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МатематикаВопрос:
Fill in the table based on the circuit shown in the image. R1 = 8 Ohm, R2 = 4 Ohm, R3 = 2 Ohm, R4 = 10 Ohm, R5 = 12 Ohm, U1 = 32 V.
Ответ:
Based on the information provided in the image, I will calculate the required data and fill in the table.
Краткое пояснение: The solution involves applying Ohm's Law (U = IR) and the rules for series and parallel circuits to find the current (I) and voltage (U) values for each resistor. First, calculate the total resistance, then find the total current. Subsequently, calculate the current through each resistor based on the voltage and resistance.
Пошаговое решение:
- Шаг 1: Calculate the equivalent resistance for the resistors R1 and R2, which are in series. \(R_{12} = R_1 + R_2 = 8 \Omega + 4 \Omega = 12 \Omega\)
- Шаг 2: Calculate the equivalent resistance for the resistors R3 and R4, which are in series. \(R_{34} = R_3 + R_4 = 2 \Omega + 10 \Omega = 12 \Omega\)
- Шаг 3: Calculate the total resistance of the parallel combination of \(R_{12}\) and \(R_{34}\). \(\frac{1}{R_{1234}} = \frac{1}{R_{12}} + \frac{1}{R_{34}} = \frac{1}{12} + \frac{1}{12} = \frac{2}{12} = \frac{1}{6}\), so \(R_{1234} = 6 \Omega\)
- Шаг 4: The total equivalent resistance of the circuit is the sum of the parallel resistance \(R_{1234}\) and resistor R5 in series. \(R_{total} = R_{1234} + R_5 = 6 \Omega + 12 \Omega = 18 \Omega\)
- Шаг 5: Calculate the total current (I) in the circuit using Ohm's Law. \(I = \frac{U}{R} = \frac{32 V}{18 \Omega} = 1.778 A\) (approximately)
- Шаг 6: Find the voltage across \(R_5\). \(U_5 = I \cdot R_5 = 1.778 A \cdot 12 \Omega = 21.336 V\)
- Шаг 7: Find the voltage across the parallel combination of \(R_{12}\) and \(R_{34}\). Since the total voltage is 32 V, and \(U_5 = 21.336 V\), then \(U_{12} = U_{34} = 32 V - 21.336 V = 10.664 V\)
- Шаг 8: Calculate the current through \(R_1\) and \(R_2\). \(I_1 = I_2 = \frac{U_{12}}{R_{12}} = \frac{10.664 V}{12 \Omega} = 0.889 A\)
- Шаг 9: Calculate the current through \(R_3\) and \(R_4\). \(I_3 = I_4 = \frac{U_{34}}{R_{34}} = \frac{10.664 V}{12 \Omega} = 0.889 A\)
- Шаг 10: Calculate the current through \(R_5\). \(I_5 = I = 1.778 A\)
- Шаг 11: Calculate the voltage across each resistor. We already calculated: \(U_5 = 21.336 V\), \(U_{12} = U_{34} = 10.664 V\), so \(U_1 = I_1 \cdot R_1 = 0.889 A \cdot 8 \Omega = 7.112 V\), \(U_2 = I_2 \cdot R_2 = 0.889 A \cdot 4 \Omega = 3.556 V\), \(U_3 = I_3 \cdot R_3 = 0.889 A \cdot 2 \Omega = 1.778 V\), \(U_4 = I_4 \cdot R_4 = 0.889 A \cdot 10 \Omega = 8.89 V\)
Answer: I1 ≈ 0.889 A, I2 ≈ 0.889 A, I3 ≈ 0.889 A, I4 ≈ 0.889 A, I5 ≈ 1.778 A, U1 ≈ 7.112 V, U2 ≈ 3.556 V, U3 ≈ 1.778 V, U4 ≈ 8.89 V, U5 ≈ 21.336 V, R ≈ 18 Ohm, I ≈ 1.778 A, U ≈ 32 V
