A straight wire carrying a $2\mathrm{.}82-$A current is placed in a uniform magnetic field of magnitude $0\mathrm{.}252$ T directed perpendicular to the wire.

$($a$)$ Find the magnitude of the magnetic force on a section of the wire having a length of $14\mathrm{.}2$ cm$.$

N

$($b$)$ Explain why you can't determine the direction of the magnetic force from the information given in the problem.

Using this result, consider a town where the Earth's magnetic field has a magnitude of $49\mathrm{.}8$T$.$ The magnetic field vector lies in a vertical plane defined by the north$-$south and up$-$down axes, and it points $60\mathrm{.}0$ below the northward direction. In this town, a storefront window lies along the north$-$south vertical plane, and in the window is a neon sign $($which is a thin current$-$carrying discharge tube$).$ The sign carries a $33\mathrm{.}9$ mA current, starting from the lower south corner of the window, and ending at the opposite corner, which is $1\mathrm{.}55$ m to the north and $0\mathrm{.}850$ m upward. The sign spells out the word "DONUTS" between the two points. What is the net vector magnetic force $($in $$N$)$ on the neon sign? $($Take east to be the $+$x$-$axis, up to be the $+$y$-$axis, and south to be the $+$z$-$axis. Do not include units in your answer.$)$