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The electric field due to small dipole, at point (r, theta), where r is distance from center of the dipole and (theta) is angle between r and dipole,is given by
E=(kp/r^3)[sqrt(1+3cos^2 theta).
If dipole moment p is zero, then E=0,but then there is no dipole which is simply defined as two equal and opposite point charges separated by finite distance. Remember that field of a dipole is obtained by superposingfields of it’s constituent charges. As long as these charges are separated, their resultant field is not zero.
From above formula for field, if 1+3cos^2(theta)=0, then field is zero. But, then cos^2(theta)=-1/3 or cos (theta) is imaginary. This is not possible and hence field of a dipole is not zero.
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