Question
Two point dipoles $$p\hat k$$ and $$\frac{p}{2}\hat k$$ are located at $$\left( {0,0,0} \right)$$ and $$\left( {1m,0,2m} \right)$$ respectively. The resultant electric field due to the two dipoles at the point $$\left( {1m,0,0} \right)$$ is
A.
$$\frac{{9p}}{{32\pi { \in _0}}}\hat k$$
B.
$$\frac{{ - 7p}}{{32\pi { \in _0}}}\hat k$$
C.
$$\frac{{7p}}{{32\pi { \in _0}}}\hat k$$
D.
$$\frac{{6p}}{{{ \in _0}}}\hat k$$
Answer :
$$\frac{{ - 7p}}{{32\pi { \in _0}}}\hat k$$
Solution :
The given point is at axis of $$\frac{{\vec p}}{2}$$ dipole and at equatorial line of $${\vec p}$$ dipole so that field at given point is $${{\vec E}_1} + {{\vec E}_2}$$
$$\eqalign{ & {{\vec E}_1} = \frac{{2k\left( {\frac{p}{2}} \right)}}{{{2^3}}} = \frac{{Kp}}{8}\left( { + \hat k} \right) \cr & {{\vec E}_2} = \frac{{Kp}}{1}\left( { - \hat k} \right) \cr & {{\vec E}_1} + {{\vec E}_2} = - \frac{7}{8}Kp\left( { - \hat k} \right) = - \frac{{7p}}{{32\pi { \in _0}}}\hat k \cr} $$
The given point is at axis of $$\frac{{\vec p}}{2}$$ dipole and at equatorial line of $${\vec p}$$ dipole so that field at given point is $${{\vec E}_1} + {{\vec E}_2}$$
$$\eqalign{ & {{\vec E}_1} = \frac{{2k\left( {\frac{p}{2}} \right)}}{{{2^3}}} = \frac{{Kp}}{8}\left( { + \hat k} \right) \cr & {{\vec E}_2} = \frac{{Kp}}{1}\left( { - \hat k} \right) \cr & {{\vec E}_1} + {{\vec E}_2} = - \frac{7}{8}Kp\left( { - \hat k} \right) = - \frac{{7p}}{{32\pi { \in _0}}}\hat k \cr} $$
