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Vector Calculus
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
where K1, K2, K3 are constants of integration. All six constants A, φ, C, K1, K2 and K3 can be evaluated from the six initial conditions given by the vector equations r(0) = r0 and v(0) = v0, but we omit this computation here. Instead, we observe that the path of the particle is a circular helix with axis parallel to B, that is, parallel to k. One can see this from equation (3.109): The z-component of r varies linearly with t, while the x- and y-components represent uniform circular motion with angular velocity λ and radius |A1λ| around the center (K1, K2). Thus, charged particles spiral around the magnetic field lines. Qualitatively, this behavior still occurs even if the magnetic field lines are curved, as in the case of the earth’s magnetic field. Charged particles trapped by the earth’s magnetic field spiral around the magnetic field lines that run from pole to pole.
Graphical Field Plotting
Published in Sivaji Chakravorti, Electric Field Analysis, 2017
ABSTRACT For a clear understanding of the concept of electric field, some method for describing it, both qualitatively and quantitatively, is needed. Graphical field plotting is one such method that is dynamic in nature and illustrates the vector nature of the electric field. Graphical field maps are commonly drawn for practical configurations, which may be considered as two-dimensional or which are axi-symmetric in nature. Typically, in graphical field plotting, electric fieldlines are drawn to provide information about the field. The strength of the field is indicated by the density of the field lines. A high density of electric fieldlines indicates a strong field and vice versa. Complementary information can also be conveyed by simultaneously drawing the equipotential lines. Field maps could also be used for obtaining approximate values of system parameters such as capacitance.
Orbital Radiation Environments
Published in John D. Cressler, H. Alan Mantooth, Extreme Environment Electronics, 2017
The LET spectra shown in Figure 10.4 are applicable to geosynchronous and interplanetary missions where there is no geomagnetic attenuation. The earth’s magnetic field, however, provides significant protection. Due to the basic interaction of charged particles with a magnetic field, the charged particles tend to follow the geomagnetic field lines. Near the equator, the field lines tend to be parallel to the earth’s surface. Thus all but the most energetic ions are deflected away. In the polar regions the field lines tend to point toward the earth’s surface, which allows much deeper penetration of the incident ions. The effect of the geomagnetic field on the incident GCR LET spectrum during solar minimum is discussed for various orbits in Ref. [4].
A multi-order nonlinear meta-analysis of bifluidic fireball sheath fluctuations
Published in Waves in Random and Complex Media, 2023
Subham Dutta, Pralay Kumar Karmakar
The graphical illustration of the bifluidic PFS instability governed by the multi-order perturbed Poisson equations is presented after a judicious numerical analysis as in Figures (1–12). The analytically developed spatial plots and colormaps manifest a negative rate of change of the plasma parameters with respect to the radial distance (). As it is evident from Figures (1–12), the highest rate of change of the plasma parameters is observed across one Debye length () only. The spatial variation of the plasma parameters is negligibly meagre exterior to one Debye length, ; where, (, , , ). The narrow spike in the electrostatic potential (Figure 5), beside the anode region (within one Debye length, ), is a consequence of the consistent higher electrostatic potential sustained owing to the anode voltage in the dissipative plasma media. The simultaneous existence of the Mach number spikes (Figure 8) manifest higher electron (ion) velocities at the periphery of the anode, originating from denser electric field lines, thereby pulling (pushing) the electron (ions) inwards (outwards). Here, the field line density determines the electrostatic force strength experienced by the charged particles. It is found that the analytically developed potential plot (Figure 5) smoothly corroborates with the experimental [2] and theoretical [3] findings available in the literature. It justifies and validates the reliability of our bifluidic plasma model conjectures for the steady state description of the PFS dynamics [2,3].
Improved magnetorheological finishing process with rectangular core tip for external cylindrical surfaces
Published in Materials and Manufacturing Processes, 2019
Manpreet Singh, Anant Kumar Singh
The magnetic field lines travel in closed loops and also, they do not intersect with each other. The iron tool cores make a pattern tracking field lines due to each fragment of the iron particle is itself a tiny dipole. The dipole tries to align itself with a magnetic field line (Mf). But, on the edge of the tool core tip surface, the field lines are much closer to each other. Closer magnetic field lines provide stronger magnetization. When magnetic field lines are far away to each other, the magnetization is weak. In the circular tool core tip surface, the behaviour of magnetic field lines is depicted in Fig. 4(a). Due to the localization of the Mf at the edge of the circular tip surface, there is an increase in magnetization. The Mf towards the central axis of the circular tip surface is far away from each other which may result in comparatively weakening the magnetization at the central portion of the circular tool core tip surface as depicted in Fig. 4(a). The localization of magnetic field lines is influenced by the edge effect phenomena which enhances the magnetization.[9] The flow of magnetic flux in the magnetic core spreads out into the surrounding and this phenomenon is known as flux fringing.[21] This edge effect depends on the geometry of the tool core tip surface.[22] In rectangular shaped tool core tip surface, the edge effect is greater than the circular shaped tool core tip surface. This is because of the rectangular shaped tool core tip surface which has four corner edges that increase the localization of the magnetic field lines as depicted in Fig. 4(b). Figure 4(a) shows the direction of magnetic field lines (Mf) and the magnetic flux density (B) generated on the circular tip surface with a circular edge effect. Figure 4(b) depicts magnetic field lines (Mf) direction and the magnetic flux density (B) generated on the rectangular tip surface with a rectangular edge effect. There is a sudden change in the edged area on the rectangular tip surface. This results in the concentration of flux density on the tip surface. In the rectangular shaped tip surface, the magnetic field lines are concentrated at one side of the tip surface while the other side distributes the magnetic field lines uniformly. The edge effect or fringing effect is clearly proven by the results as obtained from the finite element analysis and also experimentally as depicted in Figs. 1 and 2.