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329 - Fields and waves in nature and engineering the big...

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Fields and waves in nature and engineering — the big picture: Fundamental building blocks of matter — electrons and protons at atomic scales — interact with one another gravitationally and via “electromagnetic” forces. These interactions are most conveniently described in terms of suit- ably defined “vector fields” that permeate space and time, or simply the space- time ( x, y, z, t ) ( r , t ) . Interactions attributed to particle masses can be formulated by gravitational fields g ( r , t ) specified in reference frames where spatial coordinates r = ( x, y, z ) are defined. Far stronger interactions at- tributed to particle charges , on the other hand, are formulated in terms of a pair of vector fields, E ( r , t ) and B ( r , t ) , known as electric and magnetic fields , respectively. Electric and magnetic fields: A particle with charge q and mass m as well as position and velocity vectors r and v = d r dt specified at an instant t within a measurement frame (or “lab” frame) will be accelerated in accordance with m d v dt = q ( E ( r , t ) + v × B ( r , t )) , (1) 1

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which is Newton’s 2nd law of motion 1 for a particle under the influence of Lorentz force F = q ( E + v × B ) . (2) In view of (1), the operational definitions of fields E ( r , t ) and B ( r , t ) arise from particle acceleration a = d v dt that can be measured in the lab frame: the electric field E is evidently force per unit stationary charge (i.e., v = 0 ) whereas field B describes an additional force per unit current flux (e.g., q v ) that acts in a direction perpendicular to v . There are important di ff erences between gravitational and electromag- netic interactions: Gravitational interactions are always attractive indicating that particle masses m that generate the gravitational field g ( r , t ) must all have the same algebraic sign (taken to be positive by convention). Electro- magnetic interactions, on the other hand, are attractive or repulsive depend- ing on particle charges q which can be positive or negative. By convention a positive charge q = e 1 . 6 × 10 - 21 C is attributed to the fundamental particle know as proton , while, again by convention, q = - e for an electron , the sole companion of the proton within a hydrogen atom 2 . Protons and electrons are charged elementary building blocks 3 of all atoms (hydrogen as 1 Valid so long as | v | c where c is the speed of light in vacuum. 2 Hydrogen atom exists as a consequence of mutual attraction between proton and electron counterbal- anced by quantum mechanical constraints on allowed energy states. 3 Atoms can also contain in their nuclei varying numbers of an uncharged particle known as the neutron which is responsible for di ff erent isotopes of chemical elements (e.g., the hydrogen isotope known as deuterium contains a neutron in addition to a proton and an electron). While neutrons have no net charge, they consist of charged sub-nuclear particles known as quarks whose motions within the neutron establish currents and a magnetic moment.
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