Scalar Potential Calculator

Enter Field and Point Values

Choose a conservative field family, then enter coefficients and two points. The calculator evaluates scalar potential and related quantities instantly.

Field family

Potential constant C

Quick note

Scalar potential is defined up to an additive constant.

Fₓ coefficient

Fᵧ coefficient

Fz coefficient

a coefficient

b coefficient

c coefficient

Radial strength k

Point P

x₁

y₁

z₁

Point Q

x₂

y₂

z₂

Formula Used

For a conservative vector field F, the scalar potential φ satisfies F = ∇φ. This page evaluates three common families:

Constant field: F = (Fₓ, Fᵧ, Fz), so φ(x,y,z) = Fₓx + Fᵧy + Fzz + C.
Diagonal linear field: F = (ax, by, cz), so φ(x,y,z) = ½ax² + ½by² + ½cz² + C.
Radial field: F = k(x,y,z) / r³, where r = √(x² + y² + z²), so φ(x,y,z) = -k/r + C.

Potential difference is Δφ = φ(Q) − φ(P). For gradient fields, the line integral from P to Q equals the same value and does not depend on the path.

How to Use This Calculator

Select the conservative field family that matches your problem.
Enter the potential constant if your problem includes one.
Fill in the model coefficients for the chosen field.
Enter the coordinates of point P and point Q.
Press the calculation button to see scalar potentials and differences.
Review field vectors, magnitudes, midpoint potential, and line integral.
Use the CSV or PDF buttons to export the result summary.

Model	Coefficients	Point P	Point Q	φ(P)	φ(Q)	Δφ
Constant	F = (3, -2, 5), C = 4	(2, 1, -1)	(4, 0, 2)	3.0000	26.0000	23.0000
Linear	a = 2, b = 4, c = -1, C = 0	(3, 2, 1)	(1, -1, 2)	16.5000	1.0000	-15.5000
Radial	k = 12, C = 0	(3, 4, 0)	(6, 8, 0)	-2.4000	-1.2000	1.2000

FAQs

1. What does scalar potential represent?

Scalar potential is a scalar function whose gradient gives a conservative vector field. It summarizes how the field changes through space.

2. Why is there a constant C?

Adding a constant does not change the gradient. That means many scalar potentials can describe the same conservative field.

3. What is the meaning of Δφ?

Δφ is the potential at Q minus the potential at P. It shows how much the scalar potential changes between two positions.

4. Why does the line integral equal Δφ here?

For conservative fields written as gradients, the fundamental theorem for line integrals applies. The path-independent result equals the potential difference.

5. When should I use the radial model?

Use the radial model for inverse-distance potentials and inverse-square gradient fields. It is common in central-force and potential theory problems.

6. Why is the radial model undefined at the origin?

The radial formula divides by the distance from the origin. At zero distance, that denominator becomes zero, so the expression fails.

7. Can this calculator verify every vector field?

No. This tool evaluates three built-in conservative field families. It does not test arbitrary fields for conservativeness automatically.

8. What do the field vectors at P and Q show?

They show the gradient values at each point. Comparing them helps you see local direction and intensity changes across the domain.