To solve the problem of calculating the work required to increase the distance between two parallel wires with a water film between them, we will follow these steps:
### Step 1: Understand the Given Data
- Length of each wire, \( L = 10 \, \text{cm} = 0.1 \, \text{m} \)
- Separation between the wires, \( d = 0.5 \, \text{cm} = 0.005 \, \text{m} \)
- Increase in distance, \( \Delta d = 1 \, \text{mm} = 0.001 \, \text{m} \)
- Surface tension of water, \( T = 72 \times 10^{-3} \, \text{N/m} \)
### Step 2: Calculate the Initial and Final Areas
The initial area of the film between the wires is given by:
\[
A_i = L \times d = 0.1 \, \text{m} \times 0.005 \, \text{m} = 0.0005 \, \text{m}^2
\]
After increasing the distance by \( \Delta d \), the new separation becomes:
\[
d_f = d + \Delta d = 0.005 \, \text{m} + 0.001 \, \text{m} = 0.006 \, \text{m}
\]
The final area of the film is:
\[
A_f = L \times d_f = 0.1 \, \text{m} \times 0.006 \, \text{m} = 0.0006 \, \text{m}^2
\]
### Step 3: Calculate the Change in Area
The change in area \( \Delta A \) is:
\[
\Delta A = A_f - A_i = 0.0006 \, \text{m}^2 - 0.0005 \, \text{m}^2 = 0.0001 \, \text{m}^2
\]
### Step 4: Calculate the Work Done
The work done \( W \) to increase the distance between the wires is given by the formula:
\[
W = 2T \Delta A
\]
Here, the factor of 2 accounts for the two surfaces of the film.
Substituting the known values:
\[
W = 2 \times (72 \times 10^{-3} \, \text{N/m}) \times (0.0001 \, \text{m}^2)
\]
\[
W = 2 \times 72 \times 10^{-3} \times 0.0001
\]
\[
W = 2 \times 72 \times 10^{-7} = 144 \times 10^{-7} \, \text{J}
\]
\[
W = 1.44 \times 10^{-5} \, \text{J}
\]
### Final Answer
The work required to increase the distance between the wires is:
\[
\boxed{1.44 \times 10^{-5} \, \text{J}}
\]
