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Chapter 25: Q. 68 (page 713)

The arrangement of charges shown in FIGURE P25.68 is called a linear electric quadrupole. The positive charges are located aty=卤s. Notice that the net charge is zero. Find an expression for the electric potential on the y-axis at distancesy>>s. Give your answer in terms of the quadrupole moment,Q=2qs2.

Short Answer

Expert verified

The expression for electric potential ony-axisat distancey>>siskQy3

Step by step solution

01

Given Information

We have given that the positive charges are located aty=卤s.

We need to find that an expression for the electric potential on the y-axisat distancesy>>s.

02

Simplification

As the electric potential is a scalar quantity, we can write the potential at point yon the y-axiswill be given as

V=kqr,

which when considering the distances and charges

V=kq1y-s+1y+s-2y

And further expanding this expression, we will get

V=kqy(y-s)+y(y-s)-2(y+s)(y-s)y(y-s)(y+s)

One ought not to panic, as the following can be simplified as

V=kqy2+ys+y2-ys-2y2+2ys-2sy+2s2y(y-s)(y+s)=kq2s2y(y-s)(y+s)

Now, let us make the simplification that for y>>s,

y(y-s)(y+s)鈮�y3,

and using the definition of the quadrupole moment Q=2qs2, we can express the result as

V=kQy3

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