A 4 Ω resistor in series with 6 Ω and 3 Ω resistors in parallel, voltage 24 V. Find the currents and the voltage across the first resistor — Physics problems | PhysLabo
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Currents in a mixed resistor circuit

Medium 10 grade Electricity / Electric Current
Problem statement

A resistor $R_1 = 4$ Ω is connected in series with two resistors $R_2 = 6$ Ω and $R_3 = 3$ Ω connected in parallel (see the circuit). A voltage $U = 24$ V is applied to the ends of the whole section. Find the current in each resistor and the voltage across $R_1$.

Circuit: resistor R₁ in series with R₂ and R₃ connected in parallel, voltage U applied to the whole section
Given
  • $R_1 = 4$ Ω, $R_2 = 6$ Ω, $R_3 = 3$ Ω
  • $U = 24$ V
  • $I_1, I_2, I_3 = ?$
  • $U_1 = ?$
Hints
Hint 1 / 2
First find the resistance of the parallel group $R_{23} = \frac{R_2 R_3}{R_2 + R_3}$, then the total resistance of the section.
Hint 2 / 2
The current through $R_1$ equals the total current. The voltage across the parallel resistors is the same — find it as $U - U_1$.
Solution

First replace the parallel group by one equivalent resistor and find the total current; it flows through $R_1$ and then splits between $R_2$ and $R_3$. The voltages across parallel resistors are equal.

Currents and voltages in the parts of the circuit
Currents and voltages in the parts of the circuit

1. Resistance of the parallel group and of the whole section:

$$R_{23} = \frac{R_2 R_3}{R_2 + R_3} = \frac{6 \cdot 3}{6 + 3} = 2 \text{ Ω}, \qquad R = R_1 + R_{23} = 4 + 2 = 6 \text{ Ω}$$

2. The total current is also the current through $R_1$:

$$I_1 = \frac{U}{R} = \frac{24}{6} = 4 \text{ A}, \qquad U_1 = I_1 R_1 = 4 \cdot 4 = 16 \text{ V}$$

3. The voltage across the parallel group and the branch currents:

$$U_{23} = U - U_1 = 8 \text{ V}, \qquad I_2 = \frac{U_{23}}{R_2} = \frac{8}{6} \approx 1.33 \text{ A}, \qquad I_3 = \frac{U_{23}}{R_3} = \frac{8}{3} \approx 2.67 \text{ A}$$

Check: $I_2 + I_3 = 4$ A $= I_1$.

Answer

$I_1 = 4$ A, $I_2 \approx 1.33$ A, $I_3 \approx 2.67$ A; $U_1 = 16$ V

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