Math Problem Statement

Beräkna strömmarna I_TOT, I1, och I2, samt spänningarna U1 och U2.

Solution

This circuit consists of resistors R1R_1, R2R_2, and R3R_3, where R2R_2 and R3R_3 are in parallel, and R1R_1 is in series with this parallel combination. We are tasked with calculating the total current ITOTI_{TOT}, the branch currents I1I_1 and I2I_2, and the voltages across R1R_1 and the parallel combination of R2R_2 and R3R_3 (i.e., U1U_1 and U2U_2).

Step 1: Equivalent Resistance of R2R_2 and R3R_3 (Parallel combination)

The total resistance of two resistors in parallel, R2R_2 and R3R_3, is given by: 1Rparallel=1R2+1R3\frac{1}{R_{\text{parallel}}} = \frac{1}{R_2} + \frac{1}{R_3} Substituting the given values of R2=200ΩR_2 = 200 \, \Omega and R3=200ΩR_3 = 200 \, \Omega: 1Rparallel=1200+1200=2200=1100\frac{1}{R_{\text{parallel}}} = \frac{1}{200} + \frac{1}{200} = \frac{2}{200} = \frac{1}{100} Thus, Rparallel=100ΩR_{\text{parallel}} = 100 \, \Omega

Step 2: Total Resistance of the Circuit

Since R1=1000ΩR_1 = 1000 \, \Omega is in series with Rparallel=100ΩR_{\text{parallel}} = 100 \, \Omega, the total resistance RtotalR_{\text{total}} is: Rtotal=R1+Rparallel=1000Ω+100Ω=1100ΩR_{\text{total}} = R_1 + R_{\text{parallel}} = 1000 \, \Omega + 100 \, \Omega = 1100 \, \Omega

Step 3: Calculate Total Current ITOTI_{\text{TOT}}

Using Ohm's Law, the total current ITOTI_{\text{TOT}} is given by: ITOT=UTOTRtotalI_{\text{TOT}} = \frac{U_{\text{TOT}}}{R_{\text{total}}} Substituting UTOT=230VU_{\text{TOT}} = 230 \, \text{V} and Rtotal=1100ΩR_{\text{total}} = 1100 \, \Omega: ITOT=23011000.209AI_{\text{TOT}} = \frac{230}{1100} \approx 0.209 \, \text{A}

Step 4: Voltage Across R1R_1 (U1U_1)

The voltage across R1R_1, denoted U1U_1, is: U1=ITOT×R1U_1 = I_{\text{TOT}} \times R_1 Substituting ITOT0.209AI_{\text{TOT}} \approx 0.209 \, \text{A} and R1=1000ΩR_1 = 1000 \, \Omega: U1=0.209×1000=209VU_1 = 0.209 \times 1000 = 209 \, \text{V}

Step 5: Voltage Across the Parallel Branch (U2U_2)

The voltage across the parallel combination of R2R_2 and R3R_3, denoted U2U_2, is the same for both resistors because they are in parallel: U2=UTOTU1U_2 = U_{\text{TOT}} - U_1 Substituting UTOT=230VU_{\text{TOT}} = 230 \, \text{V} and U1=209VU_1 = 209 \, \text{V}: U2=230209=21VU_2 = 230 - 209 = 21 \, \text{V}

Step 6: Currents Through R2R_2 and R3R_3 (I1I_1 and I2I_2)

Since R2R_2 and R3R_3 are in parallel, the current through each resistor is given by Ohm's Law.

  • For I1I_1 through R2R_2: I1=U2R2=21200=0.105AI_1 = \frac{U_2}{R_2} = \frac{21}{200} = 0.105 \, \text{A}

  • For I2I_2 through R3R_3: I2=U2R3=21200=0.105AI_2 = \frac{U_2}{R_3} = \frac{21}{200} = 0.105 \, \text{A}

Thus, the current through both resistors is the same because R2=R3R_2 = R_3.

Final Results:

  • Total current: ITOT0.209AI_{\text{TOT}} \approx 0.209 \, \text{A}
  • Current through R2R_2 (I1I_1): I1=0.105AI_1 = 0.105 \, \text{A}
  • Current through R3R_3 (I2I_2): I2=0.105AI_2 = 0.105 \, \text{A}
  • Voltage across R1R_1 (U1U_1): U1=209VU_1 = 209 \, \text{V}
  • Voltage across the parallel branch (U2U_2): U2=21VU_2 = 21 \, \text{V}

Would you like further details or clarifications?

Here are five related questions:

  1. How does the total resistance change if R2R_2 or R3R_3 is altered?
  2. What happens to the total current if the total voltage is increased?
  3. Can you describe how to solve for power dissipation in each resistor?
  4. How does the parallel resistor configuration affect the total circuit behavior?
  5. How would the current values change if R1R_1 were decreased?

Tip: Always start by simplifying parallel and series resistances before applying Ohm's Law.

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Math Problem Analysis

Mathematical Concepts

Ohm's Law
Series and Parallel Circuits
Electrical Resistance
Current Division

Formulas

V = IR (Ohm's Law)
R_parallel = (1/R2 + 1/R3)^-1 (Parallel Resistance)
I_total = V_total / R_total (Total Current)
I_1 = V_parallel / R2 (Current in R2)
I_2 = V_parallel / R3 (Current in R3)

Theorems

Ohm's Law
Kirchhoff's Current Law

Suitable Grade Level

Undergraduate - Electrical Engineering or Physics