Physics 212 Lecture 14 Biot-Savart Law :05

Infinite Straight Wire Biot-Savart Law: We can use this law to calculate the magnetic field produced by ANY current distribution BUT Easy analytic calculations are possible only for a few distributions: Infinite Straight Wire Axis of Current Loop :05

Field at the center of a ring P points out of page for each

B from infinite line of current Magnitude: B Current I OUT r • r = distance from wire Direction: Thumb: on I Fingers: curl in direction of B :07

Currents + Charges • • |Fa |> |Fb| |Fa |= |Fb| |Fa |< |Fb| A long straight wire is carrying current from left to right. Two identical charges are moving with equal speed. Compare the magnitude of the force on charge a moving directly to the right, to the magnitude of the force on charge b moving up and to the right at the instant shown (i.e. same distance from the wire). v v • B Forces are in different directions • F (a) (b) r r I |Fa |> |Fb| |Fa |= |Fb| |Fa |< |Fb| Same q, |v|, B and q (=90) 28 :10

Comment on making T for direction, then decide which way is correct Magnetic Fields obey superposition B x Two long wires carry equal and opposite currents x Comment on making T for direction, then decide which way is correct What is the direction of the magnetic field above, and midway between the two wires carrying current – at the point marked “X”? A) Left B) Right C) Up D) Down E) Zero :13

Maybe circle muI1/2pid as B? Force between current-carrying wires B d • • I2 I1 F Conclusion: Currents in same direction attract. • I1 B d F  I2 Maybe circle muI1/2pid as B? Conclusion: Currents in opposite direction repel. :16

Uniform force at every segment of wire Checkpoint 1 X B F What is the direction of the force on wire 2 due to wire 1? A) Up B) Down C) Into Screen D) Out of screen E) Zero What is the direction of the torque on wire 2 due to wire 1? A) Up B) Down C) Into Screen D) Out of screen E) Zero Draw B and F arrows Uniform force at every segment of wire No torque about any axis :18

Checkpoint 3a What is the direction of the force on wire 2 due to wire 1? A) Up B) Down C) Into Screen D) Out of screen E) Zero :21

Consider Force on Symmetric Segments! B I F out of screen I F into screen B r r Net Force is Zero! :22

Checkpoint 3b B i F out of screen F into screen What is the direction of the torque on wire 2 due to wire 1? A) Up B) Down C) Into Screen D) Out of screen E) Zero B i F out of screen F into screen :24

Draw forces on each segment Checkpoint 2 A current carrying loop of width a and length b is placed near a current carrying wire. How does the net force on the loop compare to the net force on a single wire segment of length a carrying the same amount of current placed at the same distance from the wire? Draw forces on each segment The forces are in opposite directions The net forces are the same The net force on the loop is greater than the net force on the wire segment The net force on the loop is smaller than the net force on the wire segment There is no net force on the loop A B C D E :24

Draw forces on each segment Checkpoint 2 A current carrying loop of width a and length b is placed near a current carrying wire. How does the net force on the loop compare to the net force on a single wire segment of length a carrying the same amount of current placed at the same distance from the wire? Ftop + Fbottom =0 X B1 X B2 X B1 F2 B ~ 1/R B1 > B2 F1 Fnet Draw forces on each segment The forces are in opposite directions The net forces are the same The net force on the loop is greater than the net force on the wire segment The net force on the loop is smaller than the net force on the wire segment There is no net force on the loop A B C D E :24

B on axis from Current Loop :24

Biot-Savart works, but need to do numerically What about Off-Axis ?? Biot-Savart works, but need to do numerically See Simulation !! :24

Direction of B reverses Magnetic field of a dipole moment Direction of B reverses from inside to outside. B gets weaker S N

Two Current Loops Two identical loops are hung next to each other. Current flows in the same direction in both. The loops will: A) Attract each other B) Repel each other Two ways to see this: Like currents attract N S 2) Look like bar magnets :30

Right Hand Rule Review 1. ANY CROSS PRODUCT 2. Direction of Magnetic Moment Thumb: Magnetic Moment Fingers: Current in Loop 3. Direction of Magnetic Field from Wire Thumb: Current Fingers: Magnetic Field 06

Calculation Two parallel horizontal wires are located in the vertical (x,y) plane as shown. Each wire carries a current of I =1A flowing in the directions shown. What is the B field at point P? y y I1=1A . 4cm 3cm x z 4cm P I2=1A Front view Side view (A) (B) (C) (D) (E) y . z P What is the direction of B at P produced by the top current I1? The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. :35

Calculation Two parallel horizontal wires are located in the vertical (x,y) plane as shown. Each wire carries a current of I =1A flowing in the directions shown. What is the B field at point P? y y I1=1A . 4cm 3cm x z 4cm P I2=1A Front view Side view What is the direction of B at P produced by the bottom current I2? . z y P The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. (A) (B) (C) (D) (E) :37

Calculation . . . What is the direction of B at P? (A) (B) (C) (D) Two parallel horizontal wires are located in the vertical (x,y) plane as shown. Each wire carries a current of I =1A flowing in the directions shown. What is the B field at point P? y y I1=1A . 4cm 3cm x z 4cm P I2=1A Front view Side view What is the direction of B at P? (A) (B) (C) (D) . z y P . y 90o The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. z P :40

Calculation Two parallel horizontal wires are located in the vertical (x,y) plane as shown. Each wire carries a current of I =1A flowing in the directions shown. What is the B field at point P? y y I1=1A . 4cm 3cm x z 4cm P I2=1A Front view Side view What is the magnitude of B at P produced by the top current I1? (m0 = 4p x 10-7 T-m/A) (A) 4.0 x 10-6 T (B) 5.0 x 10-6 T (C) 6.7 x 10-6 T 3cm z y . 4cm r The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. What is r? r = distance from wire axis to P :45

Calculation Two parallel horizontal wires are located in the vertical (x,y) plane as shown. Each wire carries a current of I =1A flowing in the directions shown. What is the B field at point P? y y I1=1A . 4cm 3cm x z 4cm P Btop = 4 X 10-6 T I2=1A Front view Side view What is the magnitude of B at P? (m0 = 4p x 10-7 T-m/A) (A) 3.2 x 10-6 T (B) 4.8 x 10-6 T (C) 6.4 x 10-6 T (D) 8.0 x 10-6 T y . The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class. 5cm q q B1 4cm z q B2 3cm :49