Transcript PPT

Physics 2113
Jonathan Dowling
Lecture 28: MON 23 MAR
Magnetic Fields Due to Currents: Biot-Savart Law
Jean-Baptiste Felix Savart
Biot (1774-1862) (1791–1841)
What Are We Going to Learn?
A 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)
Electric Current:
A Source of Magnetic Field
• Orested’s Observation : an
electric current creates a
magnetic field
• Simple experiment: hold a
current-carrying wire near
a compass needle!
I
B
Wire with
current
INTO page
B
B
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.
New Right Hand Rule!
• Point your thumb along the direction of the
current in a straight wire
i
• The magnetic field created by the current
consists of circular loops directed along
B
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.
Direction of B!
i
29.2: Magnetic Field due to a Long Straight Wire:
Fig. 29-4 A right-hand rule gives the direction of the magnetic field due to a current in a wire. (a) The
magnetic field B at any point to the left of the wire is perpendicular to the dashed radial line and directed
into the page, in the direction of the fingertips, as indicated by the x. (b) If the current is reversed, at any
point to the left is still perpendicular to the dashed radial line but now is directed out of the page, as
indicated by the dot.
Superposition: ICPP
I-OUT
• 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
Ä
I-IN
currents in directions
indicated
• What is the direction of B at
the center of the square?
I-OUT
⊙
Ä
I-IN
B
“Coulomb’s” Law For E-Fields
When we computed the electric field due to charges we used
Coulomb’s law. If one had a large irregular object, one broke it
into infinitesimal pieces and computed dE,
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.
The Biot-Savart Law
For B-Fields
Jean-Baptiste
Biot (1774-1862)
Felix Savart
(1791-1841)
• 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
 0 =4×10–7 T•m/A
(permeability constant of
free space)
i
Ä
Biot-Savart Law
for B-Fields
Coulomb Law
for E-Fields
i
Ä
Biot-Savart Requires A
Right-Hand Rule
Both Are 1/r2 Laws!
The r̂ has no units.
Biot-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
m0 ids (r sin q )
dB =
4p
r3
¥
m0i ds (r sin q )
B=
3
ò
4p -¥
r
Field of a Straight Wire
m0 ids (r sin q )
dB =
3
4p
r
p -q
q -p /2
sinq = cos(q - p / 2) = R / r
¥
r = (s 2 + R 2 )1/ 2
¥
Rds
m0i ds (r sin q ) m0i
=
B=
ò
3
ò
2
2 3/ 2
4
p
4p -¥
r
- ¥ (s + R )
¥
m 0i
Rds
=
2p ò0 (s 2 + R 2 )3 / 2
¥
ù
m0iR é
s
=
ê 2 2
1/ 2 ú
2
2p êë R (s + R ) úû
0
m 0i
=
2pR
Is the B-Field From a Power Line 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?
m 0i
B=
2pR
(4p ´10-7 T × m / A)(500A)
=
2p (100m)
= 1 T
Recall that the earth’s magnetic
field is ~10–4T = 100 mT
Probably not dangerous!
Biot-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?
i
Direction of B?? Not another
right hand rule?!
m0 idsR m 0 iRdf
dB =
=
3
2
4p R
4p R
m0 idf m0iF
B=
=
ò
4p R
4p R
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!
Example, Magnetic field at the center of a circular arc of a circle pg 768:
ICPP: What is the direction of the B
field at point C?
(a) Into the board Ä ? ✔
(b) Out of the board ⊙ ?
(c) To the right ® ?
(d) To the left ¬ ?
Ä
Example, Magnetic field at the center of a circular arc of a circle.:
Example, Magnetic field off to the side of two long straight currents: