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(III) Suppose a uniformly charged wire starts at point 0 and rises vertically along the positive $y$ axis to a length $\ell$ . (a) Determine the components of the electric field $E _ { x }$ and $E _ { y }$ at point $( x , 0 ) .$ That is, calculate $\vec { \mathbf { E } }$ near one end of a long wire, in the plane perpendicular to the wire. (b) If the wire extends from $y = 0$ to $y = \infty ,$ so that $\ell = \infty ,$ show that $\vec { \mathbf { E } }$ makes a $45 ^ { \circ }$ angle to the horizontal for any $x .$ [Hint: See Example 11 of "Electric Charge and Electric Field" andFig. $29 . ]$

A. $\frac{\lambda}{4 \pi \varepsilon_{0} x \sqrt{x^{2}+\ell^{2}}}(x-\sqrt{x^{2}+\ell^{2}})$

B. $45^{\circ}$ below the $x$ axis

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