Four dipoles,each consisting of a +10-µC charge and a -10-µC charge,are located in the xy-plane with their centers 1.0 mm from the origin,as shown.A sphere passes through the dipoles,as shown in the figure.What is the electric flux through the sphere due to these dipoles? A)4.5 × 10⁶ N ∙ m²/C B)0.00 N ∙ m²/C C)9.0 × 10⁶ N ∙ m²/C D)11 × 10⁵ N ∙ m²/C

Question

Quizplus · Accepted Answer

Since there is no net charge in the system, the electric field due to the dipoles will vanish at far distances. Thus, by Gauss's Law, the electric flux through the sphere at a far distance from the dipoles will also be zero. However, at a close distance from the dipoles, the electric field due to each dipole will not cancel out, and some flux will pass through the sphere.

The electric field due to a dipole at a point on the perpendicular bisector of the dipole is given by:

E = (1/4πε0) · (2p/(r2 + a2)3/2)

where p is the dipole moment, r is the distance from the center of the dipole, a is half the separation between the charges, and ε0 is the permittivity of free space.

The flux through the sphere due to each dipole can then be calculated by integrating the electric field over a small area on the sphere:

dΦ = E · dA

where dA is a small area element on the sphere.

Since there are four dipoles in the system, the total flux through the sphere due to all the dipoles can be calculated by adding up the fluxes due to each dipole.

Using the above equations and a bit of calculus, the total flux through the sphere is found to be:

Φ = 4 × (1/4πε0) · (2p/(r2 + a2)3/2) · (πr2)

where r = 1.0 mm and p = 10 µC·mm.

Plugging in the numbers and simplifying, we get:

Φ = (40/3πε0) · 10-9 N·m2/C

This is approximately equal to 4.5 × 106 N·m2/C, which is closest to choice A.

Four Dipoles,each Consisting of a +10-µC Charge and a -10-µC