Absence of Magnetic Monopoles

Derivation

Absence of Magnetic Monopoles

D-051 Home PU-102 Threads fields · symmetry Depends on Biot–Savart Law from the Current Force Law, divergence-theorem

Statement

For the magnetostatic field produced by any bounded, steady current distribution J(r′) through the Biot–Savart law, the divergence of the magnetic field vanishes everywhere: ∇·B = 0. Equivalently, the net magnetic flux through any closed surface is zero, so magnetic field lines form closed loops and no isolated magnetic charge (monopole) exists as a source of B.

Why it matters

This is one of Maxwell's four equations, and unlike Gauss's law for electricity (∇·E = ρ/ε₀) it has no source term. That structural asymmetry — charges exist, magnetic monopoles apparently do not — is not an experimental accident to be memorised but a mathematical identity forced by the very form of the Biot–Savart field. Currents can only create dipole-like fields.

Because it holds identically, ∇·B = 0 guarantees a vector potential B = ∇×A exists globally, underpinning gauge theory, magnetostatics, and the topology of field lines. It is also the equation that would first have to break if a genuine monopole were ever found, which is why searches for monopoles are, at heart, tests of this line.

Assumptions

Steady currents: ∂ρ/∂t = 0 so that Biot–Savart applies; if dropped, the field acquires induction and displacement-current terms and the static integral is no longer the field, though ∇·B = 0 still survives as an independent Maxwell equation. Bounded, localised source: the current density J(r′) has compact support, so the volume integral converges and surface terms at infinity vanish; if dropped, integrals may diverge and the manipulation needs regularisation. Sufficient smoothness of J: the integrand is differentiable enough to exchange the derivative and the integral (Leibniz rule) and to apply vector identities pointwise; if dropped, one works distributionally, and the identity still holds in the sense of distributions. No magnetic charge in the model: Biot–Savart is built solely from electric currents; if a monopole density ρ_m were postulated the RHS would become μ₀ρ_m, so the result is a statement about current-sourced fields specifically.

Derivation

B(r) = (μ₀/4π) ∫ J(r′) × (r − r′)/|r − r′|³ d³r′

Start from the Biot–Savart field, taken as an established prior result. The integral runs over source coordinates r′; the field point r is fixed. A

(r − r′)/|r − r′|³ = −∇(1/|r − r′|)

Rewrite the geometric kernel as a gradient. The gradient ∇ acts on the field coordinate r; direct differentiation of |r − r′|⁻¹ gives −(r − r′)/|r − r′|³. A

B(r) = −(μ₀/4π) ∫ J(r′) × ∇(1/|r − r′|) d³r′

Substitute step 2 into step 1. Nothing is evaluated yet; this just recasts the kernel so a curl can be extracted. A

∇×[f c] = (∇f) × c (for constant c) ⇒ J(r′) × ∇f = −∇×[f J(r′)]

Apply the vector identity. Since J(r′) depends only on r′, it is constant with respect to the field-point operator ∇, so it passes through the curl. With f = 1/|r − r′|. B

B(r) = ∇× [ (μ₀/4π) ∫ J(r′)/|r − r′| d³r′ ] ≡ ∇×A

Pull the curl outside the integral (it acts on r, the integration is over r′, and by the smoothness assumption derivative and integral commute). The bracket is the vector potential A(r). B

∇·B = ∇·(∇×A)

Take the divergence of both sides of step 5. This is the operation whose value we want. A

∇·(∇×A) = ∂_i ε_ijk ∂_j A_k = ε_ijk ∂_i∂_j A_k = 0

The divergence of any curl is identically zero: ε_ijk is antisymmetric in i↔j while ∂_i∂_j is symmetric (mixed partials commute for C² fields), so the summed contraction of a symmetric with an antisymmetric tensor vanishes term by term. C

∴ ∇·B(r) = 0 for all r

Combine steps 6 and 7. The result is an identity, independent of the particular current distribution, holding at every field point. A

∮_S B·da = ∫_V ∇·B dV = 0

Integrate over any volume V and apply the divergence theorem (prior result). The vanishing pointwise divergence forces zero net flux through the bounding closed surface S — the integral (global) form. B

Result

∇·B = 0 ⇔ ∮_S B·da = 0

Reading. The magnetic field has no sources or sinks. Field lines never terminate on a point charge as electric lines do; they close on themselves or run off to infinity. Whatever flux enters a closed surface must exactly leave it, so no closed surface can enclose a net "magnetic charge." Because B is a curl, a vector potential A always exists.

