All Problems

Constant Electric Field in a Vacuum

Problem 3.39

Demonstrate that the potential of the field generated by a dipole with the electric moment \(\mathbf{p}\) (Fig. 3.4) may be represented as \(\varphi=\mathrm{pr} / 4 \pi \varepsilon_{0} r^{3},\) where \(\mathbf{r}\) is the radius vector. Using this expression, find the magnitude of the electric field strength vector as a function of and \(\theta\)

Reveal Answer
E=Er2+Eθ2=p4πεnr31+3cos⁡2θ, where Er is the radialcomponentofthevector(mathbfE),and(Eheta)isitscomponentperpendicularto(Er)E=\sqrt{E_{r}^{2}+E_{\theta}^{2}}=\frac{p}{4 \pi \varepsilon_{n} r^{3}} \sqrt{1+3 \cos ^{2} \theta}, \text { where } E_{r} \text { is the } radial component of the vector (mathbf{E}), and (E_{ heta}) is its component perpendicular to (E_{r})
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