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Electric dipole, systems of charges Physics 122 5/25/2017 Lecture III 1 Workshops • Due to low interest – 4 people and very limited resources I have to cancel one of the workshops: • Fridays, 4-6 pm B&L 108A • Please let me know alternative times I’ll switch you to other workshops 5/25/2017 Lecture III 2 I am running Rochester marathon • • • • This Saturday, September 17, 8:00 am http://www.rochestermarathon.com/race.htm Starts and ends at Frontier field Goes along East and returns on Park Ave – Lots of coffee shops and sit back, relax and watch people suffer 5/25/2017 Lecture III 3 Concepts • Primary concepts: – Electric field • Secondary concepts: – Electric dipole 5/25/2017 Lecture III 4 Laws • Dipole field • Dipole in electric field: energy and torque • Superposition principle for a continuous distribution of charge 5/25/2017 Lecture III 5 Skills • Calculate electric field of a system of charges 5/25/2017 Lecture III 6 Electric field Q1Q2 F k 2 r F Q1 E E + + 1 2 5/25/2017 • F – force between two charges(N) • Q – electric charge (C= Coulomb) F Q1 E Q2 Ek 2 r • E – electric field created at point 1 by charge 2 • Charge 2 has changed the property of space at point 1 • Charge 1 is experiencing this change Lecture III 7 Superposition of fields E E1 E2 E1 + Positive test charge E2 + - 1 2 5/25/2017 Principle of superposition: Net field created by a system of charges is a vector sum of fields created by individual charges: E E1 E2 E3 .... Lecture III 8 Electric dipole • Two opposite charges of equal value Q separated by distance l • Define dipole moment: p -Q l p Ql +Q p1 • A vector directed from negative charge to positive. • Example – water molecule H+ -O - p H+ p2 5/25/2017 Lecture III 9 Electric Dipole Field Lines y • Lines leave positive charge and return to negative charge What can we observe about E? x • Ex(x,0) = 0 • Ex(0,y) = 0 • Field largest in space between two charges • We derived: E y x,0 k ... for r >> L, 5/25/2017 Lecture III p x 2 p Ek 3 r 3/ 2 L 2 2 10 Torque Force makes objects move torque makes objects rotate rF rF sin 5/25/2017 Lecture III 11 How to add torques? • You have to think… – If the force acts to rotate the system • counterclockwise – torque and angular acceleration are positive + • clockwise – torque and angular acceleration are negative - • Only relative sign matters 5/25/2017 Lecture III 12 How to add torques? Axis of rotation F2 F1 Axis of rotation 1 + 2 1 - 2 F2 F1 5/25/2017 Lecture III 13 How to add torques? Axis of rotation F2 F1 1 - 2 Axis of rotation 1 + 2 F2 F1 5/25/2017 Lecture III 14 Electric dipole • Dipole in uniform E • Net force F=F+-F-=0 • Net torque l l F sin F sin Fl sin 2 2 F QE Fl sin EQl sin Ep sin E p 5/25/2017 Lecture III 15 Electric dipole • Dipole in uniform E • Energy - ? • Work done by the field 2 2 1 1 W d Ep sin d pE (cos 2 cos 1 ) U W pE cos p E 5/25/2017 Lecture III 16 Energy of dipole in electric field U pE cos p -Q p +Q +Q p +Q -Q -Q 0 90 U pE U 0 o 180 o U pE • Lowest energy state – dipole parallel to the field • In electric field dipoles line up with the field 5/25/2017 Lecture III 17 Dipole in electric field Eexternal Eint ernal +Q Eexternal Einternal Eexternal -Q Enet • In electric field dipoles line up with the field • Dipole internal field anti-parallel in external field • Net field is reduced 5/25/2017 Lecture III 18 Test question If this region is filled with pure water (an excellent insulator), does the electric field… A) Increase? B) Decrease? C) Remain the same + + + + + E - + + + + + E - The positive charge is shielded by the negative charges of the aligned dipoles (and vice versa). 5/25/2017 Lecture III 19 The Electric Field of a system of charges Bunch of Charges Charge Distribution E E1 E2 E3 ... qi Ei k 2 , ri - distance from charge i to ri point in space where E is evaluated + + - + + - - dq dE k 2 r E dE + +++ + + + + +++ + + 5/25/2017 Lecture III 20 Vectors by components Charge Distribution dE dq dE k 2 r E dE + +++ + + + + +++ r dE y dE sin dEx dE cos dq r, are different for different charges and depend on your definition of the coordinate system, So choose it wisely 5/25/2017 Lecture III 21 Symmetry and coordinate systems • Coordinate systems are there to help you • You have a choice of – System type • Cartesian • Cylindrical • Spherical – Origin (0,0), Direction of axis • A good choice (respecting the symmetry of the system) can help to simplify the calculations 5/25/2017 Lecture III 22 Ring of charge • A thing ring of radius a holds a total charge Q. Determine the electric field on its axis, a distance x from its center. r x2 a2 a Ex x Qx E k 2 2 3/ 2 (x a ) 5/25/2017 Lecture III E E 23 Charged disk • Disk of radius R, uniformly charged with Q, determine E on the axis, a distance z above the center. • Define charge density s =Q/pr2 • Reuse previous results – divide disk into rings radius r, integrate over r from 0 to R. z z E 2pks 1 2 2 1/ 2 (z R ) z R : s E 2pks 2 0 5/25/2017 Lecture III 24 Two parallel plates • Infinite plates • One positive, one negative, • Same charge density s s E 2 0 s E E E s E 2 0 0 + 5/25/2017 Lecture III 25 Long line of charge Determine the magnitude of the electric field at a distance x from a very long wire of uniformly distributed charge with linear charge density l (C/m). dq=ldy r x2 y2 y dq ldy dE k 2 k 2 2 r x y ldy dE x k 2 2 cos x y 5/25/2017 dE x x dE y l E 2k x Lecture III 26