Four copper wires of equal length are connected in series. Their cross-sectional areas are 0.96 cm^2, 2.02 cm^2, 2.90 cm^2, and 5.06 cm^2. A potential difference of 116 V is applied across the combination. Determine the voltage across the 2.02 cm^2 wire.
To determine the voltage across the 2.02 cm^2 wire, we need to understand how the potential difference is distributed in a series circuit.
In a series circuit, the total potential difference is divided among the components based on their resistances (or in our case, the resistivities of the wires). The potential difference across each wire depends on its resistance, which is influenced by its cross-sectional area.
To calculate the voltage across the 2.02 cm^2 wire, we need to find the ratio of its resistance to the total resistance of the circuit and then multiply that ratio by the total potential difference applied.
Here's how you can calculate it step by step:
Step 1: Find the resistances of the wires.
The resistance of a wire is inversely proportional to its cross-sectional area. The formula to calculate the resistance (R) is:
R = resistivity * (length / cross-sectional area)
Given that the length is the same for all wires, we can ignore it while comparing resistances.
For the 0.96 cm^2 wire:
R_0.96 = resistivity * (1 / 0.96)
For the 2.02 cm^2 wire:
R_2.02 = resistivity * (1 / 2.02)
For the 2.90 cm^2 wire:
R_2.90 = resistivity * (1 / 2.90)
For the 5.06 cm^2 wire:
R_5.06 = resistivity * (1 / 5.06)
Step 2: Find the total resistance of the circuit.
In a series circuit, the total resistance (R_total) is the sum of individual resistances.
R_total = R_0.96 + R_2.02 + R_2.90 + R_5.06
Step 3: Find the ratio of the resistance of the 2.02 cm^2 wire to the total resistance.
Let's denote the resistance of the 2.02 cm^2 wire as R_x.
Ratio = R_x / R_total
Step 4: Calculate the voltage across the 2.02 cm^2 wire.
The voltage across the 2.02 cm^2 wire (V_x) can be found using the formula:
V_x = Ratio * total potential difference
Now, let's put the values together and calculate:
1. Calculate the resistances:
R_0.96 = resistivity * (1 / 0.96)
R_2.02 = resistivity * (1 / 2.02)
R_2.90 = resistivity * (1 / 2.90)
R_5.06 = resistivity * (1 / 5.06)
2. Calculate the total resistance:
R_total = R_0.96 + R_2.02 + R_2.90 + R_5.06
3. Calculate the ratio:
Ratio = R_2.02 / R_total
4. Calculate the voltage across the 2.02 cm^2 wire:
V_x = Ratio * 116 V
Plug in the values for resistivity and calculate the resistances and the ratio. Finally, multiply the ratio by the total potential difference (116V) to find the voltage across the 2.02 cm^2 wire.