Transcript André Marie

As You Come In…
y
• What is the direction of
the magnetic field acting
on wire 1 from wire 2?
Does it apply a force to
wire 1?
z
x
I2
I1
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
µ0 I(dl × r)
•B=
4π
r3
Jean-Baptiste Biot
Félix Savart
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
Finding Magnetic Fields
• Ideal goal: find magnetic field,
anywhere, for any current source
• Challenge accepted by:
• Jean-Baptiste Biot
• Félix Savart
• Biot-Savart Law
µ0 I(dl × r)
•B=
4π
r3
Jean-Baptiste Biot
Félix Savart
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
Using Biot-Savart’s Law
y
x
z
• 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?
location
I
Using Biot-Savart’s Law
y
x
z
• Magnitude of current constant
throughout wire
• Take a portion of wire of length
dl—has direction of current
B?
x
location
• This wire portion is a distance r
from location—varies with
changing dl
I
r
• δ: angle between dl and r
dl
δ
Using Biot-Savart’s Law
y
x
z
• Applying Biot-Savart equation:
µ0 +∞I(dl × r)
B=
4π -∞ r3
• Step 1: Solve cross product
µ0 +∞I(dlrsinδ)
B=
4π -∞ r3
which simplifies to
µ0 +∞I(dlsinδ)
B=
4π -∞ r2
r and δ will vary—what to do?
B?
x
location
I
r
dl
δ
Using Biot-Savart’s Law
• Step 2: get r and δ in terms of l
𝑥
sin(δ) = so
𝑟
+∞
µ0 I(dlx)
B=
4π -∞ r3
r = l2 + x2 so
µ0 +∞ Ixdl
B=
4π -∞(l2 + x2)3/2
Is there anything you can pull
out of the integral?
y
x
z
B?
x
location
l
dl
I
r
δ
Using Biot-Savart’s Law
y
x
z
• Step 3: pull out constants
µ0 +∞ dl
B = Ix 2 2 3/2
4π -∞(l + x )
Now to do this nasty integral…
B?
x
location
l
dl
Integral table obtained from: http://integral-table.com/downloads/integral-table.pdf
I
r
δ
Using Biot-Savart’s Law
• Step 4: integrate
µ0
l
B = Ix 2 2 2 from l = -∞ to l = +∞
4π x l + x
which simplifies to
µ0I
l
B=
from l = -∞ to l = +∞
2
2
4π x l + x
Now to just determine what this does
infinitely far away….
y
x
z
B?
x
location
l
dl
I
r
δ
Using Biot-Savart’s Law
y
• Step 5: evaluate integral
At +∞, term becomes
µ0I
∞
4π x (∞)2 + x2
which simplifies to
µ0I ∞ µ0I 1
=
4π x∞ 4π x
x
z
B?
x
location
l
dl
I
r
δ
Using Biot-Savart’s Law
y
• Step 5: evaluate integral
At -∞, term becomes
µ0I
−∞
4π x (−∞)2 + x2
which simplifies to
µ0I −∞ µ0I −1
=
4π x∞ 4π x
Now to combine the terms from the
definite integral
x
z
B?
x
location
l
dl
I
r
δ
Using Biot-Savart’s Law
• Step 6: subtract terms
µ0I 1 µ0I −1
B=
4π x 4π x
which simplifies to
µ0I 1 µ0I 1 µ0I 1
B=
+
=2
4π x 4π x
4π x
which reduces to
µ0I
B=
2πx
Look familiar???
y
x
z
B?
x
location
l
dl
I
r
δ
Using Biot-Savart’s Law
• Step 7: find direction of B
Right-hand rule
Thumb in direction of I (or dl)
Fingers curl into the board/paper
B field directed inward
µ0I
• Solution: B =
inward
2πx
y
x
z
B?
x
location
l
dl
I
r
δ
For Tuesday…
• New law tomorrow: get ready for
more math
• Brush up on your torque
knowledge…
André-Marie Ampère
Image obtained from: http://www.astro.cz/_data/images/news/2013/09/27/pe_78b.jpg