(I = \frac{V}{R}\). To compare the current drawn by each arrangement, we can calculate their currents using the total resistance calculated in the previous step: For arrangement A: \(I_A = \frac{V}{R_A}\). For arrangement B: \(I_B = \frac{V}{R_B}\). For arrangement C: \(I_C = \frac{V}{R_C}\).

Compare the currents drawn by each arrangement

By observing the total resistance calculations in each arrangement, we can see that arrangement A has a smaller total resistance compared to the other arrangements. Since the current is inversely proportional to the resistance, according to Ohm's law, a smaller resistance will result in a higher current. Thus, arrangement A draws the most current from the battery. The correct answer is a) A.

Key concepts

Ohm's Law

Ohm’s Law is a fundamental principle in electric circuit analysis and is essential to understanding how circuits behave. It relates three very important values: voltage (V), current (I), and resistance (R). The simple equation, \[ I = \frac{V}{R} \] illustrates that the current (I) through a resistor in a circuit is directly proportional to the voltage (V) across the resistor and inversely proportional to the resistance (R) of the resistor. This means:

• If the voltage increases and resistance stays constant, the current increases.
• If the resistance increases while the voltage stays constant, the current decreases.

This law is crucial when calculating how much current a particular arrangement of electrical components will draw from a power source. For instance, in the problem's configurations, it's used to determine which arrangement of light bulbs will draw the most current from the battery based on their total resistance.

Parallel and Series Circuits

Understanding parallel and series circuits is vital for analyzing how electrical components are arranged and how they affect the overall circuit behavior.**Series Circuits:**In a series circuit, components are connected end to end, forming a single path for current flow. This means the same current flows through each component, and the total resistance is the sum of the individual resistances:\[ R_{total} = R_1 + R_2 + R_3 + \ldots \] In this setup, if one component fails, the entire circuit is broken. In our exercise, arrangement B exemplifies a series arrangement.**Parallel Circuits:**In parallel circuits, components are connected across the same voltage source, creating multiple paths for the current to flow. The total resistance, in this case, is found differently:\[ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \ldots \] This setup allows for each component to operate independently—failure in one doesn't stop current in others. In the provided problem, arrangement A and a part of arrangement C include this type of circuit. Generally, parallel arrangements have lower total resistance compared to series arrangements with equal components, leading to higher current flow.

Electric Current Calculation

Electric current is the flow of electric charge and is influenced by the circuit’s overall resistance and the applied voltage. Calculating the current flowing through a circuit requires understanding both the arrangement of components and applying Ohm’s Law.When calculating current through various circuit expressions:

• Identify the total resistance of the circuit using the rules for series and parallel configurations.
• Apply Ohm’s Law, where current is determined using \[ I = \frac{V}{R_{total}} \]where \( V \) is the voltage and \( R_{total} \) is the calculated total resistance.

In the provided exercise, we apply this method to compare three different arrangements of light bulbs. With arrangement A having the lowest total resistance, it draws the most current due to the inverse relationship between current and resistance in Ohm’s Law. This illustrates how different configurations impact the amount of current a circuit draws from a power source.