ABSTRACT
From the electromagnetic force equation (13.105), we can write the equation for the force density f as follows: f = ρ E + J × B . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn13a_1.jpg"/> Assume the medium is free space. We can eliminate the volume charge density ρ and the current density J by using Maxwell’s equations, and after using some vector identities, we can write [1] f = ρ E + J × B = ∇ ⋅ T ¯ − ε 0 μ 0 ∂ S ∂ t , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn13a_2.jpg"/> where the tensor T ¯ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqi13a_1.jpg"/> , called the Maxwell stress tensor, in free space medium is given by [1–3] T i j = ε 0 E i E j − 1 2 δ i j E 2 + 1 μ 0 B i B j − 1 2 δ i j B 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn13a_3.jpg"/> The first term on the right side of (13A.3) can be called the electric stress tensor [T e ]. If written out in matrix form in Cartesian coordinates, [ T e ] = ε 0 [ 1 2 ( E x 2 − E y 2 − E z 2 ) E x E y E x E z E x E y 1 2 ( E y 2 − E z 2 − E x 2 ) E y E z E x E z E y E z 1 2 ( E z 2 − E x 2 − E y 2 ) ] . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqn13a_4.jpg"/> The second term on the right side (13A.3) can be similarly written out. The term S in (13A.2) is the usual Poynting vector in free space given by (1.26).
