## Definition of electric and magnetic forces on Riemannian manifold

##### Kovanen, Tuomas (2013)

Kovanen, Tuomas

Tampere University of Technology

2013

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**Julkaisun pysyvä osoite on**

http://urn.fi/URN:ISBN:978-952-15-3110-1

##### Tiivistelmä

The precise relationship of electricity and magnetism to mechanics is becoming a critically important question in modern engineering. Although well known engineering methods of modeling forces in electric and magnetic systems have been adequate for bringing many useful technologies into our everyday life, the further development of these technologies often seems to require more extensive modeling. In the search for better modeling it is relevant to examine the foundations of the classical models.

The classical point charge definitions of electric and magnetic field quantities are not designed for the determination of forces in typical engineering problems. In the approach of this thesis the relationship of electric and magnetic field quantities to forces on macroscopic objects is built in the definitions. As a consequence, the determination of electrostatic and magnetostatic forces on macroscopic objects becomes a clear-cut issue. What makes the approach work is that the system of interacting objects is considered as a whole. This is contrary to the classical approach where an unrealistic test object is used to define electric and magnetic fields as properties of the source object only.

A limitation in the classical notions of electric field intensity and magnetic induction is that they are sufficient to determine forces on only rigid objects. To allow for deformable objects the model needs to be combined with that of continuum mechanics. The notions suggested in this thesis are more general as they allow rigidity to be considered with respect to the test object itself, which may not be rigid in the usual sense.

The required generality is obtained by using the mathematics of differential geometry as the framework for the definitions. This has the additional benefit of making clear the mathematical structures required for the definitions. Also, it allows the clear identification of the mathematical objects involved. Both of these outcomes are important for creating efficient and flexible computational codes for modeling.

The classical point charge definitions of electric and magnetic field quantities are not designed for the determination of forces in typical engineering problems. In the approach of this thesis the relationship of electric and magnetic field quantities to forces on macroscopic objects is built in the definitions. As a consequence, the determination of electrostatic and magnetostatic forces on macroscopic objects becomes a clear-cut issue. What makes the approach work is that the system of interacting objects is considered as a whole. This is contrary to the classical approach where an unrealistic test object is used to define electric and magnetic fields as properties of the source object only.

A limitation in the classical notions of electric field intensity and magnetic induction is that they are sufficient to determine forces on only rigid objects. To allow for deformable objects the model needs to be combined with that of continuum mechanics. The notions suggested in this thesis are more general as they allow rigidity to be considered with respect to the test object itself, which may not be rigid in the usual sense.

The required generality is obtained by using the mathematics of differential geometry as the framework for the definitions. This has the additional benefit of making clear the mathematical structures required for the definitions. Also, it allows the clear identification of the mathematical objects involved. Both of these outcomes are important for creating efficient and flexible computational codes for modeling.

##### Kokoelmat

- Väitöskirjat [3921]