Question

The single-loop circuit shown in the figure has \(V_{e m f, 1}=16.37 \mathrm{~V}\) \(V_{\text {emf } 2}=10.81 \mathrm{~V},\) and \(R_{1}=24.65 \Omega\). The current flowing in the circuit is 0.1600 A. What is the resistance \(R_{2} ?\)

Short answer

Answer: The resistance R_2 in the given circuit is 10.1 Ω.

Step-by-step solution

Write down Kirchhoff's loop rule formula

Kirchhoff's loop rule states that the sum of the voltage changes around a closed loop must be equal to zero. Therefore, we can write the equation: \(V_{emf_1} - I\cdot R_1 - V_{emf_2} - I\cdot R_2 = 0\) where \(I\) is the current flowing in the circuit.

Plug in the known values

We are given the following values: \(V_{emf_1} = 16.37 V\), \(V_{emf_2} = 10.81 V\), \(R_1 = 24.65 \Omega\), and \(I = 0.1600 A\). Plug these values into the equation from Step 1: \(16.37 - 0.1600 \cdot 24.65 - 10.81 - 0.1600 \cdot R_2 = 0\)

Solve the equation for \(R_2\)

Simplify and rearrange the equation from Step 2 to solve for \(R_2\): \(16.37 - 3.944 - 10.81 - 0.1600 \cdot R_2 = 0\) \(1.616 = 0.1600 \cdot R_2\) Now, we can divide both sides of the equation by 0.1600 to get the value of \(R_2\): \(R_2 = \frac{1.616}{0.1600}\) \(R_2 = 10.1 \Omega\) So, the resistance \(R_2\) is equal to 10.1 \(\Omega\).

Key concepts

Electrical Circuits

At the heart of our modern life lies the essential concept of electrical circuits, which can be visualized as the bloodstream of any electrical device. An electrical circuit is essentially a closed loop through which electric charges can flow. The significance of such circuits cannot be overstated, as they underpin every electronic device we use, from the simplest children's toy to the most complex supercomputer.

To comprehend the behavior and function of an electrical circuit, several elements must be considered. These include the power source (like batteries or generators), which creates a potential difference to push electrons through the circuit. Then, we have conducting wires that act as pathways, and components like resistors, capacitors, and transistors, which control the flow and manage the energy in the circuit. The versatility of circuits is immense, allowing them to manage energy in a multitude of ways, performing tasks such as amplifying signals, computing data, or simply lighting a bulb.

When encountering circuits in theory or practice, we also approach various terminologies and laws, such as Kirchhoff's Rules, which empower students and professionals to predict and calculate the behavior of these circuits with astonishing accuracy.

Voltage and Current

The movement of electric charges in a circuit is described by two fundamental concepts: voltage and current. To paint an analogy, imagine water flowing through pipes; voltage is akin to the water pressure that propels the water through the pipes, while current would be the flow rate of that water.

Specifically, voltage (\( V \)), also known as electric potential difference, is the force that pushes electric charges along a conductor. The higher the voltage, the greater the