Question

P5 Consider two current loops that carry $I_1$ and $I_2$, respectively. We have already learned that the force between these two current loops are: $\vec{F}_1 = \frac{\mu_0 I_1 I_2}{4\pi} \oint_{L_1} \oint_{L_2} \frac{d\vec{l}_1 \times (d\vec{l}_2 \times \vec{R}_{21})}{|\vec{R}_{21}|^3}$ $\vec{F}_2 = \frac{\mu_0 I_1 I_2}{4\pi} \oint_{L_1} \oint_{L_2} \frac{d\vec{l}_2 \times (d\vec{l}_1 \times \vec{R}_{12})}{|\vec{R}_{12}|^3}$ where $\vec{F}_1$ is the total force on current loop 1 due to current loop 2, and $\vec{F}_2$ is the total force on current loop 2 due to current loop 1. Please show that $\vec{F}_1 = -\vec{F}_2$. Hint: vector curl product satisfies $\vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \cdot \vec{C})\vec{B} - (\vec{A} \cdot \vec{B})\vec{C}$; you also need to apply Stokes's theorem which converts a closed line integral to a surface integral.

          P5 Consider two current loops that carry $I_1$ and $I_2$, respectively. We have already learned that the force between these two current loops are:

$\vec{F}_1 = \frac{\mu_0 I_1 I_2}{4\pi} \oint_{L_1} \oint_{L_2} \frac{d\vec{l}_1 \times (d\vec{l}_2 \times \vec{R}_{21})}{|\vec{R}_{21}|^3}$

$\vec{F}_2 = \frac{\mu_0 I_1 I_2}{4\pi} \oint_{L_1} \oint_{L_2} \frac{d\vec{l}_2 \times (d\vec{l}_1 \times \vec{R}_{12})}{|\vec{R}_{12}|^3}$

where $\vec{F}_1$ is the total force on current loop 1 due to current loop 2, and $\vec{F}_2$ is the total force on current loop 2 due to current loop 1. Please show that $\vec{F}_1 = -\vec{F}_2$.
Hint: vector curl product satisfies $\vec{A} \times (\vec{B} \times \vec{C}) = (\vec{A} \cdot \vec{C})\vec{B} - (\vec{A} \cdot \vec{B})\vec{C}$; you also need to apply Stokes's theorem which converts a closed line integral to a surface integral.

P5 Consider two current loops that carry I1 and I2, respectively. We have already learned that the force between these two current loops are:

F⃗1 = (μ0 I1 I2)/(4π)∮L1∮L2(dl⃗1 × (dl⃗2 ×R⃗21))/(|R⃗21|^3)

F⃗2 = (μ0 I1 I2)/(4π)∮L1∮L2(dl⃗2 × (dl⃗1 ×R⃗12))/(|R⃗12|^3)

where F⃗1 is the total force on current loop 1 due to current loop 2, and F⃗2 is the total force on current loop 2 due to current loop 1. Please show that F⃗1 = -F⃗2.
Hint: vector curl product satisfies A⃗× (B⃗×C⃗) = (A⃗·C⃗)B⃗ - (A⃗·B⃗)C⃗; you also need to apply Stokes's theorem which converts a closed line integral to a surface integral.

Added by Melvin M.

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