1. **Problem Statement:** Given a circuit with a 28.0 V battery, an unknown resistor $R$, an unknown emf $\varepsilon$, a 6.00 $\Omega$ resistor with 4.00 A current, and a 3.00 $\Omega$ resistor with 6.00 A current, find: (a) the current in resistor $R$; (b) the resistance $R$; (c) the unknown emf $\varepsilon$. 2. **Given:** - Battery voltage $V = 28.0$ V - Current through 6.00 $\Omega$ resistor $I_{6} = 4.00$ A - Current through 3.00 $\Omega$ resistor $I_{3} = 6.00$ A - Resistance of known resistors: $R_6 = 6.00\ \Omega$, $R_3 = 3.00\ \Omega$ - Unknowns: Current through $R$ ($I_R$), resistance $R$, emf $\varepsilon$ 3. **Formulas and Rules:** - Ohm's Law: $V = IR$ - Kirchhoff's Current Law (KCL): Sum of currents at a node is zero - Kirchhoff's Voltage Law (KVL): Sum of voltage changes around a closed loop is zero 4. **Step (a): Find current in resistor $R$** - At node $x$, currents entering and leaving must balance: $$I_R = I_3 - I_6 = 6.00 - 4.00 = 2.00\ \text{A}$$ 5. **Step (b): Find resistance $R$** - Voltage across resistor $R$ is the same as voltage across the 3.00 $\Omega$ resistor plus the unknown emf $\varepsilon$ minus the battery voltage, but first find voltage across $R$ using Ohm's law: - Current through $R$ is $2.00$ A (from step a). - Use KVL in the loop containing battery, $R$, and 3.00 $\Omega$ resistor: $$28.0 - I_R R - V_3 = 0 \implies I_R R = 28.0 - V_3$$ - Voltage across 3.00 $\Omega$ resistor: $$V_3 = I_3 \times R_3 = 6.00 \times 3.00 = 18.0\ \text{V}$$ - Substitute: $$2.00 R = 28.0 - 18.0 = 10.0 \implies R = \frac{10.0}{2.00} = 5.00\ \Omega$$ 6. **Step (c): Find unknown emf $\varepsilon$** - Use KVL in the loop containing $\varepsilon$, 6.00 $\Omega$ resistor, and battery: $$-28.0 + \varepsilon + V_6 = 0 \implies \varepsilon = 28.0 - V_6$$ - Voltage across 6.00 $\Omega$ resistor: $$V_6 = I_6 \times R_6 = 4.00 \times 6.00 = 24.0\ \text{V}$$ - Substitute: $$\varepsilon = 28.0 - 24.0 = 4.00\ \text{V}$$ **Final answers:** - (a) Current in resistor $R$ is $2.00$ A - (b) Resistance $R$ is $5.00\ \Omega$ - (c) Unknown emf $\varepsilon$ is $4.00$ V