**Unformatted text preview: **10/29/21, 8:23 AM Magnetic Fields, Magnetic Forces, and Conductors | Boundless Ph… Boundless Physics
Magnetism Magnetic Fields, Magnetic Forces, and
Conductors … 1/14 10/29/21, 8:23 AM Magnetic Fields, Magnetic Forces, and Conductors | Boundless Ph… The Hall Effect
When current runs through a wire exposed to a magnetic field a potential is produced across the conductor that is transverse to the current. LEARNING OBJECTIVES Express Hall voltage for a a metal containing only one type of
charge carriers KEY TAKEAWAYS Key Points The Hall effect is the phenomenon in which a voltage
difference (called the Hall voltage) is produced across
an electrical conductor that is transverse to the
conductor’s electric current when a magnetic field perpendicular to the conductor’s current is applied.
Moving charges in a wire will change trajectory in the
presence of a magnetic field, “bending” toward it. Thus,
those charges accumulate on one face of the material.
On the other face, there is left an excess of opposite
charge. Thus, an electric potential is created.
VH = − IB
net is the formula for Hall voltage (VH). It is a factor of current (I), magnetic field (B), thickness of the … 2/14 10/29/21, 8:23 AM Magnetic Fields, Magnetic Forces, and Conductors | Boundless Ph… conductor plate (t), and charge carrier density (n) of the
carrier electrons.
Key Terms elementary charge: The electric charge on a single proton.
transverse: Not tangent, so that a nondegenerate angle is formed between the two things intersecting. The Hall effect is the phenomenon in which a voltage difference (called
the Hall voltage) is produced across an electrical conductor, transverse
to the conductor’s electric current when a magnetic field perpendicular
to the conductor’s current is applied.
When a magnetic field is present that is not parallel to the motion of
moving charges within a conductor, the charges experience the Lorentz
force. In the absence of such a field, the charges follow a roughly
straight path, occasionally colliding with impurities.
In the presence of a magnetic field with a perpendicular component, the
paths charges take becomes curved such that they accumulate on one
face of the material. On the other face, there is an excess of opposite
charge remaining. Thus, an electric potential is created so long as the
charge flows. This opposes the magnetic force, eventually to the point of
cancelation, resulting in electron flow in a straight path.
For a metal containing only one
type of charge carrier (electrons),
the Hall voltage (VH) can be calculated as a factor of current (I),
magnetic field (B), thickness of
the conductor plate (t), and
charge carrier density (n) of the
carrier electrons: VH = − IB
net Hall Effect for Electrons: Initially, the
electrons are attracted by the
magnetic force and follow the … 3/14 10/29/21, 8:23 AM Magnetic Fields, Magnetic Forces, and Conductors | Boundless Ph… In this formula, e represents the
elementary charge.
The Hall coefficient (RH) is a characteristic of a conductor’s material, and is defined as the ratio of curved arrow. Eventually, when
electrons accumulate in excess on
the left side and are in deficit on the
right, an electric field ξy is created.
This force becomes strong enough
to cancel out the magnetic force, so
future electrons follow a straight
(rather than curved) path. induced electric field (Ey) to the
product of current density (jx) and applied magnetic field (B): RH = Ey
j xB = VHt
IB = − 1
ne The Hall effect is a rather ubiquitous phenomenon in physics, and appears not only in conductors, but semiconductors, ionized gases, and in
quantum spin among other applications. Magnetic Force on a Current-Carrying Conductor
When an electrical wire is exposed to a magnet, the current in that wire
will experience a force—the result of a magnet field. LEARNING OBJECTIVES Express equation used to calculate the magnetic force for an
electrical wire exposed to a magnetic field KEY TAKEAWAYS Key Points Magnetic force on current can be found by summing
the magnetic force on each of the individual charges
that make this current. … 4/14 10/29/21, 8:23 AM Magnetic Fields, Magnetic Forces, and Conductors | Boundless Ph… For a wire exposed to a magnetic field,F = IlB sin θdescribes the relationship between magnetic force (F),
current (I), length of wire (l), magnetic field (B), and angle
between field and wire (θ).
The direction of the magnetic force can be determined
using the right hand rule, as in fig [[17951]].
Key Terms drift velocity: The average velocity of the free charges in a conductor.
magnetic field: A condition in the space around a mag- net or electric current in which there is a detectable
magnetic force, and where two magnetic poles are
present. When an electrical wire is exposed to a magnet, the current in that wire
will be affected by a magnetic field. The effect comes in the form of a
force. The expression for magnetic force on current can be found by
summing the magnetic force on each of the many individual charges that
comprise the current. Since they all run in the same direction, the forces
can be added.
