2 Maxwell’s Equations James Clerk Maxwell (1837-1879) gathered all prior knowledge in electromagnetics and summoned the whole theory of electromagnetics in four equations, called the Maxwell’s equations. To evolve the Maxwell’s equations we start with the fundamental postulates of electrostatics and magnetostatics. 3) What are the two Maxwell's equations for magnetostatic fields? Explain their principles (the meaning behind each). 4) Give 3 approaches to determine the electrostatic field.
Start from Maxwell’s equations in the form
[ begin{align} & operatorname{curl}(vec{text{E}}) =-frac{partial vec{text{B}}}{partial text{t}}, label{7.1} & operatorname{div}(vec{text{B}}) =0, nonumber & operatorname{curl}(vec{text{B}}) =mu_{0} vec{text{J}}_{T}+mu_{0} epsilon_{0} frac{partial vec{text{E}}}{partial text{t}}, nonumber &
operatorname{div}(vec{text{E}}) =frac{1}{epsilon_{0}} rho_{T}, nonumber end{align}]
operatorname{div}(vec{text{E}}) =frac{1}{epsilon_{0}} rho_{T}, nonumber end{align}]
where
[rho_{T}=rho_{f}-operatorname{div}(vec{text{P}}) , label{7.2}]
and
[vec{text{J}}_{T}=vec{text{J}}_{f}+operatorname{curl}(vec{text{M}})+frac{partial vec{text{P}}}{partial text{t}} . label{7.3}]
Recall that ρf is the density of free charges, (vec{text{J}}_{f}) is the free current density due to the motion of the free charges, (vec P) is the electric dipole moment density, and (vec M) is the magnetic dipole density. It is presumed that the total charge density, ρT , and the total current density, (vec{text{J}}_{T}), are prescribed functions of position and of time. The equation div((vec B)) = 0 can be satisfied by setting
[vec{text{B}}=operatorname{curl}(vec{text{A}}). label{7.4}]
because the divergence of any curl is equal to zero. The first of Equations (ref{7.1}) becomes, with the help of Equation (ref{7.4}),
[operatorname{curl}(vec{text{E}})=-frac{partial}{partial text{t}} operatorname{curl}(vec{text{A}})=-operatorname{curl}left(frac{partial vec{text{A}}}{partial text{t}}right), nonumber ]
where it has been assumed that the order of the space and time derivatives can be interchanged. It follows that the curl of the sum of the electric field and the time derivative of the vector potential is zero,
[operatorname{curl}left(vec{text{E}}+frac{partial vec{text{A}}}{partial text{t}}right)=0 . label{7.5}]
The curl of any gradient is zero so that the requirement Equation (ref{7.5}) can be satisfied by putting
[vec{text{E}}+frac{partial vec{text{A}}}{partial text{t}}=-operatorname{grad} text{V}, nonumber ]
or
[vec{text{E}}=-operatorname{vec{text{grad}}} text{V}-frac{partial vec{text{A}}}{partial text{t}} . label{7.6} ]
The introduction of the vector potential, (vec A), and the scalar potential, (vec V), enables one to satisfy the first two of Maxwell’s equations (ref{7.1}). Write (vec E) and (vec B) in terms of the potentials in the second pair of Maxwell’s equations to obtain
[operatorname{curl} operatorname{curl}(vec{text{A}})=mu_{0} vec{text{J}}_{T}+epsilon_{0} mu_{0}left(-operatorname{vec{text{grad}}} frac{partial text{V}}{partial text{t}}-frac{partial^{2} vec{text{A}}}{partial text{t}^{2}}right) , nonumber ]
and
[-operatorname{div} operatorname{vec{text{grad}}}text{V}-frac{partial}{partial text{t}} operatorname{div}(vec{text{A}})=frac{rho_{T}}{epsilon_{0}} . nonumber ]
In cartesian co-ordinates, but only in cartesian co-ordinates, the vector operator curl curl can be written
[curl curl=-nabla^{2}+ operatorname{vec{text{grad}}} div. label{7.7}]
Using Equation (ref{7.7}) one obtains
[-nabla^{2} vec{text{A}}+epsilon_{0} mu_{0} frac{partial^{2} vec{text{A}}}{partial text{t}^{2}}+operatorname{vec{text{grad}}}left(operatorname{div}(vec{text{A}})+epsilon_{0} mu_{0} frac{partial text{V}}{partial text{t}}right)=mu_{0} vec{text{J}}_{T} . label{7.8}]
In order to completely specify a vector field one must give both its curl and its divergence. But at this point only the curl of (vec A) has been fixed by the requirement that (vec B) = curl((vec A)); one is still free to impose some constraint on the divergence of (vec A). It is convenient to choose the vector potential so that it satisfies the condition
[ operatorname{div}(vec{text{A}})+epsilon_{0} mu_{0}left(frac{partial text{V}}{partial text{t}}right)=0. label{7.9}]
This choice of div((vec A)) is called the Lorentz gauge. In the Lorentz gauge Equation (ref{7.8}) simplifies to become
![Electrostatics Electrostatics](/uploads/1/1/8/4/118402850/533586500.gif)
