ESF_lab_manual

ESF_lab_manual - Laboratory Sessions 1&2: Electrostatic...

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1 Laboratory Sessions 1&2: Electrostatic Field (6 hours) I. Theory 1. Mapping the electrostatic field (ESF). 1.1. Basic concepts. Two ways to depict the ESF with line plots exist: the electric field lines (or simply the field lines) and the equipotential lines (potential contours). Usually, both families of curves are plotted together, which creates a comprehensive field picture in the form of an orthogonal curvilinear mesh, whose density corresponds to the ESF energy density. Definition 1 : A field line is a line whose tangent at any point coincides with the direction of the E G field. E G E G Fig 1. E G field line. The equation defining the field line is or 0 Ed ττ × = G G G G & . (1.1) In an x-y plane, equation (1.1) leads to ˆˆ ,w h e r e y x E dy d xdx ydy dx E τ == + G . (1.2) E G x y d G dx dy x E y E Fig. 2. Geometrical interpretation of the field-line equation (1.2).
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2 Definition 2 : An equipotential surface of the potential 0 V is the geometrical place of all points whose electrostatic potential is equal to 0 V . It is defined by the equation ( ) 0 ,, Vxyz V = . (1.3) Usually, 2-D cross sections are plotted from a 3-D equipotential distribution. Thus, the equipotential lines (or contours) are derived. Since ˆ dV E Vn dn =−∇ =− G , (1.4) the field lines are orthogonal to the equipotential surfaces (or contours), see figures 3 and 4. ˆ n 0 V 0 VV + + ˆ n ˆ n ˆ n E G E G E G E G Fig.3. Equipotential contours and field lines. Fig. 4. Equipotential surfaces and field lines.
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3 The ESF is a potential field. In other words, it is curl free. Therefore, its field lines are open lines: they start at positive charges and end at negative charges (or infinity), as illustrated by the examples below. They never form a closed contour. + Fig.5. The field lines of the ESF of a spherical charge. Fig. 6. The field lines and the equipotential contours of a twin-lead line (cross-section).
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4 1.2. Plotting the field map. It is first assumed that the whole map is going to consist of N E number of field lines originating at a given point source. The total electric field flux Ψ over a closed surface is s Eds ε Ψ= ∫∫ G G . (1.5) The electric field flux Ψ does not depend on the choice of the surface as long as this surface encloses one of the field poles (sources). If one chooses to integrate along an equipotential surface, equation (1.5) simplifies to || s Ed s ∫∫ G , (1.6) since the surface element ds G has the same direction as the E G field. To represent correctly the field flux density DE = GG , we distribute the field lines in such a way that between every pair of lines there is the same amount of flux / E N ∆Ψ=Ψ , see figure 7. Fig. 7. Flux tubes and the flux density D G . It is also important how the equipotential lines are drawn. Let us assume that the equipotential lines are drawn at equal voltage increments of V =const. If this increment V is sufficiently small, the E G field will have a magnitude of V E τ = G (1.7) between the respective pair of equipotential lines. Here, is the orthogonal distance between these equipotential lines. The line segment is tangential to the field line. The field’s
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This note was uploaded on 02/29/2012 for the course PHYS 227 taught by Professor Rabe during the Fall '08 term at Rutgers.

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ESF_lab_manual - Laboratory Sessions 1&2: Electrostatic...

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