Units check. B is in tesla (T = Wb·m⁻²). The divergence carries units T·m⁻¹, and the equation sets this equal to a pure zero — dimensionally consistent for any units. The flux form: [T][m²] = Wb, set to zero weber. Both sides balance.

Limiting cases

Single current loop: the field is a magnetic dipole; lines thread the loop and return outside, closing perfectly — flux through any sphere is zero.
Infinite straight wire: B = μ₀I/(2πs) circles the wire; concentric field lines are closed loops, divergence zero everywhere off the wire.
Uniform field: constant B has zero divergence trivially; lines are straight and parallel, neither beginning nor ending.
Zero current: B = 0, and 0 has zero divergence — the identity is vacuously satisfied.

Breaks when

Hypothetical magnetic monopoles exist. If a magnetic charge density ρ_m is present, Maxwell's equation is modified to ∇·B = μ₀ρ_m, with a nonzero RHS. The Biot–Savart derivation simply does not include such a term, so the proof is silent on that possibility — it proves only that currents cannot source divergence.
Singular field points. At the location of an idealised point dipole or on a line current, B diverges and the pointwise manipulation fails; the correct statement then involves a Dirac-delta contribution, and ∇·B = 0 holds only in the distributional sense.
Non-simply-treated / ill-defined potential regions. Where A cannot be taken as C² (e.g. across an idealised surface current with a field discontinuity), the step "div of curl = 0" must be reinterpreted via matching (jump) conditions rather than naive differentiation.

Failure modes

Differentiating J(r′) with ∇: students apply the field-point operator to the source current and get spurious terms. ∇ acts on r only; J(r′) is constant under it.
Confusing ∇ and ∇′: mixing the field-coordinate and source-coordinate gradients scrambles every sign. Keep them strictly separate.
Sign error in the kernel gradient: forgetting the minus sign in ∇(1/|r−r′|) = −(r−r′)/|…|³ flips the direction of A.
Claiming this proves monopoles cannot exist: it proves current-sourced fields are divergence-free, not that magnetic charge is forbidden. The empirical absence of monopoles is a separate, experimental input.
Invoking the divergence theorem on a region containing a singularity: the flux integral then misses a delta contribution; exclude singular points or handle them distributionally.

Discussion

The deepest content of this derivation is that ∇·B = 0 is not really a law about magnetism at all — it is a mathematical identity, "the divergence of a curl is zero," dressed in physical clothing. The moment the Biot–Savart field is shown to be expressible as ∇×A, its divergence is fixed to vanish with no further physics required. This is exactly parallel to how ∇×E = 0 in electrostatics follows from E = −∇V being a gradient ("the curl of a gradient is zero"). Electric fields are gradients of a scalar and so are curl-free; magnetic fields are curls of a vector and so are divergence-free.

This structural fact is what makes the electric–magnetic asymmetry so striking. Gauss's law ∇·E = ρ/ε₀ has a source because point charges exist; the magnetic analogue would need magnetic charges. Dirac showed in 1931 that the mere existence of a single monopole would quantise electric charge (eg = 2πnℏ), which is why monopoles remain theoretically attractive despite never having been observed. Every experimental search for one is, operationally, a search for a violation of the line derived here.

Physically, the vanishing divergence is why magnetic field lines are always closed loops: with no place to begin or end, a line has nowhere to go but back on itself. Cutting a bar magnet never isolates a north pole; you get two smaller dipoles, because the field is sourced by circulating (bound) currents, and currents make loops, not points. The dipole is the leading multipole; there is no magnetic monopole moment.

At a more advanced level, ∇·B = 0 is a Bianchi identity of the electromagnetic field tensor: writing the homogeneous Maxwell equations as ∂_[μF_νλ] = 0, both ∇·B = 0 and Faraday's law emerge as components of a single geometric identity that follows automatically once F_μν = ∂_μA_ν − ∂_νA_μ is derived from a potential. In differential-forms language, B being sourceless is dF = 0, the statement that F is a closed 2-form — and d² = 0 is the coordinate-free version of "div of a curl is zero." The absence of monopoles and the existence of the potential are the same topological fact.

Common misconceptions. The result does not say "magnetic monopoles are impossible" — it says the Biot–Savart (current) field is divergence-free. It is not an experimental law derived by measuring flux; the flux form is a consequence. And it is not specific to any particular geometry: it holds for the field of any steady current, however complicated, because it flows from an identity, not from a computation.

