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Maxwell's Equations


Gauss' Law: 

\boldsymbol{\nabla}\cdot\mathbf{E}=\frac{\rho}{\epsilon_0}

Solenoidal Condition (Gauss' Law for Magnetism): 

\boldsymbol{\nabla}\cdot\mathbf{B}=0

Ampère's Law with Maxwell's correction:

\boldsymbol{\nabla}\wedge\mathbf{B}=\mu_0\mathbf{j}+\mu_0\epsilon_0\frac{\partial\mathf{E}}{{\partial}t}

Faraday's Law: 

\boldsymbol{\nabla}\wedge\mathbf{E}+\frac{\partial\mathf{B}}{{\partial}t}=\mathbf{0}

 


The Terms

\mathbf{E} - Electric Field [Units: N C-1 or V m-1]

\mathbf{B} - Magnetic Field [Units: C m-3]

\boldsymbol{\nabla}\cdot - Divergence Operator

\boldsymbol{\nabla}\wedge - Curl Operator

\epsilon_0 - Permittivity of Free Space [Units: F m-1]

\mu_0 - Permeability of Free Space [Units: H m-1]

\rho - Charge Density [Units: C m-3]

\mathbf{j} - Total Current Density [Units: C m-2]

\frac{\partial}{\partial t} - Partial Derivative With Respect To Time [Units: s-1]


What Do They Mean?

Gauss' Law expresses that the electric flux through any closed surface is proportional to the charge enclosed by the surface. The differential form is shown above, however this law can equivalently be expressed in an integral form,

\oint_S\mathbf{E}\cdot d\mathbf{A}=\frac{Q}{\epsilon_0},

where dA is an area element of the surface S and Q is the charge enclosed by the surface.

 

The solenoidal condition, also known as Gauss' Law for Magnetism, states that no magnetic monopoles exist; i.e. there are no isolated magnetic charges, only magnetic dipoles. The differential form is shown above, however this law can equivalently be expressed in an integral form,

\oint_S\mathbf{B}\cdot d\mathbf{A}=0.

The existence of magnetic monopoles is an area of ongoing research, and if they were to be found then this law would have to be modified accordingly.

 

Ampère's Law with Maxwell's Correction states that magnetic fields can be generated by electric currents and electric fields which vary with time. The original formulation by Ampère in 1826 was

\boldsymbol{\nabla}\wedge\mathbf{B}=\mu_0\mathbf{j},

but this has two main problems. Firstly, by taking the divergence of both sides and noting that the divergence of the curl of any vector is identically zero, we have \boldsymbol{\nabla}\cdot\mathbf{j}=0 whereas we would expect to recover the continuity equation, \boldsymbol{\nabla}\cdot\mathbf{j}=-\frac{\partial\rho}{{\partial}t} in general. The second problem was that in free space, \mathbf{j}=\mathbf{0} and so the original equation (above) would give \boldsymbol{\nabla}\wedge\mathbf{B}=\mathbf{0}. We know however that electromagnetic waves propagate in free space, obeying the wave equation such that \boldsymbol{\nabla}\wedge\mathbf{B}=-\frac{1}{c^2}\frac{\partial\mathbf{E}}{{\partial}t}. The additional displacement current term introduced by Maxwell rectifies these problems. The differential form is shown above, however this law can equivalently be expressed in an integral form,

\oint_C\mathbf{B}\cdot dl=\int\int_S\left(\mu_0\mathbf{j}+\mu_o\epsilon_0\frac{\partial\mathbf{E}}{{\partial}t}\right)\cdot d\mathbf{A}. 

 

Faraday's Law expresses that the induced electromotive force (EMF) in any closed circuit is equal to the rate of change of the magnetic flux through that circuit. An alternative form of this law is

\epsilon=-\frac{d\Phi_B}{dt},

where ε is the induced EMF in Volts and ΦB is the magnetic flux in Tesla per metres squared through the circuit. The direction of the induced EMF is given by Lenz's Law: The induced current is always in such a direction as to oppose the motion or change causing it.


Further information at Warwick

Maxwell's equations are introduced and used in the first year module "PX120 Electricity and Magnetism" and are used extensively in many other modules.