Name: Electric Field of a Sphere
Uploaded: 2019-07-19
Duration: 5 min 10 s
Description: This video demonstrates how an electric field varies as you move away from a point charge or a uniform charge distribution on a sphere. Watch this video.

Question

Derive an expression for the intensity of electric field due to an electric dipole on a point situated on its equatorial line.

Accepted Answer

Consider the dipole $A B$ shown below.

The positive charge $(q)$ is on the point $B$ and the negative charge $(- q)$ is on the point $A$ . We want the electric field at a point $O$ that is at a distance $r$ from the midpoint of the dipole. Recall that we can use the principle of superposition to find the effective electric field $E$ at the point $O$ .

Consider the electric field due to the negative charge on the point $O$ to be $E_{2}$ and the electric field due to the positive charge to be $E_{1}$ . Now, using the formula for the electric field due to a point charge, we have,

$E_{1} = \frac{k q}{O B} & E_{2} = \frac{k q}{O A} \Rightarrow E_{1} = \frac{k q}{(r^{2} + \frac{d^{2}}{4})} & E_{2} = \frac{k q}{(r^{2} + \frac{d^{2}}{4})}$

Since, the directions of these two electric fields are different we need to resolve it into their horizontal and vertical parts. The vertical part of the vectors will cancel each other and the horizontal parts will add up to equate to the net electrical field.

$\Rightarrow E = E_{1} \cos θ + E_{2} \cos θ$

We know from the $∆ O P B$ . the value of $\cos θ$ . Thus,

$\Rightarrow E = \cos θ (E_{1} + E_{2}) \Rightarrow E = \cos θ (\frac{2 k q}{(r^{2} + \frac{d^{2}}{4})}) \Rightarrow E = (\frac{\frac{d}{2}}{\sqrt{(r^{2} + \frac{d^{2}}{4})}}) (\frac{2 k q}{(r^{2} + \frac{d^{2}}{4})}) \Rightarrow E = (\frac{k q d}{{(r^{2} + \frac{d^{2}}{4})}^{\frac{3}{2}}})$

Now, we know that for an ideal dipole, $r > > > d$ . Thus, $r^{2} + \frac{d^{2}}{4} ≅ r^{2}$ .

$\Rightarrow E = (\frac{k q d}{{(r^{2})}^{\frac{3}{2}}}) \Rightarrow E = \frac{k q d}{r^{3}} = \frac{k p}{r^{3}}$

Derive an expression for the intensity of electric field due to an electric dipole on a point situated on its equatorial line.

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