Which expression gives Coulomb's law for the force between two point charges, and what are the SI units of the constant k?

Which expression gives Coulomb's law for the force between two point charges, and what are the SI units of the constant k?

• A. F = k q1 q2 / r^2; k = 1/(4π ε0) ≈ 8.99×10^9 N m^2/C^2
• B. F = k q1 q2 / r^3; k = 1/(4π ε0) ≈ 8.99×10^9 N m^2/C^2
• C. F = k |q1 − q2| / r^2; k = ε0
• D. F = k |q1 q2| / r; k = ε0

Answer: (A)

The force between two point charges in vacuum follows an inverse-square law: it is proportional to the product of the charges and falls off as the square of the distance between them. This reflects how the electric influence spreads spherically in three dimensions and is consistent with Gauss’s law. The correct expression is F = k q1 q2 / r^2, with k equal to 1/(4π ε0). Numerically, k ≈ 8.99×10^9 N m^2/C^2. The units of k are N m^2/C^2. This setup makes sense because the left-hand side is a force in newtons, the charges multiply to give Coulomb-squared, and dividing by r^2 (meters squared) yields the correct Newton units when multiplied by k. The sign of the force depends on the signs of q1 and q2: like charges repel, opposite charges attract, but the magnitude is always proportional to |q1 q2|. Why other forms don’t fit: replacing r^2 with r^3 would change the spatial decay to a different power, which doesn’t match how the field expands in space. Using the difference |q1 − q2| would imply the force depends on how charges differ rather than on their product, which isn’t how point charges interact. And using a simple ε0 in place of the full constant 1/(4π ε0) would mis-scale the magnitude, since the vacuum permittivity together with 4π factors sets the correct numerical value for the force. So the expression with the product q1 q2, divided by r^2, and scaled by 1/(4π ε0) is the correct form, with k having units of N m^2/C^2.

The force between two point charges in vacuum follows an inverse-square law: it is proportional to the product of the charges and falls off as the square of the distance between them. This reflects how the electric influence spreads spherically in three dimensions and is consistent with Gauss’s law.

The correct expression is F = k q1 q2 / r^2, with k equal to 1/(4π ε0). Numerically, k ≈ 8.99×10^9 N m^2/C^2. The units of k are N m^2/C^2. This setup makes sense because the left-hand side is a force in newtons, the charges multiply to give Coulomb-squared, and dividing by r^2 (meters squared) yields the correct Newton units when multiplied by k. The sign of the force depends on the signs of q1 and q2: like charges repel, opposite charges attract, but the magnitude is always proportional to |q1 q2|.

Why other forms don’t fit: replacing r^2 with r^3 would change the spatial decay to a different power, which doesn’t match how the field expands in space. Using the difference |q1 − q2| would imply the force depends on how charges differ rather than on their product, which isn’t how point charges interact. And using a simple ε0 in place of the full constant 1/(4π ε0) would mis-scale the magnitude, since the vacuum permittivity together with 4π factors sets the correct numerical value for the force.

So the expression with the product q1 q2, divided by r^2, and scaled by 1/(4π ε0) is the correct form, with k having units of N m^2/C^2.