Which is the correct vector form of the force on q1 due to q2?

Which is the correct vector form of the force on q1 due to q2?

• A. F12 = k q1 q2 (r1 − r2) / |r1 − r2|^3
• B. F12 = k q1 q2 (r2 − r1) / |r1 − r2|^3
• C. F12 = k q1 q2 (r2 − r1) / |r2 − r1|^3
• D. F12 = k q1 q2 (r2 − r1) / |r1 − r2|^3

Answer: (A)

The force on a test charge due to another charge points along the line from the source charge to the test charge, with a magnitude that falls off as 1 over the distance squared. In vector form this becomes F12 = k q1 q2 (r1 − r2)/|r1 − r2|^3. The vector (r1 − r2) points from q2 to q1, so this expression has the correct direction and magnitude: its magnitude is k|q1 q2|/|r1 − r2|^2, and the denominator 3-neatly turns the distance into a unit vector times the right 1/r^2 factor. If you replace (r1 − r2) with (r2 − r1), you flip the direction, giving the opposite force. Using the distance with |r2 − r1|^3 likewise keeps the magnitude the same but still flips the direction when paired with the wrong vector, so it’s not correct.

The force on a test charge due to another charge points along the line from the source charge to the test charge, with a magnitude that falls off as 1 over the distance squared. In vector form this becomes F12 = k q1 q2 (r1 − r2)/|r1 − r2|^3. The vector (r1 − r2) points from q2 to q1, so this expression has the correct direction and magnitude: its magnitude is k|q1 q2|/|r1 − r2|^2, and the denominator 3-neatly turns the distance into a unit vector times the right 1/r^2 factor. If you replace (r1 − r2) with (r2 − r1), you flip the direction, giving the opposite force. Using the distance with |r2 − r1|^3 likewise keeps the magnitude the same but still flips the direction when paired with the wrong vector, so it’s not correct.