Step 1: Conceptual Foundation:
Separating two charges from a distance 'r' to infinity necessitates work equivalent to the change in the system's electrostatic potential energy. This work is precisely the negative of the initial potential energy, as the potential energy at infinite separation is zero.
Step 2: Methodological Framework:
1. Determine the initial separation distance (r) between the charges.
2. Compute the initial potential energy (U\(_i\)) of the two-charge system using the formula:
\[ U_i = k \frac{q_1 q_2}{r} \]3. The work done (W) to achieve infinite separation is calculated as:
\[ W = U_f - U_i \] Given that the final potential energy at infinity (U\(_f\)) is 0, the equation simplifies to:
\[ W = -U_i \]
Step 3: Illustrative Calculation:
Provided Data:
Charge \( q_1 = 4 \, \mu\text{C} = 4 \times 10^{-6} \, \text{C} \)
Charge \( q_2 = -3 \, \mu\text{C} = -3 \times 10^{-6} \, \text{C} \)
Coordinates of \(q_1\): (-6 cm, 0, 0).
Coordinates of \(q_2\): (6 cm, 0, 0).
Calculate the initial distance 'r':
\[ r = 6 \, \text{cm} - (-6 \, \text{cm}) = 12 \, \text{cm} = 0.12 \, \text{m} \]Calculate the initial potential energy U\(_i\):
\[ U_i = \left(9 \times 10^9 \frac{\text{N m}^2}{\text{C}^2}\right) \times \frac{(4 \times 10^{-6} \, \text{C}) \times (-3 \times 10^{-6} \, \text{C})}{0.12 \, \text{m}} \]\[ U_i = \frac{9 \times 10^9 \times (-12 \times 10^{-12})}{0.12} \, \text{J} \]\[ U_i = \frac{-108 \times 10^{-3}}{0.12} \, \text{J} = \frac{-108 \times 10^{-3}}{12 \times 10^{-2}} \, \text{J} \]\[ U_i = -9 \times 10^{-1} \, \text{J} = -0.9 \, \text{J} \]Determine the work done for infinite separation:
\[ W = -U_i = -(-0.9 \, \text{J}) = 0.9 \, \text{J} \]
Step 4: Conclusive Result:
The work required is 0.9 J.