For a conservative electrostatic field, the potential difference ΔV between two points is related to the electric field by which expression?

For a conservative electrostatic field, the potential difference ΔV between two points is related to the electric field by which expression?

• A. ΔV = ∫_A^B E · dl
• B. ΔV = ∫_A^B E × dl
• C. ΔV = −∫_A^B E · dl
• D. ΔV = ∫_B^A E · dl

Answer: (C)

In a conservative electrostatic field, the electric field is the negative gradient of the potential, E = -∇V. For a small displacement dl, the potential changes by dV = -E · dl. Integrating along a path from point A to point B gives ΔV = V(B) - V(A) = -∫_A^B E · dl. This negative sign is essential: the line integral of E along the path equals the negative of the potential difference, and in a conservative field the result depends only on the endpoints, not the path taken. The other forms either drop the minus sign, use an invalid cross product, or reverse the integration limits, which would not match the standard definition. Therefore, ΔV = -∫_A^B E · dl is the correct relationship.

In a conservative electrostatic field, the electric field is the negative gradient of the potential, E = -∇V. For a small displacement dl, the potential changes by dV = -E · dl. Integrating along a path from point A to point B gives ΔV = V(B) - V(A) = -∫_A^B E · dl. This negative sign is essential: the line integral of E along the path equals the negative of the potential difference, and in a conservative field the result depends only on the endpoints, not the path taken. The other forms either drop the minus sign, use an invalid cross product, or reverse the integration limits, which would not match the standard definition. Therefore, ΔV = -∫_A^B E · dl is the correct relationship.