arXiv:physics/0611202v1 [physics.ed-ph] 22 Nov 2006 na dniyfrtevlm nerlo h qaeo vecto a of square the of integral volume the for identity an On eateto hsc,Uiest fWsosnMdsn M Wisconsin-Madison, of University Physics, of Department ed eako ae yA .Stewart M. A. by paper a on Remark field. oa Durand Loyal 1 ∗ dsn icni 53706 Wisconsin adison, r Gubarev, Stodolsky, and Zakharov1 have noted the following identity for vector fields that vanish sufficiently rapidly at spatial infinity,

1 1 d3xA(x) · A(x = d3xd3x′ ∇ · A(x) ∇′ · A(x′) Z Z x x′ 4π | − |h

+ ∇ × A(x) · ∇′ × A(x′) , (1)

i and used it to investigate properties of the vector potential in quantum field theory. Stewart2 has shown that the identity is also of general interest in classical electromagnetic theory. It can be used, for example, to derive easily interpreted expressions for the energies in time-dependent electric and magnetic fields. The existence of this identity is not obvious. In Ref. 1 it is not proven, but it is noted that it follows from the momentum-space relation (k ×A˜)2 = k2A˜2 −(k ·A˜)2 where A˜(k) is the Fourier transform of the vector field. The position-space derivation given by Stewart is based on the Helmholtz decomposition of a three-dimensional vector field into irrotational and solenoidal parts. We give here an alternative position-space derivation that uses only familiar operations starting from the relations

1 1 δ3(x − x′)= − ∇2 (2) 4π |x − x′|)

for the Dirac delta function, and

∇ × (∇ × A)= ∇(∇ · A) −∇2A. (3)

Both are familiar to students and are used in the solution of Poisson’s equation for the potential of a point charge and the derivation of the wave equation from Maxwell’s equations. By using Eq. (2), we obtain

d3xA(x) · A(x)= d3xd3x′A(x) · δ3(x − x′)A(x′) Z Z 1 3 3 ′A x ′2 1 A x′ = − d xd x ( ) · ∇ ′ ( ). (4) 4π Z  |x − x | If we integrate by parts over x′ and use the assumed rapid vanishing of A(x′) for |x′|→∞, we can transfer the action of the derivatives from the factor 1/|x − x′| to A(x′) without acquiring extra surface terms. We use the identity (3) for the double curl, make another

2 partial integration, and rewrite the result as

1 1 ′ d3xA(x)·A(x)= − d3xd3x′A(x) · ∇ 2A(x′) (5a) Z 4π Z |x − x′| 1 1 = − d3xd3x′A(x) · ∇′ ∇′ · A(x′) − ∇′ × ∇′ × A(x′) Z x x′ 4π | − |h

i 1 3 3 ′ A x ∇′ 1 ∇′ A x′ = d xd x ( ) · ′ ( · ( ) 4π Z h |x − x | A x ∇′ 1 ∇′ A x′ − ( ) · x x′ × × ( ) . (5b)  | − |
i Now ∇′(1/|x − x′|)= −∇(1/|x − x′|), so with some rearrangement of the vector products, Eq. (5b) becomes

1 1 d3xA(x) · A(x) = − d3xd3x′ A(x) · ∇ ∇′ · A(x′) Z Z x x′ 4π h | − |

∇ 1 A x ∇′ A x′ + x x′ × ( ) · × ( ) . (6) | − |

i A final partial integration with respect to x gives the desired identity. The derivation generalizes easily to give an identity of the same form for the volume integral of the product of two different rapidly decreasing vector fields.

∗ Electronic address: [email protected] 1 F. V. Gubarev, L. Stodolsky, and V. I. Zakarov, “On the significance of the vector potential squared,” Phys. Rev. Lett. 86 (11), 2220–2222 (2001). 2 A. M. Stewart, “On an identity for the volume integral of the square of a vector field,” Am. J. Phys. (preceding paper).

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