Maxwell stress tensor

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Electromagnetism
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The Maxwell Stress Tensor (also known as Maxwell's Stress Tensor) is used to calculate the stresses on objects in magnetic or electrical fields. It is used in many finite element programs to determine the forces on objects being analyzed.

If the magnetic and electrical fields at the surface of an object are known, the forces at that surface can be calculated, and the overall force on the object can be determined. In some cases, such as motors, the electrical fields are neglected, and stress and force calculations are made using only the magnetic fields.

The Maxwell Stress Tensor is naturally derived while studying the momentum conservation to a system of particles in presence of electromagnetic fields. A qualitative description to derive Maxwell Stress Tensor is as follows: 1) consider a system of charged particles in presence of e.m.fields 2) consider the Lorentz force observed on the particles due to the e.m. fields 3) convert the summation to a continuum integral 4) Using rate of change of momentum equals applied force and Maxwell's equations convert all variables to electric and magnetic fields 5) by extracting the momentum of the EM fields we can obtain the momentum flux into a given volume. The Maxwell stress tensor is nothing but a description of this momentum flux. Both electric and magnetic fields pull along the lines of force and push perpendicular to them.

Contents

Equation

In physics, the Maxwell stress tensor is the stress tensor of an electromagnetic field. In cgs units, it is given by:

\sigma_{ij}=\frac{1}{4\pi}\bigl(E_{i}E_{j}+H_{i}H_{j}- \tfrac{1}{2}(E^2+H^2)\delta_{ij}\bigr),

where E is the electric field, H is the magnetic field and δij is Kronecker's delta.

In SI units, it is given by:

\sigma _{ij} = \varepsilon_0 E_i E_j + \frac{1} {{\mu _0 }}B_i B_j - \frac{1}{2}\bigl( {\varepsilon_0 E^2 + \tfrac{1} {{\mu _0 }}B^2 } \bigr)\delta _{ij} ,

where ε0 and μ0 are the electric constant and the magnetic constant respectively.

The element ij of the Maxwell stress tensor has units of momentum per unit of area times time and gives the flux of momentum parallel to the ith axis crossing a surface normal to the jth axis (in the negative direction) per unit of time.

These units can also be seen as units of force per unit of area (negative pressure), and the ij element of the tensor can also be interpreted as the force parallel to the ith axis suffered by a surface normal to the jth axis per unit of area. Indeed the diagonal elements give the tension (pulling) acting on a differential area element normal to the corresponding axis. Unlike forces due to the pressure of an ideal gas, an area element in the electromagnetic field also feels a force in a direction that is not normal to the element. This shear is given by the off-diagonal elements of the stress tensor.

Magnetism only

If the field is only magnetic (which is largely true in motors, for instance), some of the terms drop out, and the equation in SI units becomes:

\sigma _{ij} = \frac{1} {{\mu _0 }}B_i B_j - \frac{1}{2}\bigl( {\tfrac{1} {{\mu _0 }}B^2 } \bigr)\delta _{ij} N/m^2

For cylindrical objects, such as the rotor of a motor, this is further simplified to:

\sigma _{rt} = \frac{1} {{\mu _0 }}B_r B_t - \frac{1}{2\mu _0} { B^2 } \delta _{rt} N/m^2

Where r is the shear in the radial (outward from the cylinder) direction, and t is the shear in the tangential (around the cylinder) direction. It is the tangential force which spins the motor. B_r is the flux density in the radial direction, and B_t is the flux density in the tangential direction.

See also

References

  • Richard Becker,"Electromagnetic Fields and Interactions",Dover Publications Inc., 1964.
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  • This page was last modified on 3 October 2008, at 07:24.

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