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Introductory Physics Homework Help
Potential vector of a oscilating dipole:
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[QUOTE="TSny, post: 6631751, member: 229090"] A straightforward, but somewhat tedious, method is to work in cartesian coordinates. Consider the ##x##-component of ##\nabla \left[ \frac {\mathbf{\hat r} \cdot \mathbf{ \dot p}(t_0)}{r} \right]##: $$\left(\nabla \left[ \frac {\mathbf{\hat r} \cdot \mathbf{ \dot p}(t_0)}{r} \right]\right)_x = \frac{\partial}{\partial x} \left[ \frac{x}{r^2} \dot p_x(t_0) + \frac{y}{r^2} \dot p_y(t_0) +\frac{z}{r^2} \dot p_z(t_0) \right]$$ Expand this out. You can drop terms that fall off faster than ##1/r##. When calculating ##\large \frac{\partial \dot p_x(t_0)}{\partial x}##, recall that ##t_0 = t-r/c =t-\sqrt{x^2+y^2+z^2}/c##. So, you can use the chain rule to write $$ \frac{\partial \dot p_x(t_0)}{\partial x} = \frac{d \dot p_x}{dt_0} \frac{\partial t_0}{\partial x}$$ Likewise for dealing with ##\large \frac{\partial \dot p_y(t_0)}{\partial x}## and ##\large \frac{\partial \dot p_z(t_0)}{\partial x}##. You should find that $$\left(\nabla \left[ \frac {\mathbf{\hat r} \cdot \mathbf{ \dot p}(t_0)}{r} \right]\right)_x = \left( \frac {\mathbf{\hat{r}}}{r} \cdot \mathbf{\ddot p}(t_0)\right) \frac{\partial t_0}{\partial x}$$ Similar results are obtained for the ##y## and ##z## components of ##\nabla \left[ \frac {\mathbf{\hat r} \cdot \mathbf{ \dot p}(t_0)}{r} \right]##. [/QUOTE]
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Potential vector of a oscilating dipole:
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