[nabla^{2} vec{text{A}}-epsilon_{0} mu_{0} frac{partial^{2} vec{text{A}}}{partial text{t}^{2}}=-mu_{0} vec{text{J}}_{T} , label{7.10} ]
or in component form
[ begin{array}{l} &nabla^{2} text{A}_{text{x}}-epsilon_{0} mu_{0} frac{partial^{2} text{A}_{text{x}}}{partial text{t}^{2}}=-left.mu_{0} text{J}_{text{T}}right|_{x}, label{7.11} & nabla^{2} text{A}_{text{y}}-epsilon_{0} mu_{0} frac{partial^{2} text{A}_{text{y}}}{partial text{t}^{2}}=-left.mu_{0} text{J}_{text{T}}right|_{y}, & nabla^{2} text{A}_{text{z}}-epsilon_{0} mu_{0} frac{partial^{2} text{A}_{text{z}}}{partial text{t}^{2}}=-left.mu_{0} text{J}_{text{T}}right|_{z}, end{array} ]
Similarly, if the last of Maxwell’s Equations (ref{7.1}) is combined with Equation (ref{7.6}) and with the Lorentz condition (ref{7.9}) one finds
![Electrostatic Electrostatic](/uploads/1/1/8/4/118402850/941866189.png)
[nabla^{2} text{V}-epsilon_{0} mu_{0} frac{partial^{2} text{V}}{partial text{t}^{2}}=-frac{rho_{T}}{epsilon_{0}} . label{7.12}]
Obviously, the four equations (ref{7.11} plus ref{7.12}) are very similar and the form of a solution that satisfies one of them must also satisfy the other three. (The fact that Ax, Ay, Az, V all satisfy equations of the same form is no accident: according to the special theory of relativity these four quantities are related to the four components of a single vector in four-dimensional space-time). Consider the homogeneous equation
[nabla^{2} text{V}-epsilon_{0} mu_{0} frac{partial^{2} text{V}}{partial text{t}^{2}}=0 ; nonumber ]
or, since c2 = 1/((epsilon)0µ0),
[nabla^{2} text{V}-frac{1}{text{c}^{2}} frac{partial^{2} text{V}}{partial text{t}^{2}}=0. label{7.13}]
This equation is called the wave equation. A spherically symmetric solution that satisfies the wave equation is
[text{V}=frac{text{f}(text{t}-[text{r} / text{c}])}{text{r}} . label{7.14}]
where f(x) is any function whatsoever. It is instructive to substitute the function (ref{7.14}) into the wave equation. Since the function does not depend upon either of the angular co-ordinates, θ or (phi), the Laplacian operator becomes
[nabla^{2} text{V}=frac{1}{text{r}^{2}} frac{partial}{partial text{r}}left(text{r}^{2} frac{partial text{V}}{partial text{r}}right) . nonumber ]
Inserting the function (7.14) one obtains
[frac{partial text{V}}{partial text{r}}=-frac{f}{text{r}^{2}}-frac{dot{text{f}}}{text{cr}} , nonumber ]
since
[frac{partial text{f}}{partial text{r}}=left(frac{partial text{f}}{partial text{t}}right)left(frac{partial}{partial text{r}}left[text{t}-frac{text{r}}{text{c}}right]right)=-frac{dot{text{f}}}{text{c}} . nonumber ]
Therefore
[r^{2} frac{partial V}{partial r}=-f-frac{r dot{f}}{c} , nonumber ]
and
[frac{partial}{partial text{r}}left(text{r}^{2} frac{partial text{V}}{partial text{r}}right)=-frac{partial text{f}}{partial text{r}}-frac{dot{text{f}}}{text{c}}+frac{text{r} ddot{text{f}}}{text{c}^{2}}=frac{text{r} ddot{text{f}}}{text{c}^{2}} , nonumber ]
thus
[nabla^{2} text{V}=frac{ddot{text{f}}}{text{rc}^{2}} . nonumber ]
But
[frac{partial^{2} V}{partial t^{2}}=frac{ddot{f}}{r c^{2}} ,nonumber ]
and therefore the wave equation (ref{7.13}) is satisfied by a potential function of the form Equation (ref{7.14}) where f(x) is an arbitrary function of its argument, x. Apart from the appearance of the retarded time, tR = t − r/c, the form of Equation (ref{7.14}) is very similar to the potential function for a point charge. It is therefore natural to suppose that the potential function that is generated by a time-varying point charge q(t) located at the origin is given by
[text{V}(text{r}, text{t})=frac{1}{4 pi epsilon_{0}} frac{text{q}(text{t}-text{r} / text{c})}{text{r}}, label{7.15}]
where the value of the charge at the retarded time must be used to calculate the potential at the time of observation, t: the retarded time must be used in order to allow for the finite time required to propagate a signal from the charge to the observer at the speed of light. The notion of a time-dependent charge is an unusual one: think of a tiny volume at the origin into which charge can flow with time. Then the potential function (ref{7.15}) describes the contribution to the potential at the position of the observer due to the charge in that tiny volume element at the origin. The potential function (ref{7.15}) goes
over into the electrostatic potential for a point charge if the observer is so close to the origin that (r/c) can be neglected, or if the charge q is independent of the time.
