Find the currents in the unbalanced Wheatstone bridge (Fig. 5.1). Assume that v0 = 1.5V,r1 = r2 = 100 ,r3 = 150,rx = 120,ra = 1000, and rs = 10 .
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To find the currents in an unbalanced Wheatstone bridge, we can follow these steps:
Step 1: Understand the Wheatstone Bridge
A Wheatstone bridge is a circuit that is used to measure unknown resistances by comparing them to known resistances. It consists of four resistors arranged in a diamond shape, with a power source connected to two opposite corners, and a galvanometer connected to the other two opposite corners.
In the given circuit (Fig. 5.1), v0 is the power source, r1 and r2 are known resistors, r3 is the unknown resistor, and rx is a variable resistor that can be adjusted to balance the bridge.
Step 2: Analyze the Circuit
To determine the currents in the circuit, we first need to calculate the equivalent resistance of the bridge.
In this case, we can use the formula for the equivalent resistance of a Wheatstone bridge:
Requivalent = [(r1 * r3) - (r2 * rx)] / (r2 + rx)
Plugging in the values, we get:
Requivalent = [(100 * 150) - (100 * 120)] / (100 + 120)
Requivalent = (15000 - 12000) / 220
Requivalent = 3000 / 220
Requivalent ≈ 13.64 Ω
Step 3: Calculate the Currents
Now that we have the equivalent resistance, we can calculate the currents in the bridge using Ohm's Law.
The current flowing from v0 to the junction between r1 and r3 is given by:
I1 = (v0 - v1) / ra
Where v1 is the voltage drop across r1, and ra is the resistance connected to the junction.
To find v1, we can use Ohm's Law:
v1 = I1 * r1
Plugging in the values:
v1 = I1 * 100
The current flowing from the junction to rx is given by:
I2 = (v1 - v2) / rs
Where v2 is the voltage drop across rx, and rs is the resistance connected to the junction.
To find v2, we can use Ohm's Law:
v2 = I2 * rx
The current flowing from rx to r2 is given by:
I3 = (v2 - v3) / r2
Where v3 is the voltage drop across r3.
To find v3, we can use Ohm's Law:
v3 = I3 * r3
Finally, the current flowing from r2 to the junction is given by:
I4 = (v3 - v0) / ra
Step 4: Substitute and Solve
Substitute the values we've calculated into the equations above and solve for the currents.
I1 = (v0 - (I1 * 100)) / 1000
I2 = ((I1 * 100) - (I2 * 120)) / 10
I3 = (I2 * 120 - (I3 * 150)) / 100
I4 = ((I3 * 150) - v0) / 1000
This is a system of equations that can be solved using various methods, such as substitution or matrix methods. Solving this system of equations will give you the currents flowing through the circuit.
Please note that it is important to check the balance of the bridge (i.e., make sure the resistance of the bridge is balanced) before proceeding with the calculations, as an unbalanced bridge may lead to inaccurate results.