Last Time Magnetic Force Motors and Generators Gauss' Law 1

Features: 1. Proportionality constant 2. Size and shape independence 3. Independence on number of charges inside 4. Charges outside contribute zero Gauss’s Law 2

Today Gauss' Law: Examples Ampere's Law 3

Example: Amount of induced charge inside a conductor Net charge Q ( > 0) on the conductor Hollow region Hollow region Field ? Net charge –q < Net charge induced = +q Net charge (Q-q) on the outside surface

iClicker question 5 A conductor has a net charge of -24  C on it. It also has a hollow region in the interior. If you place a charge of +12 μC in this hollow region, what will be the total charge on the exterior surface of the object? A.-24  C B.12  C C.-36  C D.-12  C E.5  C

Again: Continuous Charge Distribution 1: Charged Line At a point P on perpendicular axis:  x Blast from the Past 6

Infinitely long uniformly charged line r h Gauss’s Law: Same result but much less work! E 7

Long Cylindrical Capacitor 1.Put charges +q on inner cylinder of radius a, -q on outer cylindrical shell of inner radius b. 2. Calculate E by Gauss’ Law a b +q -q 3. Calculate V 4. Divide q by V C dep. log. on a, b L 8

Spherical Capacitor 1.Put charges +q on inner sphere of radius a, -q on outer shell of inner radius b. 2. Calculate E by Gauss’s Law 3. Calculate V from E 4. Divide q by V q is proportional to V C only depends on a,b (isolated sphere) 9

iClicker Question A. B. 0 C. D. 10

Uniformly charged thin, infinite sheet A h Gauss’s Law! 11

Gauss' Law for Magnetism? GAUSS' LAW for charge So far, no experiment has found a "magnetic charge" (a.k.a. magnetic monopole) Big fat ZERO!  Gauss' Law for Magnetism is simpler: GAUSS' LAW FOR MAGNETISM 12

Next Up: Ampere's Law First review Biot-Savart Law 13

Very Close to the Wire B I Very close to the wire: r << L CLOSE TO THE WIRE Blast from the Past Lecture 13 14

Very Long Wire B I Very Long wire: L >> r VERY LONG WIRE Blast from the Past Lecture 13 15

s Viewed from the end Current coming out of board Very Long Wire 16

Very Long Wire Viewed from the end Current coming out of board Cylindrical pattern of B-field  Let's take a line integral along one circle AMPERE'S LAW is along our circle 17

Gauss' Law for Point Charge s Gauss' Law for Point Charge: Works for any size sphere because r cancels Gauss' Law E-field Point Charge On each sphere: Field Surface Area Remember Me? Slide 3 Something similar is going to happen for B of a wire 18

s Current coming out of board Ampere's Law for Long Wire Biot-Savart Law B-field Long Wire On each circle: Field Circumference Works for any size circle because r cancels Ampere's Law for Long Wire: 19

s Current coming out of board Ampere's Law for Long Wire Biot-Savart Law B-field Long Wire On each circle: Field Circumference Works for any size circle because r cancels Ampere's Law for Long Wire:  In any segment, the contribution from any circle is the same. (Like a flashlight.) 20

s Ampere's Law for Long Wire Biot-Savart Law B-field Long Wire Current coming out of board In any segment, the contribution from any "circle" is the same. (Like a flashlight.) On each circle: Field Circumference 21

Ampere's Law for Long Wire Biot-Savart Law B-field Long Wire Current coming out of board In any segment, the contribution from any circle is the same. (Like a flashlight.) s Surround the wire with any shape by following different circles in different places. On each circle: Field Circumference 22

Ampere's Law for Long Wire Biot-Savart Law B-field Long Wire Current coming out of board In any segment, the contribution from any circle is the same. (Like a flashlight.) s Surround the wire with any path by following different circles in different places. Line integral aruond outer surface is always same. On each circle: Field Circumference 23

Ampere's Law for Long Wire Biot-Savart Law B-field Long Wire Current coming out of board In any segment, the contribution from any circle is the same. (Like a flashlight.) s Surround the wire with any path by following different circles in different places. Line integral around outer surface is always same. Limit of small segments  works for any smooth path On each circle: Field Circumference 24

Ampere’s Law in Magnetostatics The path integral of the dot product of magnetic field and unit vector along a closed loop, Amperian loop, is proportional to the net current encircled by the loop, Choosing a direction of integration. A current is positive if it flows along the RHR normal direction of the Amperian loop, as defined by the direction of integration. Biot-Savart’s Law can be used to derive another relation: Ampere’s Law 25

iClicker Question Three currents I 1, I 2, and I 3 are directed perpendicular to the plane of this page as shown. The value of the Ampere’s Law line integral of B∙ dl counterclockwise around the circular path is +  0 I 1. What’s the currents in I 2 and I 3 ? a. I 2 =0, I 3 can be any value b. I 2 =0, I 3 can only be zero c. I 2 =I 1, I 3 can be any value d. I 2 =2I 1, I 3 can be any value e. I 2 =0.5I 1, I 3 can be any value I1I1 I3I3 I2I2 26

iClicker Question Three currents I, 2I, and 3I are directed perpendicular to the plane of this page as shown. What is the value of the Ampere’s Law line integral of B∙ dl counterclockwise around the circular path shown? a. 4  0 I b.  2  0 I c. 2  0 I d. 6  0 I e. zero 3I 2I I 27

iClicker question Ampere’s Law: n windings per unit length 28 Use Ampere’s law to calculate the magnetic field inside a solenoid. (n is number of wraps per unite length). A. B. C. D.

Example: Magnetic field of a long wire outside the wire 29

Example: A Non-Uniform Current Distribution Insider the cylinder, the total current encircled by the Amperian loop is Long, hollow cylindrical current of current density: 30

iClicker Question outside the wire 31 Assume uniform current density, what’s the magnetic field vs. r inside the long wire. A). B). C). D).

iClicker Question outside the wire inside the wire 32