Potential is the energy bookkeeping of an electric field. It is a scalar, it falls off as 1/r, and its gradient gives back the field.
The electric potential at a point is the work done per unit positive charge in bringing it from infinity to that point, V = Q / (4πε₀r). The field is the negative gradient of the potential, E = −dV/dr, and the energy of charge q is Eₙ = qV.
Compare the 1/r potential with the 1/r² field for the same point charge. The potential drops more gently with distance, and where its curve is steepest the field is largest, because E = −dV/dr.
Potential is a scalar, so potentials from several charges add algebraically (with sign). The work done moving a charge q between two points is W = qΔV, and the potential energy of two point charges is Eₙ = Qq / (4πε₀r).
Four quick checks on potential, its 1/r form, and its link to the field. Each correct answer earns XP and lights this skill on your star map.
The electric potential at a point is:
The potential of a point charge varies with distance as:
The electric field is related to the potential by:
The work done moving a charge q through a potential difference ΔV is:
Potential falls off as 1/r, more gently than the 1/r² field, do not confuse the two. Potential is a scalar (add the contributions algebraically), while the field is a vector. The negative sign in E = −dV/dr means the field points from high to low potential.
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