![Page 1: Electricity & Magnetism Lecture 14 - SFU.ca Lecture 14 - BiotSavart.pdf(() Fundamental(law(for(determining(the(direcFon(and(magnitude(of(the(magneFc(field(due(to(an(elementof(current](https://reader033.vdocument.in/reader033/viewer/2022042212/5eb480b76f2ff516a92e7cbf/html5/thumbnails/1.jpg)
Today’s Concept:
Biot-‐Savart Law
Electricity & MagnetismLecture 14
Electricity & Magne/sm Lecture 14, Slide 1
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Your comments are important to somebody
Could you summarize all the right hand rules?
I think I dislocated my wrist while trying to apply the right hand rule.
Shouldn't all current in a wire flow on the outside surface, and therefore the magneFc field inside it be zero?
![Page 3: Electricity & Magnetism Lecture 14 - SFU.ca Lecture 14 - BiotSavart.pdf(() Fundamental(law(for(determining(the(direcFon(and(magnitude(of(the(magneFc(field(due(to(an(elementof(current](https://reader033.vdocument.in/reader033/viewer/2022042212/5eb480b76f2ff516a92e7cbf/html5/thumbnails/3.jpg)
What is it? Fundamental law for determining the
direcFon and magnitude of the magneFc field due to an element of current.
It is named for Jean-Baptiste Biot and Felix Savart who discovered this relationship in 1820.
Infinite Straight Wire
Axis of Current Loop
GOOD NEWS: Remember Gauss’ Law? Allowed us to calculate E for symmetrical charge distribuFons
NEXT TIME: Introduce Ampère’s Law Allows us to calculate B for symmetrical current distribuFons
Biot-Savart Law:
We can use this law to calculate the magneFc field produced by ANY current distribuFon BUT Easy analyFc calculaFons are possible only for a few distribuFons:
Plan for Today: Mainly use the results of these calculaFons!
Electricity & Magne/sm Lecture 14, Slide 2
Text
![Page 4: Electricity & Magnetism Lecture 14 - SFU.ca Lecture 14 - BiotSavart.pdf(() Fundamental(law(for(determining(the(direcFon(and(magnitude(of(the(magneFc(field(due(to(an(elementof(current](https://reader033.vdocument.in/reader033/viewer/2022042212/5eb480b76f2ff516a92e7cbf/html5/thumbnails/4.jpg)
µ0 = 4� ⇥ 10�7 T ·m/A
Current I OUT
r = distance from wire
r
Magnitude:B
IntegraFng gives result
Direc.on:
Thumb: on IFingers: curl in direc.on of B
B from Infinite Line of Current
•
Electricity & Magne/sm Lecture 14, Slide 3
![Page 5: Electricity & Magnetism Lecture 14 - SFU.ca Lecture 14 - BiotSavart.pdf(() Fundamental(law(for(determining(the(direcFon(and(magnitude(of(the(magneFc(field(due(to(an(elementof(current](https://reader033.vdocument.in/reader033/viewer/2022042212/5eb480b76f2ff516a92e7cbf/html5/thumbnails/5.jpg)
A long straight wire is carrying current from leD to right. Two iden/cal 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|
Currents + Charges
v
I
v(a)
r r(b)
Electricity & Magne/sm Lecture 14, Slide 4
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Two long wires carry opposite current
What is the direc/on of the magne/c field above, and midway between the two wires carrying current – at the point marked “P”?
A) Le< B) Right C) Up D) Down E) Zero
x
Adding Magnetic Fields
P
Electricity & Magne/sm Lecture 14, Slide 5
![Page 7: Electricity & Magnetism Lecture 14 - SFU.ca Lecture 14 - BiotSavart.pdf(() Fundamental(law(for(determining(the(direcFon(and(magnitude(of(the(magneFc(field(due(to(an(elementof(current](https://reader033.vdocument.in/reader033/viewer/2022042212/5eb480b76f2ff516a92e7cbf/html5/thumbnails/7.jpg)
F12 = I2Lµ0
2�dI1I towards
us • •Another I towards usF
Conclusion: Currents in same direcFon a\ract!
•
I towards us ×
Another I away from us
F
Conclusion: Currents in opposite direcFon repel!
d B
Bd
Force Between Current-Carrying Wires
Electricity & Magne/sm Lecture 14, Slide 6
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CheckPoint 2 & 4
What is the direcFon 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 direcFon of the torque on wire 2 due to wire 1?
A) Up B) Down C) Into Screen D) Out of screen E) Zero
Electricity & Magne/sm Lecture 14, Slide 7
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CheckPoint 7
What is the direcFon of the force on wire 2 due to wire 1?
A) Up B) Down C) Into Screen D) Out of screen E) Zero
Electricity & Magne/sm Lecture 14, Slide 8
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B
B
IF out of screen
Net Force is Zero!
I F into screen
rr
Consider Force on Symmetric Segments
Electricity & Magne/sm Lecture 14, Slide 9
What about torque?
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CheckPoint 9
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
Electricity & Magne/sm Lecture 14, Slide 10
What is the net torque on wire 2 due to wire 1?
