Maxwell’s Equations in Point Form

Section 42.2 Maxwell’s Equations in Point Form

Multivariate Calculus Required.

To read this section you should review the following topics from your Calculus book.

The Gradient of a Function; Variation of a function in an arbitrary direction.
The Divergence of a Vector Field; Physical Meaning of Divergence; Divergence and Flux.
The Curl of a Vector Field; Physical Meaning of Curl; Circulation of a field.

Subsection 42.2.1 Maxwell’s Equations

In our studies of the electricity and magnetism we have extensively studied two vector fields, namely the electric and magnetic fields \(\vec E\) and \(\vec B\text{.}\)

You may recall that we had introduced the concept of fields to get away from the action at a distance present in the non-field formulation, such as the Coulomb’s force law and Newton’s law of universal gravitation. We also found that experimental observations are usually interpreted in terms of laws that integrate over these fields. With the correction by Maxwell to Ampere’s Law we have following four laws of electric and magnetic field in the integral form.

\begin{align} \amp \textrm{Gauss's Law for }\vec E: \ \ \oint \vec E\cdot d\vec A = \frac {Q_{\textrm{enc}}} {\epsilon_0} \tag{42.6}\\ \amp \textrm{Gauss's Law for }\vec B:\ \ \oint \vec B\cdot d\vec A =0 \tag{42.7}\\ \amp \textrm{Faraday's Law:}\ \ \oint \vec E\cdot d\vec l = -\frac {d \Phi_B}{d t} \tag{42.8}\\ \amp \textrm{Ampere-Maxwell's Law:}\ \ \oint \vec B\cdot d\vec l =\mu_0 I_{\textrm{enc}} +\mu_0\epsilon_0\frac {d \Phi_E}{d t} \tag{42.9} \end{align}

The integral forms are written in terms of the flux and circulations of the electric and magnetic fields. We have also learned that the flux and circulations of vector fields are related to their divergence and curl. More specifically, flux per unit volume through a closed surface is equal to the divergence of the field, and circulation per unit area of a loop is equal to the component of the curl of the vector field in the direction perpendicular to the loop. We can use these results from multivariate calculus to deduce equations that applicable at each point in space and at a particular instant in time, i.e., localized or point-form.

As an example, let us apply Gauss’s law given in Eq. (42.6) to a closed surface \(S\) around an arbitrary space point P which may or may not have any charge in the volume \(V\) enclosed by the surface \(S\text{.}\) Suppose we divide both sides by the volume \(V\) and take the infinitesimal volume limit.

\begin{equation*} \lim_{V\rightarrow 0}\left[ \frac{1}{V} \oint_S \vec E\cdot d\vec A\right] = \frac{1}{\epsilon_0} \lim_{V\rightarrow 0}\left[ \frac{Q_{\textrm{enc}}}{V}\right] \end{equation*}

The limit on the left side will give the flux per unit volume at the point P which is equal to the divergence of the electric field at that point, and the limit on the right side will give charge per unit volume, which is the charge density \(\rho\) at point P.

\begin{equation*} \vec{\nabla}\cdot \vec E = \frac{\rho}{\epsilon_0} \end{equation*}

This equation is differential form of Gauss’s law for electric field. This equation is also called the point or local form of Gauss’s law. Often we say this is the Gauss’s law.

Faraday’s and Ampere-Maxwell’s laws can be converted into the differential forms using curls. I will not present the derivation but just write the final answer for all the four laws for the electric and magnetic field here since they are so pretty in the differential form.

\begin{align*} \amp \vec{\nabla}\cdot \vec E = \frac{\rho}{\epsilon_0}\\ \amp \vec{\nabla}\cdot \vec B = 0\\ \amp \vec{\nabla}\times \vec E = -\frac{\partial \vec B}{\partial t}\\ \amp \vec{\nabla}\times \vec B = \mu_0 \vec J + \mu_0\epsilon_0 \frac{\partial \vec E}{\partial t} \end{align*}

These equations together with the Lorentz force law gives a complete description of all electromagnetic phenomena. According to the Lorentz force law, the force on a point charge \(Q\) with velocity \(\vec v\) is given by

\begin{equation*} \vec F_\text{em} = Q\vec E + Q\vec v\times \vec B. \end{equation*}

We will now explore consequences of Maxwell’s equations using the differential form.

Exercises 42.2.2 Exercises

1. Practice Gradients.

Calculate the gradient of the following scalar fields: (a) \(f = x + y+ z\text{,}\) (b) \(f = xyz\text{,}\) (c) \(f = \frac{{\displaystyle1}}{{\displaystyle\sqrt{x^2+y^2+z^2}}}\text{,}\) (d) \(f = \ln \left(\sqrt{x^2+y^2} \right)\text{.}\)

Answer.

(b) \(yz \hat u_x + zx\hat u_y + xy \hat u_z\text{.}\)

Solution.

