Example 19.3 Electric Field of a Dipole Problem An electric dipole consists of a

Question

Example 19.3 Electric Field of a Dipole
Problem An electric dipole consists of a point charge q and a point charge -q separated by a distance of 2a as in Figure 19.11. Neutral atoms and molecules behave as dipoles when placed in an external electric field. Furthermore, many molecules, such as HCl, are permanent dipoles. (HCl can be effectively modeled as an H+ ion combined with a Cl? ion.)

Find the electric field due to the dipole along the y axis at the point P, which is a distance y from the origin.

Find the electric field for points y » a far from the dipole.

Strategy For , use the sum of the x components of 1 and 2 to calculate an equation for the total field . For , use your response from part A and adjust the equation for the situation given.

Figure 19.11 The total electric field at P due to two equal and opposite charges (an electric dipole) equals the vector sum 1 + 2. The field 1 is due to the positive charge q, and 2 is the field due to the negative charge -q.


Solution

At P, the fields 1 and 2 due to the two particles are equal in magnitude because P is equidistant from the two charges. The total field at P is = 1 + 2, where the magnitudes of the fields are given by the following expressions.

The y components of 1 and 2 are equal in magnitude and opposite in sign, so they cancel. The x components are equal and add because they have the same sign. The total field is therefore parallel to the x axis and has a magnitude equal to the equation. (Use the following as necessary: ke, y, a, and q.)


E =

The preceding equation gives the value of the electric field on the y axis at all values of y. For points far from the dipole, for which y » a, we can ignore a2 in the denominator and write the equation. (Use the following as necessary: ke, y, a, and q.)
E ˜
Therefore, we see that along the y axis the field of a dipole at a distant point varies as 1/r3, whereas the more slowly varying field of a point charge varies as 1/r2. (Note: In the geometry of this example, r = y.) At distant points, the fields of the two charges in the dipole almost cancel each other. The 1/r3 variation in E for the dipole is also obtained for a distant point along the x axis and for a general distant point.

Explanation / Answer

The y-components cancel, and since each charge gives kq/d^2 cos to the y-component, the field is equal to 2(kq/(y2+a2) * (a/(y2+a2)) = 2kqa/(y2+a2)3/2. (r=y).

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