![Page 12: Electricity & Magnetism Lecture 14 - SFU.ca Lecture 14 - BiotSavart.pdf(() Fundamental(law(for(determining(the(direcFon(and(magnitude(of(the(magneFc(field(due(to(an(elementof(current](https://reader033.vdocument.in/reader033/viewer/2022042212/5eb480b76f2ff516a92e7cbf/html5/thumbnails/12.jpg)
B
B
IF out of screenI F into screen
rr
Consider Force on Symmetric Segments
Electricity & Magne/sm Lecture 14, Slide 9
What about torque?
Text
τ
![Page 13: Electricity & Magnetism Lecture 14 - SFU.ca Lecture 14 - BiotSavart.pdf(() Fundamental(law(for(determining(the(direcFon(and(magnitude(of(the(magneFc(field(due(to(an(elementof(current](https://reader033.vdocument.in/reader033/viewer/2022042212/5eb480b76f2ff516a92e7cbf/html5/thumbnails/13.jpg)
B on axis from Current Loop
..
I
Resul.ng B Field
Current in Wire
Electricity & Magne/sm Lecture 14, Slide 12
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Two idenFcal loops are hung next to each other. Current flows in the same direcFon in both.
The loops will:
A) A`ract each other B) Repel each other
Two Current Loops
Electricity & Magne/sm Lecture 14, Slide 13
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1. ANY CROSS PRODUCT
2. DirecFon of MagneFc Moment
Thumb: Magne/c Moment
Fingers: Current in Loop
3. DirecFon of MagneFc Field from Wire
Thumb: Current
Fingers: Magne/c Field
Right Hand Rule Review
Electricity & Magne/sm Lecture 14, Slide 14
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Conceptual AnalysisEach wire creates a magneFc field at P
B from infinite wire: B = m0I / 2πrTotal magneFc field at P obtained from superposiFon
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
CalculationTwo parallel horizontal wires are located in the verFcal (x,y) plane as shown. Each wire carries a current of I = 1A flowing in the direcFons shown.
What is the B field at point P?
Electricity & Magne/sm Lecture 14, Slide 15
![Page 17: Electricity & Magnetism Lecture 14 - SFU.ca Lecture 14 - BiotSavart.pdf(() Fundamental(law(for(determining(the(direcFon(and(magnitude(of(the(magneFc(field(due(to(an(elementof(current](https://reader033.vdocument.in/reader033/viewer/2022042212/5eb480b76f2ff516a92e7cbf/html5/thumbnails/17.jpg)
Two parallel horizontal wires are located in the verFcal (x,y) plane as shown. Each wire carries a current of I = 1A flowing in the direcFons shown.
What is the B field at point P?
A B C D E
y
.
z
y
P
.
z
y
P
What is the direc/on of B at P produced by the top current I1?
.
z P
.
z
y
P
.
z
y
P
Calculationy
x
.
z
I1 = 1A
Front view Side view
I2 = 1A
4cm
4cm
y
P3cm
Electricity & Magne/sm Lecture 14, Slide 16
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What is the direc/on of B at P produced by the bo`om current I2?
.
z
y
P
.
z
y
P
.
z
y
P
.
z
y
P.
z
y
P
CalculationTwo parallel horizontal wires are located in the verFcal (x,y) plane as shown. Each wire carries a current of I = 1A flowing in the direcFons shown.
What is the B field at point P?
y
x
.
z
I1 = 1A
Front view Side view
I2 = 1A
4cm
4cm
y
P3cm
A B C D E
Electricity & Magne/sm Lecture 14, Slide 17
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What is the direc/on of B at P?
.
z
y
P
..
z
y
P
.
z
y
P
CalculationTwo parallel horizontal wires are located in the verFcal (x,y) plane as shown. Each wire carries a current of I = 1A flowing in the direcFons shown.
What is the B field at point P?
y
x
.
z
I1 = 1A
Front view Side view
I2 = 1A
4cm
4cm
y
P3cm
A B C D
zP
y
Electricity & Magne/sm Lecture 14, Slide 18
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Calculation
What is the magnitude of B at P produced by the top current I1?
(µ0 = 4π 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
Two parallel horizontal wires are located in the verFcal (x,y) plane as shown. Each wire carries a current of I = 1A flowing in the direcFons shown.
What is the B field at point P?
y
x
.
z
I1 = 1A
Front view Side view
I2 = 1A
4cm
4cm
y
P3cm
Electricity & Magne/sm Lecture 14, Slide 19
![Page 21: Electricity & Magnetism Lecture 14 - SFU.ca Lecture 14 - BiotSavart.pdf(() Fundamental(law(for(determining(the(direcFon(and(magnitude(of(the(magneFc(field(due(to(an(elementof(current](https://reader033.vdocument.in/reader033/viewer/2022042212/5eb480b76f2ff516a92e7cbf/html5/thumbnails/21.jpg)
y
What is the magnitude of B at P? (µ0 = 4π 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
Btop = 4 x 10−6 T
CalculationTwo parallel horizontal wires are located in the verFcal (x,y) plane as shown. Each wire carries a current of I = 1A flowing in the direcFons shown.
What is the B field at point P?
y
x
.
z
I1 = 1A
Front view
I2 = 1A
4cm
4cm
y
P3cm
Electricity & Magne/sm Lecture 14, Slide 20