As You Come In…

• What is the direction of the magnetic field acting on wire 1 from wire 2? Does it apply a force to wire 1?

I1

⨡

⨡

x

y

I2

z

A Week of Laws

• Lots of very smart people, so little time!• Many scientists made observations

about magnetism (and electricity)• Some derived equations—others,

patterns in behavior• The list:• Biot-Savart Law• Ampere’s Law• Lenz’s Law• Faraday’s Law

Finding Magnetic Fields

• Ideal goal: find magnetic field, anywhere, for any current source• Challenge accepted by:• Jean-Baptiste Biot• Félix Savart

Finding Magnetic Fields

• Ideal goal: find magnetic field, anywhere, for any current source• Challenge accepted by:• Jean-Baptiste Biot• Félix Savart

• Biot-Savart Law• B =

Images obtained from: http://en.wikipedia.org/wiki/Jean-Baptiste_Biot#mediaviewer/File:Jean_baptiste_biot.jpg, and http://cdn.timerime.com/cdn-4/upload/resized/124038/1379226/resized_image2_ebcb59e0639d0abe2ed28df58dc03baf.jpg

⨡ ⨡⨡

Jean-Baptiste Biot

Félix SavartCENSORED??

Finding Magnetic Fields

• Ideal goal: find magnetic field, anywhere, for any current source• Challenge accepted by:• Jean-Baptiste Biot• Félix Savart

• Biot-Savart Law• B =

Images obtained from: http://en.wikipedia.org/wiki/Jean-Baptiste_Biot#mediaviewer/File:Jean_baptiste_biot.jpg, and http://cdn.timerime.com/cdn-4/upload/resized/124038/1379226/resized_image2_ebcb59e0639d0abe2ed28df58dc03baf.jpg

⨡ ⨡⨡

Jean-Baptiste Biot

Félix Savart

Using Biot-Savart’s Law

• Suppose you have an infinitely long wire with current I• Want to know B at a location

that is a distance x from wire x B?

⨡x

y

z

I

⨡location

Using Biot-Savart’s Law

• Magnitude of current constant throughout wire• Take a portion of wire of length

dl—has direction of current• This wire portion is a distance r

from location—varies with changing dl• δ: angle between dl and r

x B?

⨡x

y

z

I⨡

location

dl

r

⨡

δ

Using Biot-Savart’s Law

• Applying Biot-Savart equation:B =

• Step 1: Solve cross productB = which simplifies toB = r and δ will vary—what to do?

x B?

⨡x

y

z

I⨡

location

dl

r

⨡

δ

⨡ ⨡⨡

+∞

-∞

+∞

-∞

+∞

-∞

Using Biot-Savart’s Law

• Step 2: get r and δ in terms of lsin(δ) = soB = r = soB = Is there anything you can pull out of the integral?

x B?

⨡x

y

z

I⨡

location

dl

r

⨡

δ

l+∞

-∞

+∞

-∞

Using Biot-Savart’s Law

• Step 3: pull out constantsB = IxNow to do this nasty integral… x B?

⨡x

y

z

I⨡

location

dl

r

⨡

δ

l

+∞

-∞

Integral table obtained from: http://integral-table.com/downloads/integral-table.pdf

Using Biot-Savart’s Law

• Step 4: integrateB = Ix from l = -∞ to l = +∞which simplifies toB = from l = -∞ to l = +∞Now to just determine what this does infinitely far away….

x B?

⨡x

y

z

I⨡

location

dl

r

⨡

δ

l

Using Biot-Savart’s Law

• Step 5: evaluate integralAt +∞, term becomes which simplifies to =

x B?

⨡x

y

z

I⨡

location

dl

r

⨡

δ

l

Using Biot-Savart’s Law

• Step 5: evaluate integralAt -∞, term becomes which simplifies to = Now to combine the terms from the definite integral

x B?

⨡x

y

z

I⨡

location

dl

r

⨡

δ

l

Using Biot-Savart’s Law

• Step 6: subtract termsB = - which simplifies toB = + = 2which reduces toB = Look familiar???

x B?

⨡x

y

z

I⨡

location

dl

r

⨡

δ

l

Using Biot-Savart’s Law

• Step 7: find direction of BRight-hand ruleThumb in direction of I (or dl)Fingers curl into the board/paperB field directed inward

• Solution: B = inward

x B?

⨡x

y

z

I⨡

location

dl

r

⨡

δ

l

⨡

For Tuesday…

• New law tomorrow: get ready for more math• Brush up on your torque

knowledge…

Image obtained from: http://www.astro.cz/_data/images/news/2013/09/27/pe_78b.jpg

André-Marie Ampère