Question

Two resistors, \(R_{1}=3.00 \Omega\) and \(R_{2}=5.00 \Omega\), are connected in series with a battery and an ammeter, as shown in the figure. The battery supplies \(V_{e m f}=8.00 \mathrm{~V},\) and the ammeter has the resistance \(R_{\mathrm{A}}=1.00 \Omega\). What is the current measured by the ammeter? a) \(0.500 \mathrm{~A}\) b) \(0.750 \mathrm{~A}\) c) \(0.889 \mathrm{~A}\) d) \(1.00 \mathrm{~A}\) e) \(1.50 \mathrm{~A}\)

Short answer

Answer: (c) 0.889A

Step-by-step solution

Calculate the total resistance

In a series circuit, the total resistance is the sum of the individual resistances. In this case, we have resistors \(R_1\), \(R_2\), and \(R_A\) connected in series. Therefore, we can calculate the total resistance (\(R_{\text{total}}\)) as follows: $$R_{\text{total}} = R_1 + R_2 + R_A = 3.00\Omega + 5.00\Omega + 1.00\Omega$$

Calculate the total resistance value

Now, sum up the resistance values to find the total resistance of the circuit: $$R_{\text{total}} = 9.00\Omega$$

Apply Ohm's law to find the current

Ohm's law states that the current (I) in a circuit is equal to the voltage (V) divided by the resistance (R). We can use the total resistance that we calculated (\(R_{\text{total}} = 9.00\Omega\)) and the voltage supplied by the battery (\(V_{\text{emf}} = 8.00\text{V}\)) to calculate the current flowing through the circuit: $$I = \frac{V_{\text{emf}}}{R_{\text{total}}} = \frac{8.00\text{V}}{9.00\Omega}$$

Find the current value

Divide the voltage by the total resistance to find the current: $$I = 0.889\text{A}$$ The current measured by the ammeter is \(0.889\text{A}\). So, the correct answer is (c) \(0.889\text{A}\).

Key concepts

Series Circuits

In the world of electronics, a series circuit is one of the most basic and vital configurations we encounter. It's characterized by the connection of components end-to-end, such that the same electric current flows through each component without branching off.

What makes series circuits particularly important is their effect on the total resistance. When resistors are lined up in series, their resistances add up linearly, resulting in a higher total resistance than any single resistor alone. This cumulative effect is essential for controlling the current in electronic devices and systems.

Imagine stringing together beads on a single thread; this is akin to how resistors are set up in a series circuit. No matter how many beads (resistors) you add, they're all connected by the same thread (current path). This property implies that if any of the components fail or are disconnected, the entire circuit breaks, just as the bead chain would break if the thread snapped.

Electric Current Calculation

Electric current can be thought of as the flow of electric charge through a conductor, similar to water flowing through a pipe. The calculation of current in any circuit is foundational in understanding how electrical systems work. Ohm's law, expressed as the simple equation I = V / R, where I represents the current, V the voltage, and R the resistance, is essential for this calculation.

In the context of a series circuit, this becomes particularly straightforward, as the total resistance can be found by summing up all individual resistances. Once the total resistance is known, one can use the voltage supplied to the circuit to calculate the current using Ohm's law.

It's similar to determining the amount of water flowing through a series of connected pipes based on the water pressure (voltage) and the total constriction of the pipes (resistance). The calculated current is the same through every part of the series circuit, just as the flow rate of water would be consistent throughout the connected pipes.

Resistance in Circuits

Resistance is a fundamental concept in the field of electronics as it dictates how much current will flow for a given voltage. It's the measure of opposition to the flow of electric current, often likened to the friction experienced by water in pipes.

In a circuit, each component adds its resistance to the total opposition of the circuit. Understanding how these resistances add up is vital for designing circuits with desired electrical properties. In a series circuit, we find the total resistance by simply adding up the resistance of each individual component. This sum gives us a larger total resistance, restricting the current flow more than any individual resistor would on its own.

With higher resistance, less current flows for a given voltage, just like narrower or rougher pipes reduce the flow rate of water under the same pressure. In practical terms, resistors may be used to protect sensitive components from the potentially damaging effects of excessive current, much like a valve might restrict water flow to prevent bursting a hose.