Poynting Vector

As em waves propagate through matter from their source to distant receiving point, there is a energy from source to receiver. The rate at which energy is transmitted through unit area perpendicular to the direction of propagating of energy is called Poynting vector and is represented by P. Let us derive an expression for Poynting Vector.

Two of the Maxwell’s equations are
        ∇ x H = J + ∂D/∂t                        … (1)

        ∇ x E = - ∂B/∂t                            … (2)

Taking dot product of both sides of Eqs. (1) and (2) with E and H respectively,
        E. (∇ x H) = E.J + E.∂D/∂t = E.J + ∂/∂t (1/2 εE²)            … (3)

        H. (∇ x E) = H. (-∂B/∂t) = ∂/∂t (1/2 μH²)                … (4)

Subtracting Eq. (4) from Eq. (3), we get

E. (∇ x H) – H. (∇ x E) = E.J + ∂/∂t (1/2 μH2 + 1/2 εE2)                … (5)

But ∇. (E x H) = H. (∇x E) – E. (∇ x H)
. : - ∇. (E x H) = E.J + ∂/∂t (1/2 μH² + 1/2 εE²)                    … (6)

Let us now consider a volume V enclosed by a surface S. Integrating the above relation over the volume V, we have
    ∂/∂t ∫ (μH²/2 + εE²/2) dV + ∫(E.J) dV = - ∫∇.(EXH) dv
Using Gauss divergence theorem,
    ∫ ∇.( E x H) dV = Ñ„_s (E x H) . dS                        … (7)

The first term on L.H.S of Eq. (7) represents the rate of decrease of energy stored in volume V due to electric and magnetic fields. The second term on L.H.S. represents the rate at which electromagnetic energy is lost through Joule heating. Hence the R.H.S. represents the rate of flow of energy over the surface S enclosing the volume V.

Therefore, E X H gives the rate of flow of energy through unit area enclosing the volume V. This is denoted by P, called Poynting vector.

        . :        P = E X H                    … (8)

The direction of P is perpendicular to both E and H.