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W06D2 Magnetic Forces and Sources of

Magnetic Fields

W06D2 Magnetic Force on Current Carrying Wire, Sources of Magnetic Fields: Biot-Savart Law

Reading Course Notes: Sections 8.3, 9.1-9.2

Announcements

Review Week 6 Tuesday from 9-11 pm in 26-152

PS 6 due Week 6 Tuesday at 9 pm in boxes outside 32-082 or 26-152

W06D3 PS05: Calculating Magnetic Fields and Magnetic Force Reading Course Notes: Sections 8.9, 8.10, 9.10.1, 9.11.1-.3, 9.11.7-.8

Exam 2 Thursday March 21 7:30 - 9:30 pm

Conflict Exam 2 March 22 9-11 am or 10-12 noon

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Outline

Magnetic Force on Current Carrying Wire

Sources of Magnetic Fields

Biot-Savart Law

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Magnetic Force on Current-Carrying Wire

Current is moving charges, and we know that moving charges feel a force in a magnetic field

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magd dq F v B

Magnetic Force on Current-Carrying Wire

d dqdq dq d Id

dt dt

sv s s

magd Id F s B

mag

wire

Id F s B

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Magnetic Force on Current-Carrying Wire

mag

wire

Id

F s B

If the wire is a uniform magnetic field then

If the wire is also straight then

mag ( )I F L B

Demonstration:Wire in a Magnetic Field (Jumping

Wire) G8

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magd Id F s B

http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%208&show=0

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Demonstration:

Series or Parallel Current Carrying Wires G9

You tube video of experiment showing parallel wires attracting

http://youtu.be/rU7QOukFjTo

or repelling:

http://youtu.be/bnbKdbXofEI

http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%209&show=0

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Group Problem: Current Loop

A conducting rod of uniform mass per length is suspended by two flexible wires in a uniform magnetic field of magnitude B which points out of the page and a uniform gravitational field of magnitude g pointing down. If the tension in the wires is zero, what is the magnitude and direction of the current in the rod?

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Sources of Magnetic Fields

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What Creates Magnetic Fields: Moving Charges

http://youtu.be/JmqX1GrMYnU

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Electric Field Of Point Charge

An electric charge produces an electric field:

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1ˆ

4 o

q

rE r

r̂ : unit vector directed from the charged object

to the field point P

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Magnetic Field Of Moving Charge

Moving charge with velocity v produces magnetic field:

2

ˆx

4o q

r

v r

B

:r̂ unit vector directed from charged object to P

r̂

permeability of free space 04 10 7 T m A 1

P

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Currents are Sources of Magnetic Field

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Demonstration:Field Generated by Wire G12

http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%2012&show=0

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Continuous Moving Charge Distributions:

Currents & Biot-Savart

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d dqB v From Charges to Currents?

[coulomb]

[meter]

[sec]

d dqd dq dq d Id

dt dt

sB v s s

[coulomb]

[sec][meter]

v

dq

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The Biot-Savart Law

Current element of length ds pointing in direction of current I produces a magnetic field:

02

ˆ

4

I dd

r

s rB

http://web.mit.edu/viz/EM/visualizations/magnetostatics/MagneticFieldConfigurations/CurrentElement3d/CurrentElement.htm

ds

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The Right-Hand Rule #2

ˆ ˆˆdir ( )P B z r

ˆdir d s z

P18-20

Concept Question: Biot-Savart

The magnetic field at P points towards the

1. +x direction2. +y direction3. +z direction4. -x direction5. -y direction6. -z direction7. Field is zero

P18-21

Concept Question Answer: Biot-Savart

The vertical line segment contributes nothing to the field at P (it is parallel to the displacement). The horizontal segment makes a field pointing into the page by the right-hand rule # 2 (right thumb in direction of current, fingers curl into page at P.

Answer: 6. is in the negative z direction (into page)

( )PB

dir

P18-22

Concept Question: Bent Wire

The magnetic field at P is equal to the field of:

1. a semicircle2. a semicircle plus the field of a long straight wire3. a semicircle minus the field of a long straight wire4. none of the above

P18-23

Concept Question Answer: Bent Wire

All of the wire makes into the page. The two straight parts, if put together, would make an infinite wire. The semicircle is added to this to get the magnetic field

Answer: 2. Semicircle + infinite wire

dirrB(P)

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Repeat Demonstration:Parallel & Anti-Parallel Currents G9

http://tsgphysics.mit.edu/front/?page=demo.php&letnum=G%209&show=0

P18-25

Concept Question: Parallel Wires

Consider two parallel current carrying wires. With the currents running in the opposite direction, the wires are

1. attracted (opposites attract?)

2. repelled (opposites repel?)

3. pushed another direction

4. not pushed – no net force

5. I don’t know

P18-26

Concept Q. Answer: Parallel Wires

I1 creates a magnetic field into the

page at wire 2. That makes a force on wire 2 to the right.

I2 creates a magnetic field into the

page at wire 1. That makes a force on wire 1 to the left.

Answer: 1. The wires are repelled

2 2 2 1I F L B

1 1 1 2I F L B

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Can we understand why?Whether they attract or repel can be seen in the shape of the created B field

You tube videos of field line animations showing why showing parallel wires attract: or repel:

http://youtu.be/5nKQjKgS9z0 http://youtu.be/nQX-BM3GCv4

P18-28

Concept Question: Current Carrying Coils

The above coils have1. parallel currents that attract2. parallel currents that repel3. opposite currents that attract4. opposite currents that repel

P18-29

Concept Q. Answer: Current Carrying Coils

Look at the field lines at the edge between the coils. They are jammed in, want to push out. Also, must be in opposite directions

Answer: 4. Opposite currents that repel

http://web.mit.edu/viz/EM/visualizations/magnetostatics/ForceOnCurrents/MagneticForceRepel/MagneticForceRepel.htm

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Example : Coil of Radius R

Consider a coil with radius R and current I

II

IP

Find the magnetic field B at the center (P)

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Example : Coil of Radius R

Consider a coil with radius R and current I

II

IP

1) Think about it:• Legs contribute nothing

I parallel to r• Ring makes field into page

2) Choose a ds

3) Pick your coordinates

4) Write Biot-Savart

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Animation: Magnetic Field Generated by a Current Loop

http://web.mit.edu/viz/EM/visualizations/magnetostatics/calculatingMagneticFields/RingMagInt/RingMagIntegration.htm

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Example: Coil of Radius RIn the circular part of the coil…

ˆd s r

r̂s

d

II

I

02

ˆ

4

dIdB

r

s rBiot-Savart:

ˆ| d | ds s r

0I

4ds

r 2

0I

4R dR2

0I

4dR

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II

I

Example : Coil of Radius RConsider a coil with radius R and current I

s

d

dB

0I

4dR

B dB

0I

4dR

0

2

0I

4 Rd

0

2

0I

4 R2

rB

0I

2Rdirection into page

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Group Problem:B Field from Coil of Radius R

Consider a coil with radius R and carrying a current I

What is B at point P? WARNING:

This is much harder than the previous problem. Why??