The electric field at a point due to a collection of static point charges is given by:

The electric field at a point due to a collection of static point charges is given by:

• A. E = ∑ k qi (ri − r) / |ri − r|^3
• B. E = ∑ k qi (r − ri) / |r − ri|^3
• C. E = ∑ qi / (4π ε0 r^2)
• D. E = ∑ k qi / |ri − r|^2

Answer: (B)

The electric field at a point due to several static point charges is found by treating each charge’s contribution as a vector that points from the charge to the field point and then adding those contributions. For a single point charge q_i located at r_i, the field at a point r is E_i(r) = (1/(4πϵ0)) q_i (r − r_i) / |r − r_i|^3. The direction is along the line from the charge toward the point where you’re evaluating the field, and the magnitude falls off with the square of the distance. Putting all charges together, the total field is E(r) = ∑ (1/(4πϵ0)) q_i (r − r_i) / |r − r_i|^3. This matches the correct option because it uses the vector r − r_i, not its opposite, and it includes the proper 1/|r − r_i|^3 factor to produce the correct magnitude and direction for each contribution, then sums them. The other forms miss either the vector nature or the correct direction. Using ri − r would reverse the direction of each contribution. A scalar form like qi/(4πϵ0 r^2) ignores direction entirely and doesn’t apply to multiple charges. A form like qi/(4πϵ0 |ri − r|^2) also lacks the necessary vector structure and uses the wrong power.

The electric field at a point due to several static point charges is found by treating each charge’s contribution as a vector that points from the charge to the field point and then adding those contributions.

For a single point charge q_i located at r_i, the field at a point r is E_i(r) = (1/(4πϵ0)) q_i (r − r_i) / |r − r_i|^3. The direction is along the line from the charge toward the point where you’re evaluating the field, and the magnitude falls off with the square of the distance.

Putting all charges together, the total field is E(r) = ∑ (1/(4πϵ0)) q_i (r − r_i) / |r − r_i|^3. This matches the correct option because it uses the vector r − r_i, not its opposite, and it includes the proper 1/|r − r_i|^3 factor to produce the correct magnitude and direction for each contribution, then sums them.

The other forms miss either the vector nature or the correct direction. Using ri − r would reverse the direction of each contribution. A scalar form like qi/(4πϵ0 r^2) ignores direction entirely and doesn’t apply to multiple charges. A form like qi/(4πϵ0 |ri − r|^2) also lacks the necessary vector structure and uses the wrong power.