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# Electric Fields

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1. Electric Fields

2. Electrostatic Phenomena Coulomb’s Law r 1 q q F F1 F2 = ˆ 1 2 F r 12 12 pe 2 4 r 12 o q1 Superposition Ftotal = F1+ F2 + ... q q2

3. Two balls of equal mass are suspended from the ceiling with nonconducting wire. One ball is given a charge +3q and the other is given a charge +q. +q +3q +3q +q (a) (b) g +3q +q Which of the following best represents the equilibrium positions? +3q +q (c)

4. +q +3q +q +3q +q (a) (b) (c) The force on the +3q charge due to the +qcharge must be equal and opposite to the force of the +3q charge on the +qcharge Amount of charge on each ball determines the magnitude of the force, but each ball experiences the same magnitude of force Symmetry, therefore, demands (c) P.S. Knowing the form of Coulomb’s law you can write two equations with two unknowns (T andq) Which best represents the equilibrium position? +3q Remember Newton’s Third Law!

5. Preflight 2: +10 μC +1 μC a) b) c) Two charges q = + 1 μC and Q = +10 μC are placed near each other as shown in the figure. 6) Which of the following diagrams best describes the forces acting on the charges:

6. The Electric Field A simple, yet profound observation - The net Coulomb force on a given charge is always proportional to the strength of that charge. q1 = F F1 + F2 F1 F æ ö r ˆ ˆ q q r q r q ç ÷ = + 1 1 2 2 F ç ÷ pe 2 2 4 r r è ø 0 1 2 F2 q2 test charge - Wecan now define a quantity, the electric field, which is independent of the test charge, q, and depends only on position in space: r r F The qiare the sources of the electric field º E q

7. The Electric Field r r F º E q With this concept, we can “map” the electric field anywhere in space produced by any arbitrary: F Bunch of Charges Charge Distribution + + - + + + + - + + + - + + - + + + + + + + “Net” E at origin - These charges or this charge distribution “source” the electric field throughout space

8. Example: Electric Field y 1) Notice that the fields from the top-right and bottom left cancel at the origin? a a +q +q 2) The electric field, then, is just the field from the top -left charge. It points away from the top-left charge as shown. 3) Magnitude of E-field at the origin is: a a Q x a +q kq = E 2a2 kq cosq = Ex 2a2 kq - sinq = Ey kq 1 2a2 = 2a2 2 kq 1 - = 2a2 2 What is the electric field at the origin due to this set of charges? The x and y components of the field at (0,0) are:

9. Example: Electric Field y a a q 1 +q = +q k E x 2 2 a 2 a a x Q a +q Note: If the charge Q is positive, the force will be in the direction of the electric field If the charge Q is negative, the force will be against the direction of the electric field F is F is Now, a charge, Q, is placed at the origin. What is the net force on that charge?

10. Let’s Try Some Numbers... If q = 5mC, a = 5cm, and Q = 15mC. y Then Ex= 6.364  106 N/C and Ey= -6.364  106 N/C a a +q +q a a x Fx = QEx and Fy = QEy a Fx=95.5 N Fy=-95.5 N So... +q and We also know that the magnitude of E = 9.00  106 N/C We can, therefore, calculate the magnitude of F F = |Q| E = 135N

11. y E Q1 d Q2 x Two charges, Q1 and Q2, fixed along the x-axis as shown produce an electric field, E, at a point (x,y)=(0,d)which is directed along the negative y-axis. - Which of the following is true? (a) Both chargesQ1 and Q2are positive (b) Both chargesQ1andQ2are negative (c) The chargesQ1andQ2have opposite signs

12. y E Q1 d Q2 x E E E Q1 Q1 Q2 Q2 Q1 Q2 (a) (b) (c) Two charges, Q1 and Q2, fixed along the x-axis as shown produce an electric field, E, at a point (x,y)=(0,d)which is directed along the negative y-axis. - Which of the following is true? (a) Both chargesQ1 and Q2are positive (b) Both chargesQ1andQ2are negative (c) The chargesQ1andQ2have opposite signs

13. Ways to Visualize the E Field vector map field lines + chg + chg + + Consider the E-field of a positive point charge at the origin

14. Rules for Vector Maps + chg + Direction of arrow indicates direction of field Length of arrowsµlocal magnitude of E

15. Rules for Field Lines - + Field at two white dots differs by a factor of 4 sincer differs by a factor of 2 Local density of field lines also differs by a factor of 4 (in 3D) • Lines leave (+) charges and return to (-) charges • Number of lines leaving/entering chargeµamount of charge • Tangent of line = direction of E • Local density of field linesµlocal magnitude of E

16. Preflight 2: 6) A negative charge is placed in a region of electric field as shown in the picture. Which way does it move ? a) up c) left e) it doesn't move b) down d) right

17. 7) Compare the field strengths at points A and B. a) EA > EB b) EA = EB c) EA < EB

18. y 2a +Q a a x a -Q (b) Ex(2a,a) = 0 (c) Ex(2a,a) > 0 (a) Ex(2a,a) < 0 • Consider a dipole (2 separated equal and opposite charges) with the y-axis as shown. • Which of the following statements aboutEx(2a,a)is true?

