All Problems

Constant Electric Field in a Vacuum

Problem 3.40

A point dipole with an electric moment pp oriented in the positive direction of the zz axis is located at the origin of coordinates. Find the projections EzE_{z} and E1E_{1} of the electric field strength vector (on the plane perpendicular to the zz axis at the point SS (see Fig. 3.4)). At which points is E\mathbf{E} perpendicular to p?

Reveal Answer
 3.40. Ez=p4πε03cos⁡2θ−1r3,E⊥=p4πε03sin⁡θcos⁡θr3; \text { 3.40. } E_{z}=\frac{p}{4 \pi \varepsilon_{0}} \frac{3 \cos ^{2} \theta-1}{r^{3}}, E_{\perp}=\frac{p}{4 \pi \varepsilon_{0}} \frac{3 \sin \theta \cos \theta}{r^{3}} ; E⊥p\mathbf{E} \perp \mathbf{p} at the points located on the lateral surface of a cone whose axis is directed along the zz axis and whose semi-vertex angle θ\theta is found \begin{aligned} &\text { from the relation } \cos \theta=1 / \sqrt{3}\left(\theta_{1}=54.7^{\circ}, \theta_{2}=123.5^{\circ}\right) . \text { At these }\\ &\text { points } E=E_{\perp}=\frac{p \sqrt{2}}{4 \pi \varepsilon_{0} r^{3}} \end{aligned}
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