The force (F) a magnetic field (B)
exerts on an individual charge (q)
traveling at drift velocity vd is:
F = qv dB sin θ In this instance, θ represents the
angle between the magnetic field
and the wire (magnetic force is
typically calculated as a cross
product). If B is constant throughout a wire, and is 0 elsewhere,
Right Hand Rule: Used to
determine direction of magnetic
force. then for a wire with N charge car- … 5/14 10/29/21, 8:23 AM Magnetic Fields, Magnetic Forces, and Conductors | Boundless Ph… riers in its total length l, the total
magnetic force on the wire is: F = NqvdB sin θ . Given that N=nV, where n is the number of charge carriers per unit volume and V is volume of the wire, and that this volume is calculated as
the product of the circular cross-sectional area A and length (V=Al),
yields the equation: F = (nqAvd )lB sin θ . The terms in parentheses are equal to current (I), and thus the equation
can be rewritten as: F = IlB sin θ The direction of the magnetic force can be determined using the right
hand rule, demonstrated in. The thumb is pointing in the direction of the
current, with the four other fingers parallel to the magnetic field. Curling
the fingers reveals the direction of magnetic force. Torque on a Current Loop: Rectangular and General
A current-carrying loop exposed to a magnetic field experiences a
torque, which can be used to power a motor. LEARNING OBJECTIVES Identify the general quation for the torque on a loop of any
shape KEY TAKEAWAYS … 6/14 10/29/21, 8:23 AM Magnetic Fields, Magnetic Forces, and Conductors | Boundless Ph… Key Points
τ = NIAB sin θ can be used to calculate torque (τ ) a loop of N turns and A area, carrying I current feels in the
presence of a magnetic field B.
Although the forces acting upon the loop are equal and
opposite, they both act to rotate the loop in the same
direction.
Torque experienced is independent of the loop’s shape.
What matters is the area of the loop.
Key Terms torque: A rotational or twisting effect of a force; (SI unit newton-meter or Nm; imperial unit foot-pound or ft-lb) When a current travels in a loop that is exposed to a magnetic field, that
field exerts torque on the loop. This principle is commonly used in motors, in which the loop is connected to a shaft that rotates as a result of
the torque. Thus, the electrical energy from the current is converted to
mechanical energy as the loop and shaft rotate, and this mechanical energy is then used to power another device. Torque on a Current Loop: Electrical energy from the current is converted … 7/14 10/29/21, 8:23 AM Magnetic Fields, Magnetic Forces, and Conductors | Boundless Ph… to mechanical energy as the loop and shaft rotate, and this mechanical
energy is then used to power another device. In this model, the north and south poles of magnets are denoted by N
and S, respectively. In the center is a rectangular wire loop of length l
and width w, carrying current I. The effect of magnetic field B on the current-carrying wire exerts torque τ.
To understand the torque, we must analyze the forces acting on each
segment of the loop. Assuming a constant magnetic field, we can conclude that the forces on the top and bottom parts of the loop are equal
in magnitude and opposite in direction, and thus produce no net force.
Incidentally, those forces are vertical and thus parallel to the shaft.
However, as illustrated by (a) in the figure below, the equal but opposite
forces produce a torque that acts clockwise.
Given that torque is calculated
from the equation: τ = rF sin θ where F is force on the rotating
object, r is the distance from the
pivot point that the force is applied, and θ is the angle between
r and F, we can use the sum of
two torques (the forces act on either side of the loop) to find the
total torque: τ = w
2 F sin θ + w
2 Varying torque on a charged loop
in a magnetic field: Maximum
torque occurs in (b), when is 90
degrees. Minimum torque is 0, and
occurs in (c) when θ is 0 degrees.
When loop rotates past =0, the
torque reverses (d). F sin θ = wF sin θ Note that r is equal to w/2, as illustrated.
To find torque we still must solve for F from the magnetic field B on the … 8/14 10/29/21, 8:23 AM Magnetic Fields, Magnetic Forces, and Conductors | Boundless Ph… current I. The rectangle has length l, so F=IlB. Replacing F with IlB in the
torque equation gives: τ = wIlB sin θ Note that the product of w and l is included in this equation; those terms
can be replaced with area (A) of the rectangle. If another shape of wire is
used, its area can be inserted in the equation regardless of shape
(whether circular, square, or otherwise).
Also note that this equation of torque is for a single turn. Torque increases proportionally according to number of turns (N). Thus, the general equation for torque on a loop of any shape, of N turns, each of A
area, carrying I current and exposed to a magnetic field B is a value that
fluctuates as the loop rotates, and can be calculated by: τ = NIAB sin θ Ampere’s Law: Magnetic Field Due to a Long
Straight Wire
Current running through a wire will produce a magnetic field that can be
calculated using the Biot-Savart Law. LEARNING OBJECTIVES Express the relationship between the strength of a magnetic
field and a current running through a wire in a form of equation KEY TAKEAWAYS Key Points … 9/14 10/29/21, 8:23 AM Magnetic Fields, Magnetic Forces, and Conductors | Boundless Ph… Ampere ‘s Law states that for a closed curve of length
C, magnetic field (B) is related to current (IC):
∮ C Bdℓ = μ0 IC . In this equation, dl represents the dif- ferential of length of wire in the curved wire, and μ0 is
the permeability of free space.
