Q21P - Fundamentals Of Physics

Question

Electric quadruple. Figure 22-46 shows a generic electric quadruple. It consists of two dipoles with dipole moments that are equal in magnitude but opposite in direction. Show that the value of E on the axis of the quadruple for a point P a distance z from its center (assume$\mathit{z}\mathbf{ }\mathbf{\gg }\mathit{d}$ ) is given by $\mathit{E}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{o}}}\frac{\mathbf{3}\mathbf{Q}}{{\mathbf{z}}^{\mathbf{4}}}$in which  is known as the quadruple moment $\mathit{Q}\mathbf{ }\mathbf{(}\mathbf{=}\mathbf{2}\mathbf{q}{\mathbf{d}}^{\mathbf{2}}\mathbf{)}$of the charge distribution.

Step-by-Step Solution

Answer

The value of  on the axis of the quadruple for a point P at a distance z from its center is given by$\frac{1}{4\pi {\epsilon }_o}\frac{3Q}{z^4}$ 

 

Step 1: The given data

An electric quadrupole: Two dipoles with dipole moments that are the same in magnitude but opposite in direction.

 

The given quadruple moment is given as:$Q=2q{d}^2$

Step 2: Understanding the concept of electric field

The electric quadruple is composed of two dipoles, each with a dipole moment of magnitude,. The dipole moments point in the opposite directions and produce fields in the opposite directions along the quadruple axis. Considering the point P on the axis, a distance z to the right of the quadruple center takes a rightward pointing field to be positive.

 

The electric field of an electric dipole along the dipole axis by the left dipole,

 

    $E=\frac{1}{2\pi {\epsilon }_o}\frac{qd}{{(z-d/2)}^3}$                                                      (i)

 

The electric field of an electric dipole along the dipole axis by the right dipole,

                                                               

 $E=\frac{-1}{2\pi {\epsilon }_o}\frac{qd}{{(z+d/2)}^3}$                                                                (ii)

Step 3: Calculation of the electric field

Using the binomial expansions of the denominator of equations (i) and (ii), we can get,

 ${(z–d/2)}^{–3}\approx z^{–3}+3z^{–4}(–d/2)$

${(z+d/2)}^{–3}\approx z^{–3}–3z^{–4}(d/2)$

Now, the net electric field from both the dipole points is given by substituting the above values in equations (i) and (ii) as follows:

 $\begin{array}{c}E=\frac{qd}{2\pi {\epsilon }_o}[\frac{1}{z^3}+\frac{3d}{2z^4}-\frac{1}{z^3}+\frac{3d}{2z^4}]\\ =\frac{1}{4\pi {\epsilon }_o}\frac{6q{d}^2}{z^4}\\ =\frac{1}{4\pi {\epsilon }_o}\frac{3Q}{z^4}\text{ (}\because Q=2q{d}^2\text{)}\end{array}$

Hence, the value of the electric dipole is $\frac{1}{4\pi {\epsilon }_o}\frac{3Q}{z^4}$for the given quadruple moment.