Magnetic Force on a Current-Carrying Wire — Question 4

Question 4

A bent wire in a uniform field has the same net force as its endpoint displacement

A wire carries current I from A=(0,0) to B=(L,H) through a right-angle bend in the xy plane. The applied field is uniform $B_0\hat z$.

1. Integrate the force along the path and compare it with the force on a straight wire between the same endpoints.

2. For I=2.00 A, L=.300 m, H=.200 m and B0=.500 T, find the net force vector.

Question 4 - Solution

A bent wire in a uniform field has the same net force as its endpoint displacement

(a) Method and calculation

$$\vec F=I\int_A^B d\vec\ell\times\vec B=I(\vec r_B-\vec r_A)\times\vec B.$$$$\vec F=IB_0(H\hat x-L\hat y),\quad\text{independent of path shape for uniform }\vec B.$$

(b) Interpretation and further calculation

$$F_x=IB_0H=(2)(.500)(.200)=.200\,\mathrm N.$$$$F_y=-IB_0L=-(2)(.500)(.300)=-.300\,\mathrm N.$$

Physical check

Using the total wire length in ILB would discard the changing force direction along the bend.