Determining the Value and Direction of Current in BD using Thevenin Theorem

Introduction:
The Thevenin theorem is a useful technique in electrical engineering that allows us to simplify complex circuits into an equivalent circuit, which consists of a Thevenin voltage source (Vth) and a Thevenin resistance (Rth). By doing so, we can analyze the behavior of a circuit more easily. In this case, we will apply the Thevenin theorem to determine the value and direction of current in BD.

Given Information:
- EMF of the battery (ε) = 20 V
- Internal resistance of the battery (r) = negligible
- Resistance of various arms of the bridge (R1 = 4 Ω, R2 = 6 Ω, R3 = 8 Ω, R4 = 10 Ω)

Step 1: Calculate Thevenin Voltage (Vth):
To determine the Thevenin voltage (Vth), we need to find the open-circuit voltage across the terminals AB. In this case, we can directly consider it as the EMF of the battery (ε) since there is no voltage drop across the internal resistance.

Vth = ε = 20 V

Step 2: Calculate Thevenin Resistance (Rth):
To find the Thevenin resistance (Rth), we need to determine the equivalent resistance between the terminals AB when all the sources are removed. In this case, we will remove the battery and short-circuit terminals AB.

Step 2.1: Simplify the Circuit:
By removing the battery and short-circuiting terminals AB, the circuit simplifies as follows:

```
R1 R2
A ---/\/\/\---/\/\/\--- B
|
R3
|
R4
|
|
---
-
```

Step 2.2: Calculate Rth:
To calculate Rth, we can use the series and parallel resistance formulas. Starting from terminal A, we can calculate the equivalent resistance step by step:

1. R1 and R2 are in parallel:
RP1-2 = (R1 * R2) / (R1 + R2) = (4 Ω * 6 Ω) / (4 Ω + 6 Ω) = 2.4 Ω

2. RP1-2 and R3 are in series:
RS1-3 = RP1-2 + R3 = 2.4 Ω + 8 Ω = 10.4 Ω

3. RS1-3 and R4 are in parallel:
RP1-4 = (RS1-3 * R4) / (RS1-3 + R4) = (10.4 Ω * 10 Ω) / (10.4 Ω + 10 Ω) = 5.2 Ω

Therefore, the Thevenin resistance (Rth) is 5.2 Ω.

Step 3: Determine the Current in BD:
Now that we have the Thevenin voltage (Vth = 20 V) and Thevenin resistance (Rth = 5.2 Ω), we can determine the current in