Electric potential gradient in a radial field

Expert intro: The key link is the potential gradient: electric field points toward decreasing potential.

Detailed walkthrough

Core relationship

In electrostatics, the electric field and electric potential are connected by the gradient relation

$E=-\frac{dV}{dr}$

along a chosen radial direction. The minus sign is essential: the field points from higher potential to lower potential.

In a uniform field, this reduces to the familiar idea that the potential changes linearly with distance. In a radial field, however, the field strength generally depends on position, so the potential does not change at a constant rate everywhere.

What changes in a radial field

A radial field is one where field lines point inward or outward from a source, such as around a point charge. In that case, the field magnitude is not constant; it varies with radius. For a point charge,

$E(r)=\frac{kQ}{r^2},$

and therefore the potential is

$V(r)=\frac{kQ}{r} + C.$

Differentiating gives

$-\frac{dV}{dr}=\frac{kQ}{r^2},$

which matches the field strength.

This is the central idea behind the passage: the electric field strength at any point is the negative potential gradient at that point.

Why the minus sign matters

Potential increases in the opposite direction to the electric field. If you move in the direction of the field, the potential decreases. If you move against the field, the potential increases.

That sign convention is not cosmetic; it tells you the direction of force on a positive test charge. A positive charge is pushed toward lower potential, while a negative charge experiences force in the opposite direction.

Practical takeaway

When you are given a potential function, the field comes from the slope of that function. When you are given the field, the potential is found by integrating the field with respect to distance. In radial problems, always be careful about whether the variable is a straight-line distance, a radial coordinate, or a path-dependent quantity.

💡 Pitfall guide

A very common mistake is to treat the radial field like a uniform field and write $E=V/d$ everywhere. That only works when the field is constant over the region. Another trap is to forget the minus sign in $E=-dV/dr$ and then reverse the direction of the field. Students also sometimes confuse electric potential with electric potential energy: potential is energy per unit charge, while potential energy depends on the charge value itself. Keeping those three ideas separate avoids most errors in radial-field questions.

🔄 Real-world variant

If the potential is given as $V(r)=\dfrac{A}{r}$, then the field is found directly by differentiation: $E(r)=-dV/dr=A/r^2$. If the problem instead gives two equipotential radii, you can estimate the field near that region using the local gradient. A useful variant question is: “What happens if the field is not radial but still one-dimensional?” Then the same rule $E=-dV/dx$ applies, but the coordinate changes. The method is identical; only the geometry changes.

🔍 Related terms

potential gradient, radial field, equipotential surface