Lecture 4 Electric Field Chapter 23 Electric Field 8 Electric field lines Point away from positive and towards negative Tangent to the field line is the direction of the E field at that point lines is proportional to magnitude of the charge Electric Field 9 Electric field E is the force per unit test charge in N C r r F E q0 For a point charge r q0 q F k 2 r so r q E k 2 r Electric Field 10 Direction of E direction of F E points towards away from a negative positive point charge Superposition of electric fields r r r r E E1 E2 En Checkpoint 2 Rank magnitude of net E Checkpoint 2 Rank magnitude of net E a E x k 2q k 3q k 5q i d2 d2 d2 E 5q Ey k 2 j d 50 q E E k d2 2 x 2 y Do this for the rest and find All Equal Electric Field 11 Electric dipole 2 equal magnitude opposite charged particles separated by distance d What s the electric field at point P due to the dipole Electric Field 12 E is on z axis so E E z E E giving where q q E k 2 k 2 r r d r z 2 d r z 2 Electric Field 13 Substituting and rearranging gives 2 2 kq d d E 2 1 1 z 2 z 2 z Assuming z d then expand using binomial theorem ignoring higher order terms d z 1 kq d d E 2 1 1 z z z Electric Field 14 Approximate E field for a dipole is 2 kqd E 3 z Define electric dipole moment p as where direction of p is from the negative to positive end E field along dipole axis at large distances z d is r p qd 2kp E 3 z Electric Field 15 What happens when a dipole is put in an electric field Net force from uniform E is zero But force on charged ends produces a net torque about its center of mass Electric Force 16 r r Definition of torque r F rF sin r For dipole rewrite it as xFsin d x Fsin Substitute F and d to get r r p E r Electric Field 17 Torque acting on a dipole tends to rotate p into the direction of E Associate potential energy U with the orientation of an electric dipole in an E field Dipole has least U when p is lined up with E Electric Field 18 Remember 90 90 U W d pE sin d Potential energy of a dipole r r U pE cos p E U is least greatest when p and E are in same opposite directions Checkpoint 5 Rank a magnitude of torque and b U greatest to least pE sin a Magnitudes are same U pE cos U greatest at 180 b 1 3 tie then 2 4 Electric Field 19 E field from a continuous line of charge Use calculus and a charge density instead of total charge Q Linear charge density Q Length Surface charge density Q Area Volume charge density Q Volume Electric Field 20 Ring of radius R and positive charge density Use q E k 2 r Divide ring into diff elements of charge so dq ds Electric Field 21 Differential dE at P is dq ds dE k 2 k 2 r r From trig r z R 2 2 2 Look for symmetry All cancel and point upward Electric Field 22 Parallel component dE is dE cos k z ds 2 R 2 cos Use trig to rewrite cos z z cos 2 1 2 2 r z R Electric Field 23 Substituting dE cos k z z 2 R 2 3 2 ds Integrate around the ring E dE cos kz 2 r ds z R 2 2 3 2 0 Electric Field 24 Finally get E Replace with kz 2 r z 2 R q 2 r Charge ring has E of E z kqz 2 2 3 2 R 2 3 2 Electric Field 25 Charge ring has E of E z kqz 2 R 2 3 2 Check z R then kq E 2 z From far away ring looks like point charge Electric Field 26 Also do this for charged disk But now surface charge dq A 2 rdr Integrate to get z 1 E 2 2 2 0 z R Electric Field 27 Charge disk of radius R E 2 0 1 2 2 z R z Let R then get E 2 0 Acts as infinite sheet of a nonconductor with uniform charge
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