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(a) Berechnen Sie den Gesamtwiderstand Rges der Schaltung.
(b) Wie groß sind der Strom I und die Spannung U4?
(c) Berechnen Sie die Spannung U5 in Abhängigkeit vom Gesamtwiderstand Rges. Vereinfachen Sie Ihr Ergebnis soweit wie möglich!


To answer these questions, let's go step by step, analyzing the circuit and solving each part.

The circuit diagram has four resistors \( R_1 \), \( R_2 \), \( R_3 \), and \( R_4 \), arranged as follows:

- \( R_1 \) and the combination of \( R_2 \), \( R_3 \), and \( R_4 \) are connected in series.
- \( R_2 \) and \( R_3 \) are in parallel, and this combination is in series with \( R_4 \).

Let's break down each question.

### (a) Berechnen Sie den Gesamtwiderstand \( R_{\text{ges}} \) der Schaltung.

To find the total resistance \( R_{\text{ges}} \):

1. **Calculate the equivalent resistance of \( R_2 \) and \( R_3 \) in parallel**:
   \[
   R_{23} = \frac{R_2 \cdot R_3}{R_2 + R_3}
   \]

2. **Add \( R_{23} \) and \( R_4 \) in series**:
   \[
   R_{234} = R_{23} + R_4
   \]

3. **Add \( R_1 \) and \( R_{234} \) in series** to find the total resistance:
   \[
   R_{\text{ges}} = R_1 + R_{234}
   \]

### (b) Wie groß sind der Strom \( I \) und die Spannung \( U_4 \)?

1. **Calculate the total current \( I \)** using Ohm's Law:
   \[
   I = \frac{U_0}{R_{\text{ges}}}
   \]

2. **Calculate the voltage \( U_4 \) across \( R_4 \)**. Since \( R_{234} \) (including \( R_4 \)) is in series with \( R_1 \), the voltage across \( R_{234} \) can be found as:
   \[
   U_{234} = I \cdot R_{234}
   \]
   Then, the voltage across \( R_4 \) specifically is:
   \[
   U_4 = I \cdot R_4
   \]

### (c) Berechnen Sie die Spannung \( U_5 \) in Abhängigkeit vom Gesamtwiderstand \( R_{\text{ges}} \). Vereinfachen Sie Ihr Ergebnis soweit wie möglich!

The voltage \( U_5 \) across the entire parallel section (including \( R_2 \), \( R_3 \), and \( R_4 \)) can be expressed in terms of \( R_{\text{ges}} \):

1. Using the total voltage \( U_0 \) and \( R_{\text{ges}} \), we have:
   \[
   U_5 = U_0 - (I \cdot R_1)
   \]
2. Substitute \( I = \frac{U_0}{R_{\text{ges}}} \) into the equation:
   \[
   U_5 = U_0 - \left(\frac{U_0}{R_{\text{ges}}} \cdot R_1\right) = U_0 \left(1 - \frac{R_1}{R_{\text{ges}}}\right)
   \]

This provides \( U_5 \) in terms of \( R_{\text{ges}} \) and \( U_0 \).

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If you provide values for \( R_1 \), \( R_2 \), \( R_3 \), \( R_4 \), and \( U_0 \), I can calculate the numerical answers.

Would you like further details, or have any questions? 

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Here are 5 additional questions that could help expand your understanding of this circuit:

1. How would the total resistance change if an additional resistor was added in parallel with \( R_2 \)?
2. What would happen to the current \( I \) if the voltage \( U_0 \) were doubled?
3. How does the voltage \( U_4 \) compare with the voltage across \( R_2 \) or \( R_3 \)?
4. What is the power dissipated by \( R_4 \) given the current \( I \)?
5. How would you approach solving this if all resistors were variable?

**Tip**: In circuits with both series and parallel components, simplify step-by-step by calculating equivalent resistances for parallel parts first, then adding them with series components.

(a) Berechnen Sie den Gesamtwiderstand Rges der Schaltung. (b) Wie groß sind der Strom I und die Spannung U4? (c) Berechnen Sie die Spannung U5 in Abhängigkeit vom Gesamtwiderstand Rges. Vereinfachen Sie Ihr Ergebnis soweit wie möglich!

Step-by-step solution to calculate the total resistance (Rges), current (I), and voltages (U4, U5) in a series-parallel circuit with four resistors. Explore key concepts in circuit analysis, including Ohm's Law and voltage division, suitable for university-level physics and engineering courses.

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