Determine the node voltages $v_1$ and $v_2$ for the given circuit: 
To determine the node voltages \( V_1 \) and \( V_2 \), we can use the nodal analysis method. Let's apply Kirchhoff's current law (KCL) at nodes \( V_1 \) and \( V_2 \).
Step 1: Apply KCL at Node \( V_1 \)
Consider the current entering and leaving the node:
\[ 2 = \frac{V_1}{7} + \frac{V_1 - V_2}{15} + \frac{V_1}{3} \]Multiply through by 105 (LCM of 7, 15, and 3) to clear fractions:
\[ 210 = 15V_1 + 7(V_1 - V_2) + 35V_1 \]Simplify:
\[ 210 = 15V_1 + 7V_1 - 7V_2 + 35V_1 \]Combine like terms:
\[ 210 = 57V_1 - 7V_2 \]Step 2: Apply KCL at Node \( V_2 \)
Consider the current entering and leaving the node:
\[ \frac{V_2 - V_1}{15} + \frac{V_2}{5} = 4 \]Multiply through by 15 (LCM of 15 and 5) to clear fractions:
\[ V_2 - V_1 + 3V_2 = 60 \]Simplify:
\[ 4V_2 - V_1 = 60 \]Step 3: Solve the System of Equations
We now have two equations:
\[ 57V_1 - 7V_2 = 210 \] \[ 4V_2 - V_1 = 60 \]Rearrange the second equation:
\[ V_1 = 4V_2 - 60 \]Substitute \( V_1 \) in the first equation:
\[ 57(4V_2 - 60) - 7V_2 = 210 \]Expand and simplify:
\[ 228V_2 - 3420 - 7V_2 = 210 \] \[ 221V_2 = 3630 \]Solve for \( V_2 \):
\[ V_2 = 20 \, V \]Substitute \( V_2 \) back to find \( V_1 \):
\[ V_1 = 4 \times 20 - 60 = 80 - 60 = 20 \, V \]Thus, the node voltages are \( V_1 = 20 \, V \) and \( V_2 = 20 \, V \).
Conclusion: The correct answer is \( V_1 = 20 \, V \), \( V_2 = 20 \, V \).
Important Questions from Mixed Topic (CUET PG)
- Who was the founder of Bolshevik Communist party?
- What is the key guide to statecraft in the realist tradition?
- Chronologically arrange the events in the Cold War period.
A. Berlin Wall is constructed
B. Communist China joins the UN
C. Soviet invasion of Czechoslovakia
D. Berlin Blockade
Choose the correct answer from the options given below: Morgenthau's principles of political realism are:
A. Politics is rooted in permanent and unchanging human nature which is basically self centred, self-regarding and self-interested
B. Politics is an autonomous sphere of action and cannot therefore be reduced to morals
C. International Politics is an arena of conflicting self-interests
D. The ethics of international relations is situational ethics which is very different from private morality
Choose the correct answer from the options given below:Who among the following political thinkers consider the anarchical self help system to be a compelling factor for States to maximise their relative power positions?
