Apply the superposition principle to two point charges q1 at r1 and q2 at r2: What is the electric field at position r?

Apply the superposition principle to two point charges q1 at r1 and q2 at r2: What is the electric field at position r?

• A. E = E1 − E2; E_i = (1/(4π ε0)) q_i (r − r_i)/|r − r_i|^3
• B. E = E1 × E2; E_i = (1/(4π ε0)) q_i (r_i − r)/|r − r_i|^2
• C. E = E1 + E2; E_i = (1/(4π ε0)) q_i (r_i − r)/|r − r_i|^3
• D. E = E1 + E2; E_i = (1/(4π ε0)) q_i (r − r_i)/|r − r_i|^3

Answer: (D)

The main idea here is that electric fields add linearly when multiple sources are present—the superposition principle. For a point charge q_i at position r_i, the field at r is E_i(r) = (1/(4π ε0)) q_i (r − r_i)/|r − r_i|^3. The total electric field at r from two charges is the sum of the individual fields: E(r) = E1(r) + E2(r), with E1 = (1/(4π ε0)) q1 (r − r1)/|r − r1|^3 and E2 = (1/(4π ε0)) q2 (r − r2)/|r − r2|^3. This is the correct expression because the field from each charge points along the line from the charge toward the point, with magnitude k|q|/|r − r_i|^2. The other forms misapply the direction or the operation. Subtracting the second field would not reflect superposition, a product of fields is not how fields combine, and using r_i − r would flip the direction of each field, which is inconsistent with the standard convention for E from a positive charge.

The main idea here is that electric fields add linearly when multiple sources are present—the superposition principle. For a point charge q_i at position r_i, the field at r is E_i(r) = (1/(4π ε0)) q_i (r − r_i)/|r − r_i|^3. The total electric field at r from two charges is the sum of the individual fields: E(r) = E1(r) + E2(r), with E1 = (1/(4π ε0)) q1 (r − r1)/|r − r1|^3 and E2 = (1/(4π ε0)) q2 (r − r2)/|r − r2|^3. This is the correct expression because the field from each charge points along the line from the charge toward the point, with magnitude k|q|/|r − r_i|^2.

The other forms misapply the direction or the operation. Subtracting the second field would not reflect superposition, a product of fields is not how fields combine, and using r_i − r would flip the direction of each field, which is inconsistent with the standard convention for E from a positive charge.