Question 14

A rigid wire consists of a semicircular portion of radius $$R$$ and two straight sections. The wire is partially immerged in a perpendicular magnetic field $$\vec{B} = B_0 \hat{j}$$ as shown in figure. The magnetic force on the wire if it has a current $$i$$ is :

$$\text{From the figure, the field is out of the page along the z-axis: } \vec{B} = B_0\hat{k}$$

$$\text{Immersed initial point (left wire entering field) to final point (right wire leaving field):}$$ $$\vec{L}_{\text{eff}} = 2R\hat{i}$$

$$\vec{F} = i(\vec{L}_{\text{eff}} \times \vec{B}) = i(2R\hat{i} \times B_0\hat{k})$$

$$\vec{F} = 2iRB_0(\hat{i} \times \hat{k}) = -2iRB_0\hat{j}$$

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