Summary
Lecture 16
A long wire with electric current produces a magnetic field around the wire.
B=
Magnetic field due to straight long wire
with current at distance r from the wire:
Solenoids and Electromagnets
Ampere’s Law
Torque on Current Loop
The constant μ0 is called the “permeability
of free space” or “magnetic constant”.
μ0 I
2π r
μ 0 = 4π × 10 −7 T ⋅ m / A
Magnetic force between two wires is used for definition of the units for current
(ampere) and charge (coulomb).
The magnetic force between two
long straight wires with current:
Fm =
μ 0 I1 I 2
l
2π d
Two wires with parallel currents attract each other, two wires with untiparallel
currents repel each other.
Physics 112, Spring 2010, Feb 19, Lecture 16
Physics 112, Spring 2010, Feb 19, Lecture 16
Direction of Magnetic Field in Solenoid
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Magnetic Field Produced by Solenoid
Solenoid: a long coil of wire consisting of many loops (or turns) of wire.
The direction of the magnetic field
inside the solenoid:
1) up
2) down
Solenoid has north and south poles
similarly to a permanent magnet.
3) to the left
4) to the right
S
Physics 112, Spring 2010, Feb 19, Lecture 16
N
3
Right-hand rule #2:
magnetic field of
loop with current.
Physics 112, Spring 2010, Feb 19, Lecture 16
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Magnetic Field Produced by Solenoid
Magnetic Field Produced by Solenoid
Cross-sectional view into a solenoid.
100 loops of wire form a 20-cm long solenoid. If the current in the solenoid is
2 A, what is the magnetic field inside of the solenoid?
B=
μ 0 IN
l
μ0 is the permeability of free
space” or “magnetic constant”.
Uniform magnetic field produced
inside of long solenoid:
The constant μ0 is called the “permeability
of free space” or “magnetic constant”.
B=
μ 0 = 4π × 10 −7 T ⋅ m / A = (4)(3.14) × 10 −7 T ⋅ m / A
μ 0 IN
l
B=
μ 0 = 4π × 10 −7 T ⋅ m / A
Physics 112, Spring 2010, Feb 19, Lecture 16
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Ampere’s Law
Ampere’s Law: Application to Long Straight Wire
Ampere’s law:
∑B
We will take the product of the length of
each segment times the component of
magnetic field B parallel to that segment.
Δl = μ 0 I encl
For circular path around the wire,
B// = B for any segment of the path.
Δl = μ 0 I encl
Component of B
parallel to each
segment Δl.
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Physics 112, Spring 2010, Feb 19, Lecture 16
Ampère’s law shows a general relation between a current in wires of any
shape and the magnetic field around them.
∑B
(4π ×10 −7 T ⋅ m / A)(2 A)(100)
= 1.26 ×10 −3 T
0.2m
∑B
μ 0 = 4π ×10 −7 T ⋅ m / A
B=
* surface doesn’t necessarily need to be flat.
It just has to be bounded by the closed path.
Physics 112, Spring 2010, Feb 19, Lecture 16
Δl = B ∑ Δl = B(2π r ) = μ 0 I
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Small segment
Δl.
μ0 I μ0 I
=
2πr 2π r
Physics 112, Spring 2010, Feb 19, Lecture 16
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2
Ampere’s Law: Application to Solenoid
Magnetic Force on Loop with Current (#1)
Red dashed lines indicate the
path chosen for use in
Ampere’s law.
If the coil is rectangular and aligned with the field, the forces on the
individual edges can be calculated for each wire in the coil.
Max force when the area of
the loop is parallel to B.
( BII Δl ) cd = μ 0 I encl
Zero force on
these edges of
coil (parallel to B)
( BII Δl ) cd = Bl
( BII Δl ) ab + ( BII Δl ) bc + ( BII Δl ) cd + ( BII Δl ) da = μ 0 I encl
≈
=
=
0
0
0
Uniform magnetic field produced
inside of long solenoid:
Bl = μ 0 I
Bl = μ 0 NI
B=
The loop experiences a torque.
What is a torque?
μ 0 IN
Max force on these edges of coil
(perpendicular to B):
l
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Physics 112, Spring 2010, Feb 19, Lecture 16
Magnetic Force on Loop with Current (#2)
r
B
Physics 112, Spring 2010, Feb 19, Lecture 16
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If a coil of wire is placed in a magnetic field and a current (I ) flows in the
wire, there may be a torque on the loop.
1. What is the direction of the magnetic force acting on the loop?
2. What is the torque acting on the loop?
r
F1
Fmax = IaBN
Torque on Current Loop: Magnetic Dipole Moment
The loop with current is placed in a magnetic field as shown below.
zero
Fmax = IaB
The coil with N loops and area A and a uniform field B experiences the
torque of:
up and down
I
τ = NIAB sin θ
NIA = “magnetic dipole moment”, M.
M is a vector perpendicular to the face
of the coil with magnitude:
Axis
M = NIA
I
r
F2
Physics 112, Spring 2010, Feb 19, Lecture 16
θ is the angle between M and B.
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Physics 112, Spring 2010, Feb 19, Lecture 16
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Torque and Force on Current Loop (#1)
Torque and Force on Current Loop (#2)
A rectangular coil of wire with a = 30 cm, b = 20 cm, and contains 10
loops. The current in each loop is 3.00 A, and the coil is placed in a 2.00-T
magnetic field. Find the maximum torque and force exerted on the coil by
the magnetic field.
A rectangular coil of wire with a = 30 cm, b = 20 cm, and contains 10
loops. The current in each loop is 3.00 A, and the coil is placed in a 2.00-T
magnetic field. Find the maximum torque and force exerted on the coil by
the magnetic field.
B
b
1. The area of one loop of the coil:
−2
A = ab = (0.300 m)(0.200 m) = 6 × 10 m
a
2
τ = Fl
b/2
τ max = 2 Fmax
I
2. The maximum torque occurs when the
coil’s face is parallel to the magnetic filed
(θ=900, sin900=1):
B
b
3. The maximum force on the coil:
I
Fmax =
τ max
b
=
b
= Fmaxb
2
a
3.60 N ⋅ m
= 18 N
0.20 m
b/2
I
I
or
τ max = NIAB sin θ = (10)(3.00 A)(6 ×10 m )(2.00 T )(1) = 3.60 N ⋅ m
−2
2
Fmax = IaBN = (3.00 A)(0.30 m)(2.00 T )(10) = 18 N
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Physics 112, Spring 2010, Feb 19, Lecture 16
Magnetic Field
Physics 112, Spring 2010, Feb 19, Lecture 16
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Magnetic Field
We have a loop made of one wire with current. The loop is placed in a space
without any magnetic field. How we can define its magnetic dipole moment?
The source of a magnetic field are:
This is:
1) both positive and negative electric charges at rest
1) a vector parallel to the loop area with the magnitude equals to
the product (current)x(area)
2) only positive electric charges in motion
2) a vector perpendicular to the loop area with the magnitude
equals to the ratio (current)/(area)
3) both positive and negative charges in motion
3) a vector perpendicular to the loop area with the magnitude
equals to the product (current)x(area)
4) only negative charges in motion
4) there is no magnetic dipole moment because there is no
external magnetic field
Note:
The electric charge is a source of the electric field while
moving electric charge is a source of the electric field and
magnetic field!
NIA = “magnetic dipole moment”, M.
M is a vector perpendicular to the face
of the coil with magnitude:
Physics 112, Spring 2010, Feb 19, Lecture 16
M = NIA
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Physics 112, Spring 2010, Feb 19, Lecture 16
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