Work required to assemble two point charges q1 and q2 separated by distance r from infinity to a fixed separation r equals:

Work required to assemble two point charges q1 and q2 separated by distance r from infinity to a fixed separation r equals:

• A. U = k q1 q2 / r
• B. U = (1/2) k q1 q2 / r
• C. U = k (q1 + q2) / r
• D. U = k q1 q2 r

Answer: (A)

The work required to assemble two point charges from infinity to a fixed separation is the Coulomb potential energy of the pair, which is proportional to the product of the charges and inversely proportional to their separation: U = k q1 q2 / r. Imagine bringing the second charge in from infinity to a distance r from the first. The force between them is F = k q1 q2 / d^2, where d is the current separation. The small amount of work to move it a little closer is dW = F dd, and integrating from infinity down to r gives W = ∫∞^r k q1 q2 / d^2 dd = k q1 q2 / r. That final value is the energy stored in the field, i.e., the work you must do to assemble the configuration. The sign depends on the charges: like signs give a positive energy, opposite signs give negative energy. The other forms don’t fit because they either change how the energy scales with distance or involve the sum of the charges instead of their product, and they include a factor or dependence that isn’t part of the Coulomb interaction.

The work required to assemble two point charges from infinity to a fixed separation is the Coulomb potential energy of the pair, which is proportional to the product of the charges and inversely proportional to their separation: U = k q1 q2 / r.

Imagine bringing the second charge in from infinity to a distance r from the first. The force between them is F = k q1 q2 / d^2, where d is the current separation. The small amount of work to move it a little closer is dW = F dd, and integrating from infinity down to r gives W = ∫∞^r k q1 q2 / d^2 dd = k q1 q2 / r. That final value is the energy stored in the field, i.e., the work you must do to assemble the configuration. The sign depends on the charges: like signs give a positive energy, opposite signs give negative energy.

The other forms don’t fit because they either change how the energy scales with distance or involve the sum of the charges instead of their product, and they include a factor or dependence that isn’t part of the Coulomb interaction.