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## Homework Statement

Consider a closed semi-circular loop lying in the xy plane carrying a current I in the

counterclockwise direction, as shown in Figure attached.

A uniform magnetic field pointing in the +y direction is applied. Find the magnetic force acting on the straight segment and the semicircular arc.

## Homework Equations

The force in xx axis [itex]\vec{F1}[/itex] is easy to see that it has the normal direction [itex]\hat{i}[/itex]

so that force :

[itex] \vec{F} = I 2R\hat{i}\times \vec{B} \hat{j}= 2IRB \hat{k} [/itex]

where [itex]\hat{k}[/itex] is directed out the page.

Now the force along the arc.

The solution says:

To evaluate [itex]\vec{F2}[/itex] , we first note that the differential length element [itex]d\vec{s}[/itex] on the semicircle can be written as:

[itex]d\vec{s} = ds\hat{\theta } = IRd\theta (-sin \theta \hat{i} + cos \theta \hat{j})[/itex]

I know that [itex]s = R \theta[/itex], but i don't know where [itex]-sin \theta \hat{i} + cos \theta \hat{j} [/itex] come from.

Some tips ??