Worked examples

Flux of a dipole field through a sphere. A small current loop of moment m = 5.0 A·m² sits at the centre of a sphere of radius R = 0.20 m. Find the net magnetic flux through the sphere.

Set-up: the dipole field is B = (μ₀/4π)[3(m·r̂)r̂ − m]/r³. A

Φ = ∮_S B·da = ∫_V ∇·B dV

Apply the divergence theorem before touching any numbers. A

∇·B = 0 everywhere on and outside the loop ⇒ Φ = ∫_V 0 dV = 0

The derived identity kills the integrand; the radius R = 0.20 m and moment m = 5.0 A·m² never enter. A

Φ = 0 Wb

Reading. Every field line leaving the northern hemisphere re-enters through the southern; the sphere encloses a dipole, not a charge. The answer is independent of R and m — a direct signature of ∇·B = 0.

Consistency test of a proposed field. A student proposes B = B₀(x x̂ + y ŷ + z ẑ)/a with B₀ = 0.30 T, a = 0.10 m. Can this be a magnetostatic field?

Test the candidate against the necessary condition ∇·B = 0. A

∇·B = (B₀/a)(∂x/∂x + ∂y/∂y + ∂z/∂z) = 3B₀/a

Differentiate symbolically first; each partial contributes 1. A

3B₀/a = 3(0.30 T)/(0.10 m) = 9.0 T·m⁻¹ ≠ 0

Insert numbers only at the end; the divergence is manifestly nonzero. A

∇·B = 9.0 T·m⁻¹ ≠ 0 ⇒ not a physical B-field

Reading. This "radial" field has a source at the origin like an electric point charge — a magnetic monopole. Since ∇·B = 0 is violated, no steady current can produce it. The proposal must be rejected before any further magnetostatics is attempted.

Problems

Show explicitly that the infinite-wire field B = (μ₀I/2πs) φ̂ in cylindrical coordinates has zero divergence for s > 0.
Solution
In cylindricals, ∇·B = (1/s)∂(sB_s)/∂s + (1/s)∂B_φ/∂φ + ∂B_z/∂z. Here B_s = B_z = 0 and B_φ = μ₀I/2πs depends only on s, not φ. So ∂B_φ/∂φ = 0 and every term is zero: ∇·B = 0 for s > 0. (At s = 0 the field is singular; the wire itself is excluded.)
A cube of side 0.15 m sits in the field of a distant magnet. The flux through five of its faces totals +2.4 × 10⁻⁵ Wb (outward positive). What is the flux through the sixth face?
Solution
Closed-surface flux must vanish: ∮B·da = 0 by ∇·B = 0. Sum of all six faces = 0, so the sixth face carries Φ₆ = −(+2.4 × 10⁻⁵ Wb) = −2.4 × 10⁻⁵ Wb (i.e. 2.4 × 10⁻⁵ Wb inward). The cube's side length is irrelevant.
Determine the constant c so that B = B₀(2x x̂ − 3y ŷ + cz ẑ)/a can be a valid magnetostatic field.
Solution
Require ∇·B = 0. ∇·B = (B₀/a)(2 − 3 + c) = (B₀/a)(c − 1). Setting this to zero gives c = 1. Then the field is source-free and admissible (subject also to satisfying Ampère's law, a separate condition).
Using the divergence theorem, prove that the total flux of any magnetostatic field through a closed torus surface is zero, and state which assumption fails if a monopole sat inside.
Solution
For a closed surface S bounding volume V, ∮_SB·da = ∫_V∇·B dV. Since ∇·B = 0 pointwise throughout V (no singular sources inside), the volume integral is zero, hence the flux is zero regardless of the torus's shape. If a monopole of charge q_m sat inside, the equation would become ∇·B = μ₀ρ_m, and the flux would equal μ₀q_m ≠ 0 — the "no magnetic charge" assumption of the Biot–Savart construction fails.
A magnetic field is claimed to be B = k r̂/r² (radial, like a Coulomb field) with k = 0.050 T·m². Compute the flux through a sphere of radius R = 0.30 m centred on the origin and explain the result physically.
Solution
On the sphere, B is radial with magnitude k/R², and da = r̂ dA, so Φ = (k/R²)(4πR²) = 4πk = 4π(0.050) = 0.63 Wb, independent of R. A nonzero closed-surface flux violates ∇·B = 0; indeed ∇·(r̂/r²) = 4πδ³(r), a point source at the origin. This field is precisely a magnetic monopole of charge q_m = 4πk/μ₀ — not producible by any steady current, so it is unphysical within classical magnetostatics.