College Physics 150 Chapter 19 – Magnetic Forces and Fields
• Magnetic Fields
• Magnetic Force on a Point Charge
• Motion of a Charged Particle in a Magnetic Field
• Crossed E and B fields
• Magnetic Forces on Current Carrying Wires
• Torque on a Current Loop
• Magnetic Field Due to a Current
• Ampère’s Law
• Magnetic Materials
Magnetic Fields
Magnetic Dipole
All magnets have at least one north pole and one south pole.
Field lines emerge from north poles and enter through south poles.
Opposite magnetic poles aLract and like magnetic poles repel.
If a magnet is broken in half you just end up with two magnets.
Magnets exert forces on one another.
Magnetic field lines are closed loops. There is no (known!) source of magnetic field lines. (No magnetic monopoles)
This means that you can never have only a north or south pole…they always come in pairs.
Magnetism typically comes from moving charges or electron spin.
e-­‐‑
Local magnetism – material magnetism
e-­‐‑
Non-­‐‑local magnetism – electrical magnetism
Near the surface of the Earth, the magnetic field is that of a dipole.
The Earth’s magnetic field is due to the flow of charged plasma in the core, not due to the iron ore.
Note the orientation of the magnetic poles!
Magnetic Force on a Point Charge
The magnetic force on a point charge is:
FB = q(v × B)
The unit of magnetic field (B) is the tesla (1T = 1 N/Am).
The magnitude of FB is:
FB = qB(v sin θ )
where vsinθ is the component of the velocity perpendicular to the direction of the magnetic field. θ represents the angle between v and B.
v
θ
Draw the vectors tail-­‐‑to-­‐‑tail to determine θ.
B
v
θ
B
The direction of FB is found from the right-­‐‑hand rule.
C = A×B
For a general cross product:
The right-­‐‑hand rule is: using your right hand, point your fingers in the direction of A and curl them in the direction of B. Your thumb points in the direction of C.
Note : C = A × B ≠ B × A
Example (text problem 19.20): An electron moves with speed 2.0×105 m/s in a 1.2 Tesla uniform magnetic field. At one instant, the electron is moving due west and experiences an upward magnetic force of 3.2×10-­‐‑14 N. What is the direction of the magnetic field? FB = qBv sin θ
FB
sin θ =
= 0.8323
qBv
θ = 56°
y
θ
v (west)
The angle can be either north of west OR north of east.
θ
F (up)
x
Charged Particle Moving Perpendicular to a Uniform B-­‐‑field
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A positively charged particle has a velocity v (orange arrow) as shown. The magnetic field is into the page.
The magnetic force, at this instant, is shown in blue. In this region of space this positive charge will move CCW in a circular path.
∑F = F
Applying Newton’s 2nd Law to the charge:
B
= mar
v2
qvB = m
r
Example: How long does it take an electron to complete one revolution if the radius of its path is r (see the figure on slide 11)?
The distance traveled by the electron during one revolution is d = 2πr. The electron moves at constant speed so d = vT as well. The speed of the electron can be obtained using the result of the previous slide.
2π r 2π r 2π me
T=
=
=
eBr
v
eB
me
Is the period of the electron’s motion.
Mass Spectrometer
A charged particle is shot into a region of known magnetic field.
B
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Detector
2
Here,
v
qvB = m
r
or qBr = mv
V
Particles of different mass will travel different distances before striking the detector. (v, B, and q can be controlled.)
Motion of a Charged Particle in a Uniform B-­‐‑field
If a charged particle has a component of its velocity perpendicular to B, then its path will be a circle. If it also a component of v parallel to B, then it will move forward as well. This resulting path is a helix. Crossed E and B Fields
If a charged particle enters a region of space with both electric and magnetic fields present, the force on the particle will be F = Fe + FB
= qE + q(v × B ).
Charge q> 0 with velocity v
Consider a region of space with crossed electric and magnetic fields.
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B (into page)
E
F = Fe + FB
= qE + q(v × B ).
Charge q> 0 with velocity v
Consider a region of space with crossed electric and magnetic fields.
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B (into page)
E
F = Fe + FB = 0.
The value of the charge’s speed can be adjusted so that net
The net force equal zero will occur when v = E/B.
This region of space (with crossed E and B fields) is called a “velocity selector”. It can be used as part of a mass spectrometer.
Magnetic Force on a Current Carrying Wire
The force on a current carrying wire in an external magnetic field is
F = I (L × B)
L is a vector that points in the direction of the current flow. Its magnitude is the length of the wire.
F = ILB sin θ
The magnitude is and its direction is given by the right-­‐‑hand rule.
Example (text problem 19.50): A 20.0 cm by 30.0 cm loop of wire carries 1.0 A of current clockwise.
(a) Find the magnetic force on each side of the loop if the magnetic field is 2.5 T to the left.
I = 1.0 A
Left: F out of page
Top: no force
B Right: F into page
BoLom: no force
The magnitudes of the nonzero forces are:
F = ILB sin θ
= (1.0 A )(0.20 m )(2.5 T )sin 90°
= 0.50 N
(b) What is the net force on the loop?
Fnet = 0
Torque on a Current Loop
Consider a current carrying loop in a magnetic field. The net force on this loop is zero, but the net torque is not.
Axis
Force into page
Force out of page
B
L/2
L/2
Axis
Force into page
Force out of page
B
L/2
L/2
τ = NIAB sin θ
The net torque on the current loop is:
N = number of turns of wire in the loop.
