Green's reciprocity theorem about current density and magnetic field

AI Thread Summary
The discussion focuses on proving three equations related to Green's reciprocity theorem, particularly concerning current density and magnetic fields. The first two equations were addressed using the definition of the electric field, but the third equation poses a challenge due to its reliance on a cross product. A participant explains that the third equation can be proven as a vector identity by expanding the cross products, leading to a conclusion that both sides of the equation equal zero. The reasoning hinges on the properties of dot products and the constancy of the current densities involved. This highlights the interconnectedness of these equations with concepts like Newton's third law and the Lorentz force.
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Homework Statement
I have to prove the Green's reciprocity theorem about the Electric field, electric potential, and magnetic field.
Relevant Equations
Intgral over all space J_2 cross B_1 = -Integral over all space J_1 cross B_2
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I have to prove three equations above.
For first two equations, I've been thought and made reasonable answer by using a definition of the electricfield.
However, for third, I can't use a definition of a magnetic field due to the cross product
Like J_2 X J_1 X (r_2 - r_1).
I think three of 'em are kinda related to Newton's third law of lorentz force.
Does anyone know about the proof?
Or would you think you can prove these?
 
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A:The third equation is simply a vector identity, and can be proved by expanding the cross products. $$\mathbf{J}_2 \times \mathbf{J}_1 \times (\mathbf{r}_2 - \mathbf{r}_1) = (\mathbf{r}_2 - \mathbf{r}_1)\cdot(\mathbf{J}_2 \times \mathbf{J}_1) - (\mathbf{J}_2 \cdot (\mathbf{r}_2 - \mathbf{r}_1))\mathbf{J}_1 + (\mathbf{J}_1 \cdot (\mathbf{r}_2 - \mathbf{r}_1)) \mathbf{J}_2$$The left-hand side of the equation is clearly zero, so the two terms on the right-hand side must also be equal to zero. The first term is equal to zero because the dot product of two vectors perpendicular to each other is zero. The second and third terms are equal to zero because $\mathbf{J}_2$ and $\mathbf{J}_1$ are constant, so they cannot depend on $\mathbf{r}_2 - \mathbf{r}_1$.
 
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