The figure below shows a segment of a wire of length \( 2 a \) on the \( y \)-axis and the current \( I \) in the wire is in the positive \( y \) direction. Find the magnetic field vector at a point \( P \) located on the \( x z \) plane where \( x=d \). Show that this is equivalent to the magnetic field of the infinite wire when \( d=0 \)
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(II) A segment of wire of length $d$ carries a current $I$ as shown in Fig. 49 . (a) Show that for points along the positive $x$ axis (the axis of the wire), such $x$ axis (the axis of the wire), such as point $Q,$ the magnetic field $\vec{\mathbf{B}}$ is zero. $(b)$ Determine a formula for the field at points along the $y$ axis, such as point $\mathrm{P} .$
(II) Consider a straight section of wire of length $d,$ as in Fig. $48,$ which carries a current $I$ (a) Show that the magnetic field at a point $\mathrm{P}$ a distance $R$ from the wire along its perpendicular bisector is $$B=\frac{\mu_{0} I}{2 \pi R} \frac{d}{\left(d^{2}+4 R^{2}\right) \frac{1}{2}}$$ (b) Show that this is consistent with Example 11 of Sources of Magnetic Field for an infinite wire.
In Fig. $30-28$, a straight wire of length $L$ carries current $i$. Show that $$ B=\frac{\mu_{0}|i|}{4 \pi R} \frac{L}{\left(L^{2}+R^{2}\right)^{1 / 2}} $$ gives the magnitude of the magnetic field $\vec{B}$ produced by the wire at $P_{2}$, a perpendicular distance $R$ from one end of the wire.
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