1. Problem 1a: Find $v_o$ and $i_o$ in the given circuit (Fig.01). Step 1: Identify components and given values: - Current source: $6A$ - Resistors: $2\Omega$, $8\Omega$, and the branch carrying $\frac{i_o}{4}$ current - Desired: $v_o$ across $8\Omega$ resistor and current $i_o$ through that branch. Step 2: Apply Kirchhoff's Current Law (KCL) or Voltage Law (KVL) to find relationships. - Current entering node splits into $i_o$ through $8\Omega$ and $\frac{i_o}{4}$ through another branch. - Total current after split: $i_o + \frac{i_o}{4} = \frac{5}{4}i_o$ Step 3: Since total supplying current is $6A$, set equal: $$6 = \frac{5}{4}i_o \implies i_o = \frac{6 \times 4}{5} = 4.8A$$ Step 4: Find $v_o$ across the $8\Omega$ resistor using Ohm's Law: $$v_o = i_o \times 8 = 4.8 \times 8 = 38.4V$$ 2. Problem 1b: Find current, voltage, and power across the $20k\Omega$ resistor in circuit (Fig.02). Step 1: Identify components and given values: - Current source: $5mA = 0.005A$ - Resistors: $10k\Omega$, $5k\Omega$, and $20k\Omega$ - Voltage $v_o$ at node between $10k\Omega$ and $5k\Omega/20k\Omega$ branch. Step 2: Voltage across $20k\Omega$ resistor is given as $0.01 v_o$ Step 3: Current through $20k\Omega$ resistor using Ohm's law: $$i_{20k} = \frac{v_{20k}}{20k} = \frac{0.01 v_o}{20,000} = 5 \times 10^{-7} v_o$$ Step 4: Sum currents at node with current source $5mA$ and current through $10k\Omega$ resistor ($i_{10k} = \frac{v_o}{10,000}$) and $20k\Omega$ resistor: $$5 \times 10^{-3} = \frac{v_o}{10,000} + 5 \times 10^{-7} v_o$$ Step 5: Simplify equations: $$5 \times 10^{-3} = v_o \left( \frac{1}{10,000} + 5 \times 10^{-7} \right) = v_o \left(0.0001 + 0.0000005 \right) = 0.0001005 v_o$$ Step 6: Solve for $v_o$: $$v_o = \frac{5 \times 10^{-3}}{0.0001005} \approx 49.75V$$ Step 7: Find current through $20k\Omega$ resistor: $$i_{20k} = 5 \times 10^{-7} \times 49.75 = 2.49 \times 10^{-5} A = 24.9 \mu A$$ Step 8: Compute power dissipated by $20k\Omega$ resistor: $$P = i^2 R = (2.49 \times 10^{-5})^2 \times 20,000 \approx 1.24 \times 10^{-5} W$$ Final answers: 1a) $i_o = 4.8A$, $v_o = 38.4V$ 1b) Current through $20k\Omega$ resistor $= 24.9 \mu A$, voltage $v_{20k} = 0.01 v_o = 0.4975 V$, power $= 1.24 \times 10^{-5} W$