vec$(B)=\frac{{}_0\text{I}}{4}{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{d(vec(s))(hat(r))}{r^2}$

Problem $2$: Let's examine what the vectors involved in the Biot$-$Savart Law represent, think about how we might write them down using proper vector notation, and apply what we know about cross products.

A long straight wire with a $45\mathrm{.}0$ bend in it carries a current I from left to right. Starting from left to right in the figure, three infinitesimally short pieces of that wire can be represented as vectors dvec$(s{)}_1($green$),$ dvec$(s{)}_2($red$),$ and dvec$(s{)}_3($blue$).$

a$.$ Write an expression for each of these three vectors in terms of their magnitudes $(d{s}_n),$ and the unit vectors hat$()$ and hat$().$

dvec$(s{)}_1=$

dvec$(s{)}_2=$

dvec$(s{)}_3=$

b$.$ The location of the three pieces of wire can be expressed in component form as:

$1(green)$: $(x,y)=(-2d,-2d),2(red)$: $(x,y)=(0,-d),3(blue)$: $(x,y)=(+d,-d)$

The distance between each piece of wire and the origin $($indicated by a black dot$)$ can be represented as vectors vec$(r{)}_1,$vec$(r{)}_2,$ and vec$(r{)}_3.$ Write an expression for the unit vector associated with each of these vectors $,$ and $($:hat$(r{)}_3\}.$

hat$(r{)}_1=$

hat$(r{)}_2=$

hat$(r{)}_3=$

c$.$ Finally, apply what you know about the right$-$hand rule and cross$-$products to find the magnitude and direction of dvec$(s)$hat$(r)$ for each of the three pieces of wire. Note that the magnitude should still include the infinitesimal lengths $d{s}_1,d{s}_2,$ and $d{s}_3,$

dvec$(s{)}_1$hat$(r{)}_1=$

dvec$(s{)}_2$hat$(r{)}_2=$

dvec$(s{)}_3$hat$(r{)}_3=$