The elementary solution (ref{7.15}) of the wave equation can be used, together with the principle of superposition, to construct a particular solution of the wave equation given a space and time varying distribution of charge density (see Figure (7.2.1)):
[text{V}(vec{text{R}}, text{t})=frac{1}{4 pi epsilon_{0}} int int int_{S p a c e} text{d} tau frac{rho_{T}left(vec{text{r}}, text{t}_{text{R}}right)}{|vec{text{R}}-vec{text{r}}|} , label{7.16}]
where d(tau) is the element of volume and the retarded time is given by
[text{t}_{text{R}}=text{t}-frac{|vec{text{R}}-vec{text{r}}|}{text{c}} , label{7.17}]
It may be helpful to write out Equation (ref{7.16}) explicitly in cartesian co-ordinates (see Figure (7.2.1):
[text{V}_{text{P}}(text{X}, text{Y}, text{Z}, text{t})=frac{1}{4 pi epsilon_{0}} int int int_{S p a alpha e} operatorname{dxdydz} frac{rho_{T}left(text{x}, text{y}, text{z}, text{t}_{text{R}}right)}{sqrt{(text{X}-text{x})^{2}+(text{Y}-text{y})^{2}+(text{Z}-text{z})^{2}}} , nonumber ]
Maxwell's Two Equations For Electrostatic Fields Two
where
[text{t}_{text{R}}=text{t}-frac{sqrt{(text{X}-text{x})^{2}+(text{Y}-text{y})^{2}+(text{Z}-text{z})^{2}}}{text{c}}, nonumber ]
If Equation (ref{7.16}) is the required solution of the inhomogeneous wave equation (ref{7.12}) for the potential function (text{V}(vec{text{R}}, text{t})), then by analogy the solution of each of the three Equations (ref{7.11}) must have the same form. The particular solution for the vector potential that is generated by the current density (vec{text{J}}_{T}(vec{text{r}}, t)) is given by
[vec{text{A}}(vec{text{R}}, text{t})=frac{mu_{0}}{4 pi} int int int_{S p a c e} text{d} tau frac{vec{text{J}}_{T}left(vec{text{r}}, text{t}_{text{R}}right)}{|vec{text{R}}-vec{text{r}}|} . label{7.18}]
Here again tR is the retarded time. These solutions, which satisfy Maxwell’s equations for the case in which the charge and current distributions depend upon time, have exactly the same form as the solution for the electrostatic potential, Equation (2.2.4), and the solution for the magnetostatic vector potential, Equation (4.1.13), except that the retarded time must be used in the source terms. The presence of the retarded time in the integrals makes the calculation of the scalar and vector potentials much more complicated than the equivalent calculations for the static limit. It can be shown, after much work, that the potential functions (ref{7.16}) and (ref{7.18}) satisfy the Lorentz condition, Equation (ref{7.9}).
The Electromagnetic Field Tensor
The transformation of electric and magnetic fields under a Lorentz boost we established even beforeEinstein developed the theory of relativity.We know that E-fields can transform into B-fields and vice versa.For example, a point charge at rest gives an Electric field.If we boost to a frame in which the charge is moving, there is an Electric and a Magnetic field.This means that the E-field cannot be a Lorentz vector.We need to put the Electric and Magnetic fields together into one (tensor) object to properly handle Lorentz transformationsand to write our equations in a covariant way.
The simplest way and the correct way to do this is to makethe Electric and Magnetic fields components of a rank 2 (antisymmetric) tensor.
The fields can simply be written in terms of the vector potential, (which is a Lorentz vector) .
Note that this is automatically antisymmetric under the interchange of the indices.As before, the first two (sourceless) Maxwell equations are automatically satisfied for fields derived from a vector potential.We may write the other two Maxwell equations in terms of the 4-vector .
Which is why the T-shirt given to every MIT freshman when they take Electricity and Magnetism should say
``... and God said and there was light.'
Of course he or she hadn't yet quantized the theory in that statement.
For some peace of mind, lets verify a few terms in the equations.Clearly all the diagonal terms in the field tensor are zero by antisymmetry.Lets take some example off-diagonal terms in the field tensor, checking the (old) definition of the fields in terms of the potential.
Electrostatic Work Equation
Lets also check what the Maxwell equation says for the last row in the tensor.
Maxwell's Two Equations For Electrostatic Fields Worksheet
Electrostatic Field Energy
Electrostatics Equation Sheet
We will not bother to check the Lorentz transformation of the fields here. Its right.
Jim Branson2013-04-22