Magnetic Force on a Current-Carrying Wire — Question 6

Question 6

A field gradient shifts the effective point of application of the force on a straight wire

A straight wire from x=0 to L carries current I along +x. The external field is $\vec B(x)=B_0(1+\alpha x/L)\hat z$, with B0 positive and alpha greater than -1.

1. Find the total force and torque about the origin.

2. Find the x coordinate at which a single equivalent transverse force would act. Evaluate that coordinate for alpha=2.

Question 6 - Solution

A field gradient shifts the effective point of application of the force on a straight wire

(a) Method and calculation

$$dF_y=-IB_0(1+\alpha x/L)\,dx\Rightarrow F_y=-IB_0L(1+\alpha/2).$$$$\tau_z=\int_0^Lx\,dF_y=-IB_0L^2(1/2+\alpha/3).$$

(b) Interpretation and further calculation

$$x_F=\tau_z/F_y=L\frac{1/2+\alpha/3}{1+\alpha/2}.$$$$\alpha=2:\quad x_F=\frac7{12}L>L/2.$$

Physical check

The resultant acts closer to the end where the magnetic field is stronger.