I = the current carried by the loop.
A = area of the loop.
B = the magnetic field strength.
θ = the angle between A and B.
Axis
Force into page
Force out of page
B
L/2
L/2
τ = NIAB sin θ
The net torque on the current loop is:
The direction of A is defined with a right-­‐‑hand rule. Curl the fingers of your right hand in the direction of the current flow around a loop and your thumb will point in the direction of A.
Because there is a torque on the current loop, it must have both a north and south pole. A current loop is a magnetic dipole.
(Your thumb, using the above RHR, points from south to north.)
Magnetic Field due to a Current
Moving charges (a current) create magnetic fields.
The magnetic field at a distance r from a long, straight wire carrying current I is µ0 I
B=
2πr
where µ0 = 4π×10-­‐‑7 Tm/A is the permeability of free space.
The direction of the B-­‐‑field lines is given by a right-­‐‑hand rule. Point the thumb of your right hand in the direction of the current flow while wrapping your hand around the wire; your fingers will curl in the direction of the magnetic field lines.
Magnetic Field due to a Current
Moving charges (a current) create magnetic fields.
The magnetic field at a distance r from a long, straight wire carrying current I is µ0 I
B=
2πr
where µ0 = 4π×10-­‐‑7 Tm/A is the permeability of free space.
A wire carries current I out of the page.
The B-­‐‑field lines of this wire are CCW. Note: The field (B) is tangent to the field lines.
Example (text problem 19.72): Two parallel wires in a horizontal plane carry currents I1 and I2 to the right. The wires each have a length L and are separated by a distance d. 1
I
d
2
I
(a) What are the magnitude and direction of the B-­‐‑field of wire 1 at the location of wire 2?
µ 0 I1
B1 =
2πd
Into the page
Example (text problem 19.72): Two parallel wires in a horizontal plane carry currents I1 and I2 to the right. The wires each have a length L and are separated by a distance d. 1
I
d
2
I
(b) What are the magnitude and direction of the magnetic force on wire 2 due to wire 1?
F12 = I 2 LB1 sin θ
µ 0 I1 I 2 L
= I 2 LB1 =
2πd
F12 toward top of page (toward wire 1)
Example (text problem 19.72): Two parallel wires in a horizontal plane carry currents I1 and I2 to the right. The wires each have a length L and are separated by a distance d. 1
I
d
2
I
(c) What are the magnitude and direction of the B-­‐‑field of wire 2 at the location of wire 1?
µ0 I 2
B2 =
2πd
Out of the page
Example (text problem 19.72): Two parallel wires in a horizontal plane carry currents I1 and I2 to the right. The wires each have a length L and are separated by a distance d. 1
I
d
2
I
(d) What are the magnitude and direction of the magnetic force on wire 1 due to wire 2?
F21 = I1 LB2 sin θ
µ 0 I1 I 2 L
= I1 LB2 =
2πd
F21 toward boLom of page (toward wire 2)
Example (text problem 19.72): Two parallel wires in a horizontal plane carry currents I1 and I2 to the right. The wires each have a length L and are separated by a distance d. 1
I
d
2
(e) Do parallel currents aLract or repel? They aLract.
(f) Do antiparallel currents aLract or repel? They repel. I
The magnetic field of a current loop:
The strength of the B-­‐‑field at the center of the (single) wire loop is: 0
B=
µI
2R
The magnetic field of a solenoid:
The magnetic field of a solenoid:
The field inside a solenoid is nearly uniform (if you stay away from the ends) and has a strength:
B = µ0 nI
Where n = N/L is the number of turns of wire (N) per unit length (L) and I is the current in the wire.
Ampère’s Law
Ampère’s Law relates the magnetic field on a path to the net current cuLing through the path.
Example (text problem 19.80): A number of wires carry current into or out of the page as indicated.
(a) What is the net current though the interior of loop 1?
Example (text problem 19.80): A number of wires carry current into or out of the page as indicated.
(a) What is the net current though the interior of loop 1?
Assume currents into the page are negative and current out of the page are positive.
Loop 1 encloses currents -­‐‑3I, +14I, and -­‐‑6I. The net current is +5I or 5I out of the page.
Example (text problem 19.80): A number of wires carry current into or out of the page as indicated.
(b) What is the net current though the interior of loop 2?
Loop 2 encloses currents -­‐‑16I and +14I. The net current is -­‐‑2I or 2I into the page.
Consider a wire carrying current into the page. Draw a closed path around the wire.
circulation = ∑ B||Δl
Here the B-­‐‑field is tangent to the path everywhere (hence the choice of a circular path). The circulation is
∑ B Δl = B(2πr ).
||
∑ B||Δl = µ0 I
Ampère’s Law is
where I is the net current that cuts through the circular path.
If the wire from the previous page carries a current I then the magnetic field at distance r from the wire is µ0 I
B=
.
2πr
Magnetic Materials
Ferromagnetic materials have domains, regions in which its atomic dipoles are aligned, giving the region a strong dipole moment.
When the domains are oriented randomly there will be no net magnetization of the object.
When the domains are aligned, the material will have a net magnetization.
Hall Voltage (VH)
B
If an electric current flows through a conductor in a magnetic field, the magnetic field exerts a transverse force on the moving charge carriers which tends to push them to one side of the conductor.
IB
VH =
ned
n = carrier density
d = thickness