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Lecture 22: MON 09 MAR Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Ch.29.1–2: Magnetic Fields Due to Currents Currents Biot-Savart Law Biot-Savart Law Physics 2102 Jonathan Dowling Jean- Baptiste Biot (1774- Felix Savart (1791–1841) QuickTime™ and a decompressor are needed to see this picture.

Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

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Physics 2102 Jonathan Dowling. Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents. Biot-Savart Law. Felix Savart (1791–1841). Jean-Baptiste Biot (1774-1862). What Are We Going to Learn? A Road Map. - PowerPoint PPT Presentation

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Page 1: Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

Lecture 22: MON 09 MARLecture 22: MON 09 MARCh.29.1–2: Magnetic Fields Due to Ch.29.1–2: Magnetic Fields Due to

CurrentsCurrentsBiot-Savart Law Biot-Savart Law

Physics 2102

Jonathan Dowling

Jean-Baptiste

Biot (1774-1862)

Felix Savart (1791–1841)

QuickTime™ and a decompressor

are needed to see this picture.

Page 2: Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

What Are We Going to Learn?What Are We Going to Learn?A Road MapA Road Map

• Electric charge Electric force on other electric charges Electric field, and electric potential

• Moving electric charges : current • Electronic circuit components: batteries, resistors, capacitors• Electric currents Magnetic field

Magnetic force on moving charges• Time-varying magnetic field Electric Field• More circuit components: inductors. • Electromagnetic waves light waves• Geometrical Optics (light rays). • Physical optics (light waves)

Page 3: Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

Electric Current: Electric Current: A Source of Magnetic FieldA Source of Magnetic Field

IB

Wire withcurrentINTO page

B

• Observation: an electric current creates a magnetic field

• Simple experiment: hold a current-carrying wire near a compass needle!

QuickTi

me™

and

a

dec

ompre

ssor

are n

eeded

to se

e th

is p

icture

.

B

Page 4: Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

Hans Christian Oersted was a professor of science at Copenhagen University. In 1820 he arranged in his home a science demonstration to friends and students. He planned to demonstrate the heating of a wire by an electric current, and also to carry out demonstrations of magnetism, for which he provided a compass needle mounted on a wooden stand.

While performing his electric demonstration, Oersted noted to his surprise that every time the electric current was switched on, the compass needle moved. He kept quiet and finished the demonstrations, but in the months that followed worked hard trying to make sense out of the new phenomenon.

QuickTime™ and a decompressor

are needed to see this picture.

QuickTime™ and a decompressor

are needed to see this picture.

Page 5: Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

Casey’s Right Hand Casey’s Right Hand Rule!Rule!

• Point your thumb along the direction

of the current in a straight wire

• The magnetic field created by the

current consists of circular loops

directed along your curled fingers.

• The magnetic field gets weaker with

distance: For long wire it’s a 1/R

Law!

• You can apply this to ANY straight

wire (even a small differential

element!)

• What if you have a curved wire?

Break into small elements.

iB

Direction of B! i

Page 6: Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

SuperpositionSuperposition• Magnetic fields (like electric fields) can be “superimposed” -- just do a vector sum of B from different sources

• The figure shows four wires located at the 4 corners of a square. They carry equal currents in directions indicated

• What is the direction of B at the center of the square?

B

I-OUT I-OUT

I-IN I-IN

Page 7: Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

When we computed the electric field due to charges we usedCoulomb’s law. If one had a large irregular object, one broke itinto infinitesimal pieces and computed,

rrdq

Ed ˆ4

12

0πε=

rr

rdq

Ed ˆ4

12

0πε=

r

Which we write as,

If you wish to compute the magnetic field due to a current in a wire, you use the law of Biot and Savart.

