lecture 14 biot-savart law p ids r u Ö db 4s r

Physics 212 Lecture 14, Slide 1

Physics 212 Lecture 14

Biot-Savart Law

0

2

ˆ

4

I ds rdB

r

:05


Biot-Savart Law:

:05

0

2

ˆ

4

I ds rdB

r

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

Field at the center of a ring

d s

r̂d B

Pd B points out of page for each d s

0 0

2 24 4P P

I Id s r dd B d B

r r

2 2

0 0

0 04 2

P

I IB d B d

r r

P

d sr

I

ds r dd

0

2

ˆˆ

4

I d s x r rd B r

r r


B from infinite line of current

Current I OUT

•

r = distance from wire

7

0 4 10 /Tm A

0

2

IB

r

r

Magnitude:

B

:07

Direction:

Thumb: on I Fingers: curl in direction of B

Current in WireCurrent in Wire

Resulting B fieldResulting B field

II


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).

a) |Fa |> |Fb|

b) |Fa |= |Fb|

c) |Fa |< |Fb|

v

I

v (a)

r r

(b)

28

Currents + Charges

F qv B

| | sinF qvB

Same q, |v|, B and (=90)

:10

• B

Forces are in different directions

• F

F


Two long wires carry equal and opposite currents

What is the direction of the magnetic field above, and midway between the two wires carrying current – at the point marked “X”?

x

Magnetic Fields obey superposition

A) Left B) Right C) Up D) Down E) Zero

B

x

:13


Force between current-carrying wires

• I1 F

Conclusion: Currents in same direction attract.

•

I1

I2

F

Conclusion: Currents in opposite direction repel.

:16

B d

B d

21 2 1F I L B 21 2 12

oF I L Id

• I2


Checkpoint 1

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

:18

X B

F

Uniform force at every segment of wire

No torque about any axis

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



Checkpoint 3a

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


:21


B

B

I F out of screen

Net Force is Zero!

:22

Consider Force on Symmetric Segments!

I F into screen

r r


Checkpoint 3b

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


B

B

i i

F out of screen F into screen

:24

Checkpoint 2

:24

A B C D E

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?

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 Physics 212 Lecture 14, Slide 12

Checkpoint 2

:24

A B C D E

X B1 X

B2 X B1

F2

B ~ 1/R B1 > B2

Ftop + Fbottom =0

F1 F1

Fnet

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?

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 Physics 212 Lecture 14, Slide 13


B on axis from Current Loop

:24


II



II


3

1B z R

z


What about Off-Axis ??

:24

Biot-Savart works, but need to do numerically

See Simulation !!

http://online.physics.uiuc.edu/courses/phys212/gtm/simulations/BFP.html

B

Magnetic field of a dipole moment

Direction of B reverses

from inside to outside.

B gets weaker

S N


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:

1) Like currents attract

Two Current Loops

:30

N S N S

2) Look like bar magnets


Right Hand Rule Review

06

1. ANY CROSS PRODUCT

BvqF

BLIF

Fr

B

0

2

ˆ

4

I ds rdB

r

2. Direction of Magnetic Moment

Thumb: Magnetic Moment

Fingers: Current in Loop

3. Direction of Magnetic Field from Wire

Thumb: Current

Fingers: Magnetic Field


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

x

.

z

y

P 3cm

I1=1A

Front view

I2=1A

4cm

4cm

Side view

(A) (B) (C) (D) (E)

y

.

z

y

P

.

z

y

P

• What is the direction of B at P produced by the top current I1?

.

z P

.

z

y

P

.

z

y

P

:35



• What is the direction of B at P produced by the bottom current I2?

y

x

.

z

Side view

y

P 3cm

I1=1A

Front view

I2=1A

4cm

4cm

(A) (B) (C) (D) (E)

.

z

y

P

.

z

y

P

.

z

y

P

.

z

y

P

.

z

y

P

:37



• What is the direction of B at P?

y

x

.

z

Side view

y

P 3cm

I1=1A

Front view

I2=1A

4cm

4cm

(A) (B) (C) (D)

.

z

y

P

.

z

y

P

.

z

y

P

.

z

y

P z P

.

y

90o

:40



y

x

.

z

Side view

y

P 3cm

I1=1A

Front view

I2=1A

4cm

4cm

• What is the magnitude of B at P produced by the top current I1? (0 4 x 107 T-m/A)

(A) 4.0 x 106 T (B) 5.0 x 106 T (C) 6.7 x 106 T

0

2

IB

r

• What is r? – r = distance from wire axis to P

3cm z

y

.

4cm r 2 23 4 5r cm

770 (4 10 ) 1

40 102 2 .05

IB

r

:45



y

x

.

z

Side view

y

P 3cm

• What is the magnitude of B at P? (0 4 x 107 T-m/A)

(A) 3.2 x 106 T (B) 4.8 x 106 T (C) 6.4 x 106 T (D) 8.0 x 106 T

I1=1A

Front view

I2=1A

4cm

4cm

3cm z

y

.

4cm 5cm 1 1 cosxB B

B1

B2

2 2 cosxB B

6 6

1

42 cos 2 4 10 6.4 10

5xB B

:49

Btop = 4 X 10-6 T

lecture 14 biot-savart law p ids r u Ö db 4s r

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