Two ideal electric dipoles $$A$$ and $$B$$, having their dipole moment $$p_1$$ and $$p_2$$ respectively are placed on a plane with their centres at $$O$$ as shown in the figure. At point $$C$$ on the axis of dipole $$A$$, the resultant electric field is making an angle of 37° with the axis. The ratio of the dipole moment of $$A$$ and $$B$$, $$\frac{p_1}{p_2}$$ is: (take sin 37° = $$\frac{3}{5}$$)

We need to find the ratio of the dipole moments of two ideal electric dipoles, $$\frac{p_1}{p_2}$$, based on the direction of the net electric field at point $$C$$.

1. Analyze the Orientation and Fields at Point $$C$$

From the diagram :

• Dipole $$A$$ (with moment $$p_1$$):
  Point $$C$$ lies along the axial line of dipole $$A$$. The electric field ($$E_1$$) produced by an axial dipole points along its axis:

  $$E_1 = \frac{1}{4\pi\varepsilon_0} \frac{2p_1}{r^3}$$

• Dipole $$B$$ (with moment $$p_2$$):
  Point $$C$$ lies along the equatorial line of dipole $$B$$. The electric field ($$E_2$$) produced by an equatorial dipole is perpendicular to the axial line and points opposite to the direction of the dipole moment:

  $$E_2 = \frac{1}{4\pi\varepsilon_0} \frac{p_2}{r^3}$$

2. Relate the Fields to the Angle of the Resultant Vector

The problem states that the net resultant electric field at point $$C$$ makes an angle of $$37^\circ$$ with the axis of dipole $$A$$. Using vector components:

$$\tan(37^\circ) = \frac{E_2}{E_1}$$

Given that $$\sin(37^\circ) = \frac{3}{5}$$, we know from a standard $$3-4-5$$ right triangle that:

$$\tan(37^\circ) = \frac{3}{4}$$

3. Calculate the Ratio $$\frac{p_1}{p_2}$$

Substitute the expressions for $$E_1$$ and $$E_2$$ into the tangent relationship:

$$\frac{3}{4} = \frac{\frac{1}{4\pi\varepsilon_0} \frac{p_2}{r^3}}{\frac{1}{4\pi\varepsilon_0} \frac{2p_1}{r^3}}$$

Cancel out the common constant factors ($$4\pi\varepsilon_0$$ and $$r^3$$):

$$\frac{3}{4} = \frac{p_2}{2p_1}$$

Cross-multiply to isolate the desired ratio:

$$\frac{p_1}{p_2} = \frac{4}{2 \times 3} = \frac{4}{6} = \frac{2}{3}$$

Conclusion

The ratio of the dipole moments $$\frac{p_1}{p_2}$$ is $$\frac{2}{3}$$, which corresponds exactly to Option C.