In Fig. 4.6, p1 and p2 are (perfect) dipoles a distance r apart. What is the torque on p1 due to p2? What is the torque on p2 due to p1? [In each case, I want the torque on the dipole about its own center. If it bothers you that the answers are not equal and opposite, see Prob. 4.29.]

Question

In Fig. 4.6, [latex]P _{1}[/latex] and [latex]P _{2}[/latex] are (perfect) dipoles a distance r apart. What is the torque on [latex]P _{1}[/latex] due to [latex]P _{2}[/latex]? What is the torque on [latex]P _{2}[/latex] due to [latex]P _{1}[/latex]? [In each case, I want the torque on the dipole about its own center. If it bothers you that the answers are not equal and opposite, see Prob. 4.29.]

Accepted Answer

Field of [latex]p _{1} ext { at } p _{2}( heta=\pi / 2 ext { in Eq. } 3.103): E _{1}=\frac{p_{1}}{4 \pi \epsilon_{0} r^{3}} \hat{ heta }[/latex] (points down).

[latex]E _{ dip }(r, heta)=\frac{p}{4 \pi \epsilon_{0} r^{3}}(2 \cos heta \hat{ r }+\sin heta \hat{ heta })[/latex] (3.103)

Torque on [latex]p _{2}: N _{2}= p _{2} imes E _{1}=p_{2} E_{1} \sin 90^{\circ}=p_{2} E_{1}=\frac{p_{1} p_{2}}{4 \pi \epsilon_{0} r^{3}}[/latex] (points into the page).

Field of [latex]p _{2} ext { at } p _{1}( heta=\pi ext { in Eq. } 3.103): E _{2}=\frac{p_{2}}{4 \pi \epsilon_{0} r^{3}}(-2 \hat{ r })[/latex] (points to the right).

Torque on [latex]p _{1}: N _{1}= p _{1} imes E _{2}=\frac{2 p_{1} p_{2}}{4 \pi \epsilon_{0} r^{3}}[/latex] (points into the page).

Question 4.5: In Fig. 4.6, p1 and p2 are (perfect) dipoles a distance r ap...

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