PX157 - C2 - force of current loops

force on a wire

\vec{F} = q {\vec{v}}_{d} \times \vec{B}

{\vec{v}}_{d} ∥ d \vec{l} ⟹ {\vec{v}}_{d} \cdot d \vec{l} = v_{d} d \vec{l}

\begin{aligned} d \vec{F} & = n A d l q {\vec{v}}_{d} \times \vec{B} \\ = n A q v_{d} d \vec{l} \times \vec{B} \\ ∴ d \vec{F} & = I d \vec{l} \times \vec{B} \end{aligned}

\vec{F} = \int I d \vec{l} \times \vec{B}

\begin{aligned} d \vec{F} & = I d \vec{l} \times \vec{B} \\ = I d l B \hat{y} \times \hat{z} \\ = I d l B \hat{x} \end{aligned}

\vec{F} = \int I d l B \hat{x} = I L B \hat{x}

\vec{F} = \sum {\vec{F}}_{m n} = 0

d \vec{τ} = \vec{r} \times d {\vec{F}}_{41} + (- \vec{r}) \times d {\vec{F}}_{23}

\begin{aligned} d \vec{τ} & = \vec{r} \times d {\vec{F}}_{41} - \vec{r} \times (- d {\vec{F}}_{41}) \\ = 2 \vec{r} \times d {\vec{F}}_{41} \\ = 2 \frac{b}{2} d \hat{r} \times I B d l \hat{y} \\ = d l b I B \hat{r} \times \hat{y} \\ d \vec{τ} & = d l b I B \hat{n} \times \hat{x} \end{aligned}

\vec{τ} = \int d l b I B \hat{n} \times \hat{x} = a b I B \hat{n} \times \hat{x}

\vec{τ} = \vec{μ} \times \vec{B}; \vec{μ} = I A \hat{n}; A = a b

where, $\vec{μ} =$ magnetic dipole moment, units: $C m^{2} s^{- 1}$

U = - \vec{μ} \cdot \vec{B}