The Unapologetic Mathematician

Mathematics for the interested outsider

Maxwell’s Equations (Integral Form)

It is sometimes easier to understand Maxwell’s equations in their integral form; the version we outlined last time is the differential form.

For Gauss’ law and Gauss’ law for magnetism, we’ve actually already done this. First, we write them in differential form:

\displaystyle\begin{aligned}\nabla\cdot E&=\frac{1}{\epsilon_0}\rho\\\nabla\cdot B&=0\end{aligned}

We pick any region V we want and integrate both sides of each equation over that region:

\displaystyle\begin{aligned}\int\limits_V\nabla\cdot E\,dV&=\int\limits_V\frac{1}{\epsilon_0}\rho\,dV\\\int\limits_V\nabla\cdot B\,dV&=\int\limits_V0\,dV\end{aligned}

On the left-hand sides we can use the divergence theorem, while the right sides can simply be evaluated:

\displaystyle\begin{aligned}\int\limits_{\partial V}E\cdot dS&=\frac{1}{\epsilon_0}Q(V)\\\int\limits_{\partial V}B\cdot dS&=0\end{aligned}

where Q(V) is the total charge contained within the region V. Gauss’ law tells us that the flux of the electric field out through a closed surface is (basically) equal to the charge contained inside the surface, while Gauss’ law for magnetism tells us that there is no such thing as a magnetic charge.

Faraday’s law was basically given to us in integral form, but we can get it back from the differential form:

\displaystyle\nabla\times E=-\frac{\partial B}{\partial t}

We pick any surface S and integrate the flux of both sides through it:

\displaystyle\int\limits_S\nabla\times E\cdot dS=\int\limits_S-\frac{\partial B}{\partial t}\cdot dS

On the left we can use Stokes’ theorem, while on the right we can pull the derivative outside the integral:

\displaystyle\int\limits_{\partial S}E\cdot dr=-\frac{\partial}{\partial t}\Phi_S(B)

where \Phi_S(B) is the flux of the magnetic field B through the surface S. Faraday’s law tells us that a changing magnetic field induces a current around a circuit.

A similar analysis helps with Ampère’s law:

\displaystyle\nabla\times B=\mu_0J+\epsilon_0\mu_0\frac{\partial E}{\partial t}

We pick a surface and integrate:

\displaystyle\int\limits_S\nabla\times B\cdot dS=\int\limits_S\mu_0J\cdot dS+\int\limits_S\epsilon_0\mu_0\frac{\partial E}{\partial t}\cdot dS

Then we simplify each side.

\displaystyle\int\limits_{\partial S}B\cdot dr=\mu_0I_S+\epsilon_0\mu_0\frac{\partial}{\partial t}\Phi_S(E)

where \Phi_S(E) is the flux of the electric field E through the surface S, and I_S is the total current flowing through the surface S. Ampère’s law tells us that a flowing current induces a magnetic field around the current, and Maxwell’s correction tells us that a changing electric field behaves just like a current made of moving charges.

We collect these together into the integral form of Maxwell’s equations:

\displaystyle\begin{aligned}\int\limits_{\partial V}E\cdot dS&=\frac{1}{\epsilon_0}Q(V)\\\int\limits_{\partial V}B\cdot dS&=0\\\int\limits_{\partial S}E\cdot dr&=-\frac{\partial}{\partial t}\Phi_S(B)\\\int\limits_{\partial S}B\cdot dr&=\mu_0I_S+\epsilon_0\mu_0\frac{\partial}{\partial t}\Phi_S(E)\end{aligned}

February 2, 2012 - Posted by | Electromagnetism, Mathematical Physics


  1. […] law, we’re already done, since it’s exactly the third of Maxwell’s equations in integral form. So far, so […]

    Pingback by Deriving Physics from Maxwell’s Equations « The Unapologetic Mathematician | February 3, 2012 | Reply

  2. who is maxwell and why did he decide to come with his equations that are difficult to understand

    Comment by TPL | May 17, 2012 | Reply

    • Blame the universe’s electromagnetism for being most easily calculable with vector calculus.

      Comment by Jose | February 12, 2013 | Reply

  3. That note is clear and good prapared

    Comment by Habamenshi Pierre Claver | November 23, 2012 | Reply

  4. why are these equations called maxwell’s equations while none of them are derived or proved by mexwell ,all of them were alread present

    Comment by abdur rahim | December 26, 2012 | Reply

    • Because Maxwell was a thug, and saw the relations underlying all of them. Gauss was extremely gifted, Ampere almost had it right, and Faraday did his own thing, but Maxwell connected it all. All the laws have their own names, individually, for the people who did formulate them originally.

      Comment by Carrie Elliott | November 8, 2016 | Reply

  5. where on earth is maxwell from?

    Comment by Uc | June 20, 2013 | Reply

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