Question 2.34: Formally demonstrate that the sum of the forces acting on th...

The magnetic flux density is B = B_ou_z.       Using the definition of the magnetic force given in (2.146), we write the sum of the forces that act on the four sides as

$\textbf F _{\textbf {magnetic }}=\mathbf{-\int B \times I d I}$          (2.146)

$\textbf F _{\textbf {magnetic }}=-\int_{1}^{2} \textbf B \times \textbf I \textbf d \textbf I -\int_{2}^{3} \textbf B \times\textbf I \textbf d \textbf I -\int_{3}^{4} \textbf B \times \textbf I \textbf d \textbf I -\int_{4}^{1} \textbf B \times \textbf I \textbf d \textbf I$

$=-\int_{+\Delta x / 2}^{-\Delta x / 2} I B_{0} \textbf u _{ \textbf z } \times d x \textbf u _{ \textbf x }-\int_{+\Delta y / 2}^{-\Delta y / 2}I B_{0} \textbf u _{ \textbf z } \times d y \textbf u _{ \textbf y }-\int_{-\Delta x / 2}^{+\Delta x / 2} I B_{0} \textbf u _{ \textbf z } \times d x \textbf u _{ \textbf x }-\int_{-\Delta y / 2}^{+\Delta y / 2}I B_{0} \textbf u _{ \textbf z } \times d y \textbf u _{ \textbf y }=0$

Recall that the sign of the integral is determined by the limits of the integration.