Biot-Savart LawBiot-Savart Law

Page 8: Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

The Biot-Savart LawThe Biot-Savart Law

• Quantitative rule for computing the magnetic field from any electric current

• Choose a differential element of wire of length dL and carrying a current i

• The field dB from this element at a point located by the vector r is given by the Biot-Savart Law

30

4 r

rLidBd

rrr ×=

πμ

30

4 r

rLidBd

rrr ×=

πμ

μ0 =4πx10–7 T•m/A(permeability constant)

μ0 =4πx10–7 T•m/A(permeability constant)

Jean-Baptiste Biot (1774-1862)

Felix Savart (1791-1841)Ld

r

rr

i

dB

Page 9: Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

dr E =

1

4πε0

dq

r2

) r

dr E =

1

4πε0

dq

r2

) r

Ldr

rr

i

dB

dr B =

μ0

idr L ×

) r

r2

dr B =

μ0

idr L ×

) r

r2

Biot-Savart Law for B-Fields

Coulomb Law for E-Fields

Biot-Savart Requires A Right-Hand Rule

Both Are 1/r2 Laws!The has no units.

) r

Page 10: Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

Biot-Savart LawBiot-Savart Law

• An infinitely long straight wire carries a current i.

• Determine the magnetic field generated at a point located at a perpendicular distance R from the wire.

• Choose an element ds as shown

• Biot-Savart Law: dB points INTO the page

• Integrate over all such elements

30

4 r

rsidBd

rrr ×=

πμ

30

4 r

rsidBd

rrr ×=

πμ

30 )sin(

4 r

ridsdB

θπμ

= 30 )sin(

4 r

ridsdB

θπμ

=

∫∞

∞−

=3

0 )sin(

4 r

rdsiB

θ

π

μ∫∞

∞−

=3

0 )sin(

4 r

rdsiB

θ

π

μ

Page 11: Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

( )∫∞

∞− += 2/322

0

4 Rs

Rdsi

π

μ

( )∫∞

+=

02/322

0

2 Rs

Rdsi

π

μ

( )

⎥⎥⎦

⎢⎢⎣

+=

0

2/1222

0

2 RsR

siR

π

μ

R

i

πμ2

0=

rR /sin =θ rR /sin =θ 2/122 )( Rsr += 2/122 )( Rsr +=

∫∞

∞−

=3

0 )sin(

4 r

rdsiB

θ

π

μ

Field of a Straight Field of a Straight WireWire

30

4 r

rsidBd

rrr ×=

πμ

30

4 r

rsidBd

rrr ×=

πμ

30 )sin(

4 r

ridsdB

θπμ

= 30 )sin(

4 r

ridsdB

θπμ

=

Page 12: Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

Is the B-Field From a Power Line Is the B-Field From a Power Line Dangerous?Dangerous?

A power line carries a current of 500 A.

What is the magnetic field in a house located 100 m away from the power line?

A power line carries a current of 500 A.

What is the magnetic field in a house located 100 m away from the power line?

R

iB

πμ2

0=R

iB

πμ2

0=

=(4π ×10−7T ⋅m / A)(500A)

2π (100m)

=(4π ×10−7T ⋅m / A)(500A)

2π (100m)

= 1 μT= 1 μT

Recall that the earth’s magnetic field is ~10–4T = 100 μTRecall that the earth’s magnetic field is ~10–4T = 100 μT

Probably not dangerous!

Page 13: Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

QuickTime™ and a decompressor

are needed to see this picture.

Page 14: Lecture 22: MON 09 MAR Ch.29.1–2: Magnetic Fields Due to Currents

Biot-Savart LawBiot-Savart Law• A circular arc of wire of radius R carries a current i.

• What is the magnetic field at the center of the loop?

30

4 r

rdsiBd

rr ×=

πμ

30

4 r

rdsiBd

rr ×=

πμ

20

30

44 R

iRd

R

idsRdB

φπμ

πμ

== 20

30

44 R

iRd

R

idsRdB

φπμ

πμ

==

0 0

4 4

iidB

R R

μ μφπ π

Φ= =∫0 0

4 4

iidB

R R

μ μφπ π

Φ= =∫

Direction of B?? Not another right hand rule?!

TWO right hand rules!:If your thumb points along the CURRENT, your fingers will point in the same direction as the FIELD.

If you curl our fingers around direction of CURRENT, your thumb points along FIELD!

i