Let \(r = \sqrt{x^2+y^2+z^2}\) and \(\rho=\sqrt{x^2+y^2}\text{.}\)

\begin{align*} \text{(a)}\quad \vec \nabla f \amp = \hat u_x + \hat u_y + \hat u_z.\\ \text{(b)}\quad \vec \nabla f \amp = yz\hat u_x + zx\hat u_y + xy\hat u_z.\\ \text{(c)}\quad \vec \nabla f \amp = \frac{x}{r^3}\:\hat u_x + \frac{y}{r^3}\:\hat u_y + \frac{z}{r^3}\:\hat u_z.\\ \text{(d)}\quad \vec \nabla f \amp = \frac{x}{\rho^2}\:\hat u_x + \frac{y}{\rho^2}\:\hat u_y. \end{align*}

2. Practice Divergence and Curl.

Calculate the divergence and curl of the following vector fields. (a) \(\vec F_1 = x\hat u_x + y\hat u_y + z\hat u_z.\) (b) \(\vec F_2 = \frac{{\displaystyle x\hat u_x + y\hat u_y}}{{\displaystyle\left( x^2 + y^2\right)^{3/2} }}.\) (c) \(\vec F_3 = \frac{{\displaystyle x\hat u_x + y\hat u_y+ z\hat u_z}}{{\displaystyle \left( x^2 + y^2+z^2\right)^{3/2} }}.\) (d) \(\vec F_4 = -y\hat u_x + x\hat u_y.\)

Answer.

Divergences: (a) 3, (d) 0.

Solution.

The divergences are

\begin{align*} \text{(a)}\quad\vec \nabla\cdot\vec F_1\amp = 1 + 1 + 1 = 3.\\ \text{(b)}\quad \vec \nabla\cdot\vec F_2\amp = \frac{\partial}{\partial x}\left[ \frac{{\displaystyle x}}{{\displaystyle\left( x^2 + y^2\right)^{3/2} }}\right] \\ \amp \quad\quad + \frac{\partial}{\partial y}\left[ \frac{{\displaystyle y}}{{\displaystyle\left( x^2 + y^2\right)^{3/2} }}\right]\\ \text{(c)}\quad \vec \nabla\cdot\vec F_3 \amp = \frac{\partial}{\partial x}\left[ \frac{{\displaystyle x}}{{\displaystyle \left( x^2 + y^2+z^2\right)^{3/2} }} \right] + \frac{\partial}{\partial y}\left[ \frac{{\displaystyle y}}{{\displaystyle \left( x^2 + y^2+z^2\right)^{3/2} }} \right]\\ \amp \quad\quad +\frac{\partial}{\partial z}\left[ \frac{{\displaystyle z}}{{\displaystyle \left( x^2 + y^2+z^2\right)^{3/2} }} \right] \\ \text{(d)}\quad \vec \nabla\cdot\vec F_4\amp = 0. \end{align*}

The Curl is left as an exercise.

3. Static Electric Field from Static Potential.

Find the static electric field from the electric potential of a fixed point charge \(q\) at the origin by using the following relation of the electric field to the static electric potential.

\begin{equation*} \vec E = - \vec{\nabla} V\ \ \textrm{(static)} \end{equation*}

Solution.

For the given charge \(q\) at the origin the electric potential at a point \(P\) with coordinates \((x,y,z)\) is given by

\begin{equation*} V(x,y,z) = \dfrac{1}{4\pi\epsilon_0}\:\dfrac{q}{\sqrt{x^2 + y^2 + z^2}}. \end{equation*}

The negative gradient of this function is

\begin{equation*} \vec E = - \vec \nabla V =\dfrac{q}{4\pi\epsilon_0}\left[ \dfrac{x}{r^3}\:\hat u_x + \dfrac{y}{r^3}\:\hat u_y + \dfrac{z}{r^3}\:\hat u_z\right], \end{equation*}

with \(r = \sqrt{x^2+y^2+z^2}\text{.}\)

4. Curl of a Static Magnetic Field of a Steady Current.

A straight wire carries a current \(I_0\text{.}\) Find the magnetic field at an arbitrary point and calculate the curl of the magnetic field there. Make any observations about the domain of validity of your answer.

Answer.

\(\vec\nabla\times\vec B = 0\quad (x,y,z)\) not on wire.

Solution.

The magnetic field of a straight wire carrying a constant current is written more compactly in the cylindrical coordinates. Let the wire be along the \(z\) axis with the current flowing towards the positive \(z\)-axis. Let \(s\) be the distance of the field point P from the \(z\)-axis. The, the magnetic field \(\vec B\) at P will be

\begin{equation*} \vec B(s,\phi,z) = \dfrac{\mu_0I_0}{2\pi}\:\dfrac{1}{s}\:\hat u_{\phi}, \end{equation*}

where \(\hat u_{\phi}\) is the unit vector tangent to the circles around the \(z\)-axis. We could perform the calculations of the curl in cylindrical coordinates or utilize a small arbitrary Amperian loop about the point P and find that

\begin{equation*} \dfrac{\mu_0I_{\textrm{enc}}}{\Delta A} = 0 =\dfrac{\oint \:\vec B\cdot d\vec l }{\Delta A}, \end{equation*}

where \(\Delta A\) is the area of the loop. By taking \(\Delta A\rightarrow 0\) limit we see that

\begin{equation*} \vec\nabla\times\vec B = 0. \end{equation*}

The conclusion will be valid for all points except the points on the \(z\)-axis since then all loops will include the current at the axis.

5. Divergence of a Static Electric Field of a Point Charge.

Find the divergence of electric field of a point charge at the origin. Make any observations about the domain of validity of your answer.

Hint.

Show by a similar argument as given for the curl of \(\vec B\) in the last problem, but this time for the divergence of \(\vec E\text{,}\) that \(\vec\nabla\cdot\vec E = 0\) at all points except the point where the charge is located.

Answer.

\(\vec\nabla\cdot\vec E = 0\quad (x,y,z)\) not on charge, excluding the origin.

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