Today’s Concept:What Causes Magnetic Fields

Physics 2112Unit 14

20 ˆ

4 r

rsdIBd

Unit 14, Slide 1

Compare to Electric Fields

Unit 14, Slide 2

212

120 ˆ

4 r

rvqB

212

12ˆ

4

1

r

rqE

o

In the same direction as r12 Perpendicular to r12

v out of the screen

ATm /104 70

Unit 14, Slide 3

20 ˆ

4 r

rsdIBd

212

120 ˆ

4 r

rvqB

Biot-Savart Law

B field from one moving charge

avgqnAvI But remember from previous slides

B field from tiny of current

carrying wire.sd

What is the magnetic field a distance yo away from a infinitely long wire of current I?

Example 14.1 (Infinite wire of current)

20 ˆ

4 r

rsdIBd

Unit 14, Slide 4

Conceptual Plan

Strategic AnalysisDone in prelecture in detail

Integrate

(Similar to E field for infinite line of charge)

Use Biot-Savart Law

Unit 14, Slide 5

Main Idea

20 ˆ

4 r

rsdIB

)(

)sin(*

4 22

0

oyx

dxI

f

Q

r̂sd

sd

Current I OUTr

Magnitude:B

Example 14.1 (answer)

r

IB

2

0 •

Unit 14, Slide 6

ATm /104 70

Remember:

rE

o 2

Example 14.2 (B field from hexagon)

Unit 14, Slide 7

b

A current, I, flows clockwise through a hexagonal loop of wire. The perpendicular distance between each side and the center of the loop is b.

What is the magnetic field in the center of the loop?

f

Q r̂sd

sd

120o

Example 14.3 (From Loop)

Unit 14, Slide 8

yo

A current, I, flows clockwise through a circular loop of wire. The loop has a radius a.

What is the magnetic field at a point P a distance yo above the plane of the loop in the center?a

P

x

x

Q

BcosQ

Q

Force Between Current-Carrying Wires

BLIF

212

1212 2I

dLIF o

Unit 14, Slide 9

I1 r

IB

2

10X X X X X X X X X X X X X X X X X X XX X X X X X X X X X X X X

X X X X X X X X X X X X

X X X X X X X X X X

. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

I2

F

I towards us • • Another I towards usF

Conclusion: Currents in same direction attract!

•

I towards us

Another I away from usF

Conclusion: Currents in opposite direction repel!

d B

Bd

Force Between Current-Carrying Wires

BLIF

212 1212 2I

dLIF o

Unit 14, Slide 10

Example 14.4 (Two Current Carrying Wires)

Two current carrying wires a 10cm apart for a length for 50cm. Wire 1 carries 5A and Wire 2 carries 10A with both current to the left.

What is the magnitude and direction of the force on wire 2 due to wire 1?

Unit 14, Slide 11

I1 = 5A

I2 = 10A

10cm

50cm

CheckPoint 1A

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

XB

F

Unit 14, Slide 12

CheckPoint 1B

What is the direction of the torque on wire 2 due to wire 1?

Uniform force at every segment of wire

No torque about any axis

Unit 14, Slide 13

A) Up B) Down C) Into Screen D) Out of screen E) Zero

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

Unit 14, Slide 14

CheckPoint 3B

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

Unit 14, Slide 15

LET’S DRAW A PICTURE!

Checkpoint 2: Force on a loop

A. the forces are in opposite directions

B. the net forces are the same.

C. the net force on the loop is greater than the net force on the wire segment

D. the net force on the loop is smaller than the net force on the wire segment

E. there is no net force on the loop

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?

Checkpoint question

Current flows in a loop as shown in the diagram at the right. The direction is such that someone standing at point a and looking toward point b would see the current flow clockwise.

What is the orientation of the magnetic field produced by the loop at points a and b on the axis?

(A)

(B)

(C)

(D)

B on axis from Current Loop

..I

Resulting B Field

Current in Wire

Electricity & Magnetism Lecture 14, Slide 18

What about Off-Axis ?

Biot-Savart Works, but need to do numerically

See Simulation!

Unit 14, Slide 19

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 C) There is no force between them

Two Current Loops

Unit 14, Slide 20

Two ways to see this:

1) Like currents attract

N S N S

2) Look like bar magnets

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

Right Hand Rule Review

BvqF

BLIF

Fr

B

20 ˆ

4 r

rsdIBd

Unit 14, Slide 21

Conceptual AnalysisEach wire creates a magnetic field at P

B from infinite wire: B = m0I / 2prTotal magnetic field at P obtained from superposition

y

x

.

z

I1 = 1A

Front view Side view

Strategic AnalysisCalculate B at P from each wire separatelyTotal B = vector sum of individual B fields

I2 = 1A

4cm

4cm

y

P3cm

Example 14.2

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?

Unit 14, Slide 22

If I = 6A, what is the magnitude of the magnetic field at point P?

Example 14.5 (Curved Loop of Wire)

P

20cm

12cm Conceptual Plan

Strategic Analysis

Integrate both loops

Note straight sections cancel out.

20 ˆ

4 r

rsdIBd

Use Biot-Savart Law

Good News!!!!!

Unit 14, Slide 24

Remember how we used Gauss’ Law to avoid doing integral in E

field?

We got similar law for B fields!