Ampere’s Law can be related to the Biot-Savart law,
which holds for a short, straight length of conductor:
dB = μ0
4π Idl×r
r 3 . In this equation, partial magnetic field (dB) is expressed as a function of current for an infinitesimally small segment of wire (dl) at a point r distance
away from the conductor.
After integrating, the direction of the magnetic field according to the Biot-Savart Law can be determined using
the right hand rule.
Key Terms electric field: A region of space around a charged parti- cle, or between two voltages; it exerts a force on
charged objects in its vicinity.
magnetic field: A condition in the space around a mag- net or electric current in which there is a detectable
magnetic force, and where two magnetic poles are
present. Current running through a wire will produce both an electric field and a
magnetic field. For a closed curve of length C, magnetic field (B) is related to current (IC) as in Ampere’s Law, stated mathematically as:
∮ C Bdℓ = μ0 IC In this equation, dl represents the differential of length of wire in the
curved wire, and μ0 is the permeability of free space. This can be related
to the Biot-Savart law. For a short, straight length of conductor (typically
a wire) this law generally calculates partial magnetic field (dB) as a function of current for an infinitesimally small segment of wire (dl) at a point r
distance away from the conductor: … 10/14 10/29/21, 8:23 AM Magnetic Fields, Magnetic Forces, and Conductors | Boundless Ph… dB = μ0 Idl×r 4π r 3 . In this equation, the r vector can
be written as r̂ (the unit vector in
direction of r), if the r3 term in the
denominator is reduced to r2 (this
is simply reducing like terms in a
fraction). Integrating the previous
differential equation, we find:
Direction of magnetic field: The
direction of the magnetic field can
be determined by the right hand
rule. B = μ0
4π ∮ Idl×^
r
C r 2 . This relationship holds for constant current in a straight wire, in which magnetic field at a point due to
all current elements comprising the straight wire is the same. As illustrated in the direction of the magnetic field can be determined using the
right hand rule—pointing one’s thumb in the direction of current, the curl
of one’s fingers indicates the direction of the magnetic field around the
straight wire. Magnetic Force Between Two Parallel Conductors
Parallel wires carrying current produce significant magnetic fields, which
in turn produce significant forces on currents. LEARNING OBJECTIVES Express the magnetic force felt by a pair of wires in a form of
an equation KEY TAKEAWAYS Key Points … 11/14 10/29/21, 8:23 AM Magnetic Fields, Magnetic Forces, and Conductors | Boundless Ph… The field (B1) that that current (I1) from a wire creates can
be calculated as a function of current and wire separation (r): B 1 = F = IlB sin θ μ0 I1
2πr μ0 is a constant. describes the magnetic force felt by a pair of wires. If they are parallel the equation is simplified as
the sine function is 1.
The force felt between two parallel conductive wires is
used to define the ampere —the standard unit of
current.
Key Terms ampere: A unit of electrical current; the standard base unit in the International System of Units. Abbreviation:
amp. Symbol: A.
current: The time rate of flow of electric charge.
magnetic field: A condition in the space around a mag- net or electric current in which there is a detectable
magnetic force, and where two magnetic poles are
present. Parallel wires carrying current produce significant magnetic fields, which
in turn produce significant forces on currents. The force felt between the
wires is used to define the the standard unit of current, know as an
amphere.
In, the field (B1) that I1 creates can be calculated as a function of current
and wire separation (r): B1 = μ0 I1
2πr The field B1 exerts a force on the
wire containing I2. In the figure,
this force is denoted as F2.
The force F2 exerts on wire 2 can … 12/14 10/29/21, 8:23 AM Magnetic Fields, Magnetic Forces, and Conductors | Boundless Ph… be calculated as: Magnetic fields and force exerted
by parallel current-carrying wires.:
Currents I1 and I2 flow in the same
direction, separated by a distance of
r. F2 = I2 lB1 sin θ Given that the field is uniform
along and perpendicular to wire 2, sin θ = sin 90 derees = 1. Thus the
force simplifies to: F2=I2lB1
According to Newton’s Third Law (F1=-F2), the forces on the two wires
will be equal in magnitude and opposite in direction, so to simply we can
use F instead of F2. Given that wires are often very long, it’s often convenient to solve for force per unit length. Rearranging the previous equation and using the definition of B1 gives:
F
l = μ0 I1 I2
2πr If the currents are in the same direction, the force attracts the wires. If
the currents are in opposite directions, the force repels the wires.
The force between current-carrying wires is used as part of the operational definition of the ampere. For parallel wires placed one meter away
from one another, each carrying one ampere, the force per meter is: F
l = (4π⋅10 −7 T⋅m/A)(1A) (2π)(1m) 2 = 2 ⋅ 10 −7 N/m The final units come from replacing T with 1N/(A×m).
Incidentally, this value is the basis of the operational definition of the ampere. This means that one ampere of current through two infinitely long
parallel conductors (separated by one meter in empty space and free of
any other magnetic fields) causes a force of 2×10-7 N/m on each
conductor. … 13/14 10/29/21, 8:23 AM Magnetic Fields, Magnetic Forces, and Conductors | Boundless Ph… Previous Next Privacy Policy … 14/14 ...

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