Question

how did we get the following equation (1.9) from maxwells equations

at e at where p is the density of free charges and j is the density of currents at a point where the electric and magnetic fields are evaluated. The parameters and are constants that determine the property of the vacuum and are called the electric permittivity and magnetic permeability respectively The parameter c-1/olo and its numerical value is equal to the speed of light in vacuum,c 3 x 10 (ms In the Maxwells equations, the fields E and B depend on (F. t the charge and current densities also depend on (G. tIt is not explicitly stated in the above equations, but we shall remember about this dependence in the following calculations The fields E and B produced by the source charges p and currents j are found by solving the Maxwells equations (1.1)-(14) Note that the Maxwelrs equations involve two fields that satisty a system of four coupled differential equations. Generally, we do not find fields E and B by a direct integration of the Mawell's equations. We rather first compute scalar and vector potentials from which the fields may be found Let us illustrate the conoept of vector and scalar potentials in the solution of the Maxwel's equations. First, note that the field B always has zero divergence, V-0, and hence we can always write where A is the vector potential Since V × VФ-0, where Φ is an arbitrary scalar function (scalar potential), we find from the Maxwelř's equation (1.3) thatthe electric field can be written as at The electric field (1.6) depends on the specific choice of the potentials However, the Maxwell's equations should be independent of the specific choice of the potentials Substituting Eq. (1.6) into Eq (1.1), we get Hence, the electric field (1.6) will satisfy the Maxwell's equa tion (1.1) when ar If we now substitute Eqs. (1.5) and (1.6 into E (1.4) and expand the double curl V x(V xA) to give V(-A)-VA, we obtain a three- wave equation for the vector

Answer 1

C212