Mathematical Physics Vol 1

4.2 Vector field

95

∂ B ∂ t

∂ E ∂ t

1 ε o ρ , div B = 0 , rot E = −

rot B = µ o j + ε o µ o

div E =

,

,

where E ( x , y , z , t ) and B ( x , y , z , t ) are the strength of the electric field and the induction of the magnetic field, respectively, ε 0 is the dielectric constant in the vacuum, µ 0 is the magnetic permeability of the vacuum, and ρ ( x , y , z , t ) and j ( x , y , z , t ) are the charge density and current density, respectively. Observe the second and third equations (which are called sourceless equations in the literature, because they do not include the charge density and current density, which characterize the field sources). Since the divergence of the rotor, of any vector, is identically equal to zero (4.71), we can write that B = rot A div B = div ( rot A )= 0 , where A = A ( x , y , z , t ) . Replacing this in the third Maxwell’s equation, we obtain rot E = − ∂ ∂ t rot A = rot − ∂ A ∂ t . (4.66) As the rotor of the gradient of any scalar function id identically equal to zero (4.69), the values E and ∂ A ∂ t can differ for the gradient of a scalar function Φ , where Φ = Φ ( x , y , z , t ) . Thus E = − ∂ A ∂ t − grad Φ . (4.67) The vector function A ( x , y , z , t ) and scalar function Φ ( x , y , z , t ) are called vector and scalar potential , respectively. In order to identify the physical meaning of the scalar potential, let us assume that the electromagnetic field is stationary, i.e. that it does not to change over time. Then ∂ A ∂ t =0, and thus E = − grad Φ . (4.68) By scalar multiplication by the displacement vector r we obtain E · d r = − grad Φ · d r = − ∂ Φ ∂ x d x − ∂ Φ ∂ y d y − ∂ Φ ∂ z d z = − d Φ , and then by integrating along some path from infinity to the point of space in which we observe the field, we obtain Φ ( x , y , z )= − Z ( x , y , z ) ∞ E · d r . Thus, for a stationary electromagnetic field, the scalar potential represents the work that some external force needs to perform against the electric field in order to bring the unit charge of the same sign as the field source from infinity to the observed point ( x , y , z ) . The value of the scalar potential at infinity is assumed to be zero. Obviously, in the case of a time-varying field, this conclusion is no longer valid. The vector potential A ( x , y , z ) itself has no direct physical interpretation, as opposed to its line integral along some closed contour L . Namely, as Z L A · d l = x S rot A · d S = x S B · d S , according to which changes in the electric field cause changes in the magnetic field and vice versa. He formulated the law of distribution of the velocity of molecules in a gas. He is considered one of the founders of the kinetic theory of gases, along with L. Boltzmann and R. Clausius.