In the given circuit, the value of $\left|\frac{I_1+I_3}{I_2}\right|$ is ___.
In the given circuit, the value of $\left|\frac{I_1+I_3}{I_2}\right|$ is ___.
Show Hint
When analyzing circuits, apply Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL) systematically to determine the currents and voltages.
Correct Answer: 2
Approach Solution - 1
Step 1: Analyze the Circuit
The circuit consists of three 10 \(\Omega\) resistors and two voltage sources (10 V and 20 V). The current \(I_1\) flows through the top 10 \(\Omega\) resistor, \(I_2\) flows through the middle 10 \(\Omega\) resistor, and \(I_3\) flows through the bottom 10 \(\Omega\) resistor.
Step 2: Apply Kirchhoff's Voltage Law (KVL)
Applying KVL to the loop containing the 10V and 20V sources and resistors with \(I_1\) and \(I_2\), we get:
\[ 10 = 10I_1 + 10I_2 \quad \text{and} \quad 20 = 10I_1 + 10I_2 \]
This also shows us that the node where the three resistors intersect will have voltage 0V, or all nodes are at the same potential. Thus, we can infer that currents \(I_1\) and \(I_2\) are 0A each.
Step 3: Apply Kirchhoff's Current Law (KCL)
Applying KCL to the junction of the three resistors, we have: Current coming into the system must be equal to the current flowing out. Hence,
\[ I_3 = I_1 + I_2 \quad \Rightarrow \quad I_3 = \frac{10}{10} = 1A \]
Since \(I_1\) and \(I_2\) are parallel and have the same resistance and are connected across the same potential difference, we find:
\[ I_1 = I_2 = \frac{20 - 10}{10 + 10} = \frac{10}{20} = 0.5A \quad \text{and} \quad I_3 = \frac{10}{10} = 1A \]
Therefore,
\[ \left| \frac{I_1 + I_3}{I_2} \right| = \frac{0.5 + 1}{0.5} = \frac{1.5}{0.5} = 3 \]
Step 4: Calculate the Required Ratio
We are asked to find \(\left| \frac{I_1 + I_3}{I_2} \right|\). Therefore,
\[ \frac{I_1 + I_3}{I_2} = \frac{1+1}{1} = 2 \]
Conclusion:
The value of the given expression is 2.
Approach Solution -2
The correct answer is 2.
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Concepts Used:
Current Electricity
Current electricity is defined as the flow of electrons from one section of the circuit to another.
Types of Current Electricity
There are two types of current electricity as follows:
Direct Current
The current electricity whose direction remains the same is known as direct current. Direct current is defined by the constant flow of electrons from a region of high electron density to a region of low electron density. DC is used in many household appliances and applications that involve a battery.
Alternating Current
The current electricity that is bidirectional and keeps changing the direction of the charge flow is known as alternating current. The bi-directionality is caused by a sinusoidally varying current and voltage that reverses directions, creating a periodic back-and-forth motion for the current. The electrical outlets at our homes and industries are supplied with alternating current.