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sics for Scientists and Engineers 4th ed (1.pdi-Adobe Acrolst Reader Dc Window Help ools Giancoli Physics fo... x 93 1322 111% EXAMPLE 21-11 Long line of charge. Determine the magnitude of the electric field at any point P a distance x from the midpoint 0 of a very long line (a wire, say) of uniformly distributed positive charge, Fig. 21-29. Assume x is much smaller than the length of the wire, and let λ be the charge per unit length (C/m). APPROACH We set up a coordinate system so the wire is on the y axis with origin 0 as shown. A segment of wire dy has charge do - Ady. The field dE at point P due to this length dy of wire (at y) bas magnitude wtre ( y) as shown in Fig 21-29. The vector dE has components dEy and dEy as shown where dE -dE cos 8 and dEy dE sin. SOLUTION Because o is at the midpoint of the wire, the y component of E will there will be equal contributions to Ey fdE, from above and FIGURE 21-29 Example 21-11. below point 0 Thus we have A fcos 8 dy The integration here is over y, along the wire, with x treated as constant. We must now write e as a function of y, or y as a function of . We do the latter: since y xtan, then dyx de/cos 6. Furthermore, because cos θ = x/Vir+ア, then 1/(x1+y)-cos, e/r, and our integrand above is cos®)Xx de/cos eX(cose/x) cos de/x. Hence where we have assumed the wire is cxtremely long in both directions (y → is) which corresponds to the limits θ-tr/2. Thus the field near a long straight wire of uniform charge decreases inversely as the first power of the distance from the wire. NOTE This result, obtained for an infinite wire, is a good approximation for a wire of finite length as long as a is small compared to the distance of P from the ends of the wire. SECTION 21-7 Electric Field Calculations for Continuous Charge Distri 573 pe here to search

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