19. y 2a +Q a a x a -Q (b) Ex(2a,a) = 0 (c) Ex(2a,a) > 0 (a) Ex(2a,a) < 0 Ex Solution: Draw some field lines according to our rules. • Consider a dipole (2 separated equal and opposite charges) with the y-axis as shown. • Which of the following statements aboutEx(2a,a)is true?

20. Preflight 2: 3) What is the direction of the electric field at point A? a) up b) down c) left d) right e) zero 4) What is the direction of the electric field at point B? a) up b) down c) left d) right e) zero Two equal, but opposite charges are placed on the x axis. The positive charge is placed at x = -5 m and the negative charge is placed at x = +5m as shown in the figure above.

21. Field Lines From Two Opposite Charges Dipole Dipoles are central to our existence!

22. The Electric Dipole y +Q a x q a E E r -Q Ex = ?? Ey = ?? Symmetry Ex(x,0) = 0 see the appendix for further information What is the E-field generated by this arrangement of charges? Calculate for a point along x-axis: (x, 0)

23. Electric Dipole Field Lines What can we observe about E? Ex(x,0) = 0 Ex(0,y) = 0 We derived: ... for r >> a, Lines leave positive chargeand return to negative charge Field largest in space between two charges

24. Field Lines From Two Like Charges 4 • There is a zero halfway between the two charges • r >> a: looks like the field of point charge (+2q) at origin

25. y + + + + + + + + + R + + + + x + + + + + + + + + Consider a circular ring with total charge +Q. The charge is spread uniformly around the ring, as shown, so there is λ = Q/2pR charge per unit length. The electric field at the origin is (b) (a) zero (c)

26. y + + + + + + + + + R + + + + x + + + + + + + + + The key thing to remember here is that the total field at the origin is given by the VECTOR SUM of the contributions from all bits of charge. If the total field were given by the ALGEBRAIC SUM, then (b) would be correct. (exercise for the student). Note that the electric field at the origin produced by one bit of charge is exactly cancelled by that produced by the bit of charge diametrically opposite!! Therefore, the VECTOR SUM of all these contributions is ZERO!! Consider a circular ring with total charge +Q. The charge is spread uniformly around the ring, as shown, so there is λ = Q/2pR charge per unit length. The electric field at the origin is (b) (a) zero (c)

27. Electric Field inside a Conductor 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ 2+ There is never a net electric field inside a conductor – the free charges always move to exactly cancel it out. • A two electron atom, e.g., Ca • heavy ion core • two valence electrons • An array of these atoms • microscopically crystalline • ions are immobile • electrons can move easily • Viewed macroscopically: • neutral

28. Summary Define E-Field in terms of force on “test charge” How to think about fields Electric Field Lines Example Calculation: Electric Dipole

29. Appendix A: Other ways to Visualize the E Field Graphs Ex, Ey, Ezas a function of(x, y, z) Er, Eq, Efas a function of(r, q, f) Field Lines + chg Ex(x,0,0) + x Consider a point charge at the origin

30. Appendix A- “ACT” y Consider a point charge fixed at the origin of a coordinate system as shown. –Which of the following graphs best represent the functional dependence of the Electric Field for fixed radius r? r f x Q Er Er Er 3A Fixed r > 0 2p 0 0 f 2p 0 f f 2p (a) (b) (c) Ex Ex Ex 3B Fixed r > 0 0 0 0 f f 2p f 2p 2p

31. Appendix A “ACT” y Consider a point charge fixed at the origin of a coordinate system as shown. – Which of the following graphs best represent the functional dependence of the Electric Field for fixed radiusr? r f x Q Er Er Er 3A Fixed r > 0 2p 0 0 f 2p 0 f f 2p (a) (b) (c) • At fixed r, the radial component of the field is a constant, independent off!! • For r>0, this constant is > 0. (note: the azimuthal component Ef is, however, zero)

32. Appendix A “ACT” Ex Ex Ex At fixed r, the horizontal component of the field Ex is given by: 3B Fixed r > 0 0 0 0 f f 2p f 2p 2p y Consider a point charge fixed at the origin of a coordinate system as shown. –Which of the following graphs best represent the functional dependence of the Electric Field for fixed radius r? r f x Q (a) (b) (c)

33. Appendix B: Electric Dipole y +Q a x q a E E r -Q What is the E-field generated by this arrangement of charges? Calculate for a point along x-axis: (x, 0) Ex = ?? Ey = ?? Symmetry Ex(x,0) = 0

34. Coulomb Force Radial E y Electric Dipole +Q What is the Electric Field generated by this charge arrangement? a x a -Q Now calculate for a pt along y-axis: (0,y) Ey = ?? Ex = ??

35. y Electric Dipole r +Q a x a Case of special interest: (antennas, molecules) r > > a -Q For pts along x-axis: For pts along y-axis: For r >>a For r >>a,