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  3. (a)  Derive an expression for the electric field at any point on the equatorial line of an electric dipole. (b)  Two identical point charges, each, are kept
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(a)  Derive an expression for the electric field at any point on the equatorial line of an electric dipole.

(b)  Two identical point charges, q each, are kept 2m apart in air. A third point        charge Q of unknown magnitude and sign is placed on the line joining the        charges such that the system remains in equilibrium. Find the position and         nature of Q.

 

 

 

 
 
 
 
 

Answers (1)
R rishi.raj

(a) Let us consider the two charges q and -q, be seperated by a distance 

       2a.

On the equatorial line of the dipole let any point P.at a distance r from the line joining the two charges.

Now,

    d=\sqrt{r^{2}+a^{2}}

The electric field at P due to the charge is : for q, the electric field :

    E_{q}=\frac{1}{4\pi \epsilon _{0}}\frac{q}{d^{2}}=\frac{1}{4\pi \epsilon _{0}}\frac{q}{r^{2}+a^{2}}

For -q, the electric field :

    E_{-q}=\frac{1}{4\pi \epsilon _{0}}\frac{q}{d^{2}}=\frac{1}{4\pi \epsilon _{0}}\frac{q}{r^{2}+a^{2}}

Adding horizontal components, such that vertical will cancel out.

Hence, 

    \vec{E}=(\vec{E_{q}}+\vec{E_{-q}})i\cos \theta

Let us suppose, \hat{P} be the unit vector in the direction of the dipole moment.

    \vec{E}=(\vec{E_{q}}+\vec{E_{-q}})\cos \theta\hat{P}

    E=2\; \frac{1}{4\pi \epsilon _{0}}\; \frac{q}{r^{2}+a^{2}}\cos \theta \hat{P}

    \cos \theta =\frac{a}{d}=\frac{a}{\sqrt{r^{2}+a^{2}}}

    \vec{E}=\frac{1}{4\pi \epsilon _{0}}\frac{2qa}{(r^{2}+a^{2})\frac{3}{2}}\hat{P}

    \vec{E}=\frac{1}{4\pi \epsilon _{0}}\frac{2\vec{P}}{(r^{2}+a^{2})\frac{3}{2}}

If, r> > a

Then, E=\frac{1}{4\pi \epsilon _{0}}\frac{2\vec{P}}{r^{3}}

(b) According to the Coulomb's law, All the charges must be in equilibrium and the force depends upon the distance and hence, the position of Q  must be in the mddle of the two charges.

Now, the nature of Q  must have opposite sign to that of q, to balance the forces.

 

 

 

 

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