Example 19.4 Electric Field of a Dipole Problem An electric dipole consists of a point charge 7 and a point charge -q separated by a distance of 2a as in the figure. Neutral atoms and molecules behave as dipoles when placed in an external electric field. Furthermore, many molecules, such as HCI, are permanent dipoles. (HCI can be effectively modeled as an H* ion combined with a Cl- ion.) A Find the electric field E due to the dipole along the P y axis at the point P, which is a distance y from the origin. B Find the electric field for points y > a far from the dipole. Strategy For LA, use the sum of the x components of E, and Ez to calculate an equation for the total field E. For LB], use your response from part (A) and adjust the equation for the situation given. The total electric field E at P due to two charges of equal magnitude and opposite sign (an electric dipole) equals the vector sum E, + E2. The field E, is due to the positive charge q, and Ez is the field due to the negative charge -q. Solution A Find the electric field E due to the dipole along the y axis at the point P, which is a distance y from the origin. At P, the fields E, and Ez due to the two particles are equal in magnitude because P is equidistant from the two charges. The total field at Pis E = E, + E2, where the magnitudes of the fields are given by the following expressions. E1 = E2 = ke = = ke- yz ta ? The y components of E, and Ez 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 E is therefore parallel to the x axis and has a magnitude equal to the equation. (1) E = 2ke 72 4 22 COS B From the geometry in the figure we see that cos e = a/r = a/(12 + a2)1/2. Substitute this result into Equation (1): (Use the following as necessary: Ke, y, a, and q.) E = 2ke ( x2 + 2 2 ) ( 12 + 2 2) 1/2 (2) B Find the electric field for points y > a far from the dipole. Equation (2) 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, neglect as in the denominator and write the expression for E in this case: (Use the following as necessary: Ke, y, a, and q.) Therefore, we see that along the y axis the field of a dipole at a distant point varies as 1//, whereas the more slowly varying field of a point charge varies as 1//. (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// variation in E for the dipole s also obtained for a distant point along the x axis and for a general distant point. Exercise 19.4 A piece of aluminum foil of mass 2.7 x 10"2 kg is suspended by a string in an electric field directed vertically upward. If the charge on the foil is 2.2 uC, find the strength of the field that will reduce the tension in the string to zero. N/C