LECTURE 3 DIPOLES & ELECTRIC FIELD FOR CONTINUOUS CHARGE DISTRIBUTION Lecture 3 2 ¨ Reading chapter 21-6 to 22-1. ¤ Dipoles ¤ Coulomb’s law for continuous charge distributions Electric dipoles 3 ¨ An electric dipole is a system of two equal and opposite charges separated by some distance. ¨ For an electric dipole with a negative charge, -q, and a positive charge, q, separated by a distance L, its dipole moment is defined by ¨ The electric field on the dipole axis at a field point, P, a long way away from the dipole is in the same direction as the dipole, and its magnitude is P |x| Torque on a dipole in an electric field 4 ¨ The torque on a dipole is given by ¨ The magnitude of the torque is given by Quiz: 1 5 Microwave cooking 6 ¨ A microwave oven produces a rapidly oscillating electric field. ¨ Water molecules have permanent dipole moment. ¨ The oscillating field exerts a torque on water molecules in the microwave. ¨ Water molecules often form groups of molecules, the energy from the rotation can causes these groups to break apart, and the broken apart molecules have increased thermal energy. ¨ The water molecules will then form more groups, releasing their electric potential energy as thermal energy, and the process continues. Nonpolar molecules in an electric field 7 ¨ A nonpolar molecule has no permanent dipole moment. ¨ In an external electric field, the positive and negative charges can separate creating a dipole, or polarized. ¨ This is how a charged object can attract a neutral one. ¨ In a nonuniform electric field this can cause the molecule to accelerate. Example: 1 8 ¨ A dipole of moment p = 50. e·nm is placed in a uniform electric field that has a magnitude of E = 4.0×104 NC-1. What is the magnitude of the torque on the dipole when the direction of dipole makes an angle of θ = 30.° with the direction of electric field? Demo: 1 9 ¨ Wooden “needle” ¤ Demonstration of polarization Quiz: 2 10 Continuous charge distribution 11 ¨ If we have a charge Q uniformly distributed over a volume, V, the volume charge density is given by ¨ Similarly, for Q uniformly distributed over an area, A, the area charge density is given by ¨ If we have a charge Q uniformly distributed over a length, L, the linear charge density is given by Example: 2 12 ¨ Calculate the total charge on a thin rod starting at x = 0 and finishing at x = L that has a linear charge density of � � = �� %. Symmetries 13 ¨ An object has: ¤ Rotational symmetry if it does not change under an arbitrary rotation ¤ Translational symmetry if it does not change under an arbitrary translation ¤ Reflection symmetry if it does not change under a reflection ¨ This circle has rotational symmetry about the center of the circle. ¨ This is the same circle rotated 45 degrees. Can you tell? ¨ Does this object have rotational symmetry about the center? ¨ The electric field for an object that has one of these symmetries must have the same symmetry. Quiz: 3 14 E due to continuous charge distribution 15 ¨ The electric field from a small charge element, dq, is given by ¨ The total electric field E due to a continuous charge distribution is calculated by where the integral is over the entire charge distribution. E field from an infinite line of charge 16 ¨ The E field from an infinite line of charge with linear charge density λ: ¨ From rotational symmetry of the charge, the direction of E is radial. Direction: radial dE1+dE2 P r1 R r2 + + + + + + + + + ++ + + dq1 dq2 E field from a ring of charge 17 ¨ The E field along the axis of a circular charge ring with radius a and linear charge density λ or total charge Q: ¨ From rotational symmetry of the charge, E cannot have component perpendicular to the axis. Direction: along axis E field from a disk of charge 18 ¨ The E field along the axis of a circular charge disk with radius b and area charge density σ: ¤ Think of a disk of charge as a collection of thin rings of charge, and integrate from a = 0 to a = b. Direction: normal to disk E field from an infinite plane of charge 19 ¨ The magnitude of the E field from an infinite plane of charge with area charge density σ: ¤ Think of an infinite plane of charge as a disk of charge with an infinite radius. Direction: normal to plane ¤ Note that the magnitude of E does not depend on the distance from the plane. Quiz: 4 20 Example: 3 21 ¨ A non-conducting rod of length L has charge -q uniformly distributed along its length. a) What is the linear charge density of the rod? b) What is the electric field at point P, a distance a from the end of the rod? c) If P were very far from the rod compared to L, the rod would look like a point charge. Show that your answer to b) reduces to the electric field of a point charge for a >> L. -q P - - - - - - - - - - - - - - - - L a.