Which is true in the above circuit?

Steps to Solve

1. Apply Kirchhoff's Current Law (KCL) at Nodes

For any junction in an electrical circuit, the sum of current entering the junction equals the sum of currents leaving the junction. We will apply KCL at different nodes in the circuit.

2. Identify Currents at Nodes

Let's identify the currents at the different nodes:

• At the node between $I_1$, $I_2$, and $I_3$:
  $$ I = I_1 + I_2 + I_3 $$

• At the node between $I_3$, $I_4$, $I_5$, and $I_6$:
  $$ I_3 = I_4 + I_5 + I_6 $$

3. Set Up Equations from KCL

Using the relationships from KCL, we can set up equations, starting with the first node:

• From the first node: $$ I = I_1 + I_2 + I_3 $$

4. Express Currents in Terms of Other Currents

We can express $I_3$ in terms of $I_1$ and $I_2$ using the first node relation and the second node:
Rearranging the first node gives:
$$ I_3 = I - I_1 - I_2 $$
Substituting this into the KCL equation for the second node yields: $$ I - I_1 - I_2 = I_4 + I_5 + I_6 $$

5. Simplify the Equation

Now, we want to isolate the currents to test the options given in the question. Rearranging gives:
$$ I - I_1 - I_2 - I_6 = I_4 + I_5 $$

6. Evaluate Given Options

Now, we can evaluate the options:

• a. ( I_1 - I_2 = I_4 + I_5 )
• b. ( I_1 + I_2 = I_4 + I_5 )
• c. ( I_1 + I_2 = I_4 - I_5 )
• d. ( I_1 - I_2 = I_4 - I_5 )

Compare all equations derived with the options to find the correct one.