# ch22-p027 - 27(a The linear charge density is the charge...

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Let dx be an infinitesimal length of rod at x . The charge in this segment is dq dx . The charge dq may be considered to be a point charge. The electric field it produces at point P has only an x component and this component is given by dE dx Lax x = +− 1 4 0 2 π ε λ bg . The total electric field produced at P by the whole rod is the integral () () () 2 0 0 00 0 00 11 1 44 4 , 44 L L x dx E Lax a La Lax Lq aL a aL a εε ε εε λλ λ ⎛⎞ == = ⎜⎟ π + ⎝⎠ +− λ1 == π+ π+ upon substituting qL λ −= . With q = 4.23 × 10 15 C, L =0.0815 m and a = 0.120 m, we obtain 3 1.57 10 N/C x E =− × , or 3 | | 1.57 10 N/C x E . (c) The negative sign in x E indicates that the field points in the – x direction, or 180 ° counterclockwise form the + x axis. (d) If a is much larger than L , the quantity L + a in the denominator can be approximated by a and the expression for the electric field becomes E q a x =− 4 0 2 π ε . Since 50 m 0.0815 m, aL == ± the above approximation applies and we have 8 1.52 10 N/C
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