Magnetic fields can be defined as the regions surrounding a magnet where another magnet or a moving electric charge will feel a force of attraction or repulsion. They are created by moving charges. The strengths of magnetic fields are measured in a unit called Teslas, T. In formulas, the strength of a magnetic field is represented by the variable B. In a later lesson we will learn that a tesla is equivalent to
1 tesla = 1 newton per meter per amp
This definition is based on the fact that if two 1-meter long current-carrying wire segments, each having 1 ampere of current, are either attracted to or repelled from each other by one newton of force, then the strength of the magnetic field produced by either current-carrying segment equals 1 tesla at the location of the other wire.
As shown below, magnetic field lines emerge from the North pole of a magnet and enter the South pole.
When magnetic fields are perpendicular to the plane of the page, "x's" are used to represent those flux lines which point into the plane (-z) of the page and " 's" are represent those which point out of the plane (+z) of the page.
These definitions can be remembered by imagining an arrow. If the front, or tip, of the arrow was coming "at you" then you would see a round point. If the tail feathers of an arrow were going away from you, then you would see the crossed-feathers.
Field lines can be visualized by sprinkling small iron filings over a magnet. Remember that if arrows were to be present indicating direction, these lines would begin on the North pole and terminate on the South pole.
When a compass (or any freely floating bar magnet) points north, it is actually aligning its north pole to the Earth's magnetic south pole. Yes, the Earth's geographic north pole is a magnetic south pole!
Magnetic field lines surrounding poles that have the same polarity repel - just like those between similarly charged particles - and are hyperbolic in shape. Conversely, field lines between poles having the opposite polarity attract and are elliptical in shape.
Forces on Moving Charges
As mentioned earlier, magnetic fields can only affect moving charged particles. The formula used to calculate the magnetic force on a moving charged particle through a magnetic field is a vector cross-product, where F, v and B are all mutually perpendicular vectors.
In this diagram, if the magnetic field, B, were pointed in the positive x-direction (fingers), and a positively charged particle were traveling in the positive y-direction (thumb), then the magnetic force F (palm) would point in the negative z-direction, or into the plane of the page.
As indicated in the formula, magnetic forces are maximized when the angle between the charge’s velocity and the magnetic field through which it is moving is 90º. If B is not perpendicular to the particle's velocity, you may take components of either vector: B or v. Usually the component B sin(θ) is perpendicular to the particle's velocity. In both of the diagrams shown below, the magnetic force would equal zero, since B and v have no components which are perpendicular to each other.
The right hand rule, RHR, for determining the direction of the force experienced by a moving positive charge in a magnetic field is:
thumb points in the direction of a moving positive charge's velocity, v fingers point in the direction of magnetic field, B
palm faces in the direction of the magnetic force, F
This right hand rule only applies to positive charges. You would need to use an equivalent left hand rule for electrons. Or just remember that if the force would be "up" for a positive charge, then the force will be "down" for a negative charge. That is, the force on a negative charge will always act 180º in the opposite direction.
right palm faces OUT OF the plane of the page, or in the (+z)
direction
right palm faces INTO the plane of the page, or in the (-z)
direction
In most of our problems, this 90º angle is achieved by having the magnetic field (B) point along either the positive/negative z-axis while the charged particle moves in the xy-plane.
acceleration. Magnetic forces only cause charged particles to change their direction of motion. Magnetic forces on moving charged particles are centripetal forces and result in the charged particles traveling in circular paths..
In addition, magnetic forces do NOT do any work on moving charges. That is, they do not cause a particle to gain kinetic energy. Remember that work = Fs cos(θ). For magnetic forces, FB and s (and the particle's
tangential velocity, v) are always at right angles and cos 90º = 0.
The formula that allows you to calculate the radius of these circular paths is:
solving for r yields
Note that the radius is directly proportional to the particle's momentum and inversely proportional to the magnitude of the charge and the strength of the magnetic field.
Magnetic fields generated by current-carrying wires
Circular magnetic fields are generated around current carrying wires. The strength of these fields varies directly with the size of the current flowing through the wire and inversely to the distance from the wire.
In this diagram, the solid teal circle in the center represents a cross-section of a current-carrying wire in which the current is coming out of the plane of the paper.
The concentric circles surrounding the wire's cross-section represent magnetic field lines.
The rule to determine the direction of the magnetic field lines is called the right hand curl rule.
In this rule, your
thumb points in the direction of the current fingers curl in the direction of B
The equation to calculate the strength of the magnetic field around a current-carrying wire is:
where
µo, permeability of free space = 4π x 10-7 Tm/A
I, current flowing through the wire, measured in amps
B, magnetic field strength, measured in Tesla
r, distance from the wire, measured in meters
Forces on current-carrying wires
When a segment of a current-carrying wire is placed in an external magnetic field, the interaction between the magnetic field of the wire and the external magnetic field is exhibited by a force which is calculated with the formula:
F = Bperpendicular IL
where
B is the external, perpendicular magnetic field measured in Tesla, I is the current measured in amps, and
L is the length of the current segment (in meters) that lies in the external magnetic field, B.
The direction of this force also obeys the RHR where your
thumb points in the direction of the current, I
fingers point in the direction of the external magnetic field, B
Forces between two current-carrying wires
If two current carrying wires are parallel to each other, their respective magnetic fields either attract or repel each other.
As you can see in the diagram above, if two parallel wires have currents traveling in opposite directions, the magnetic fields generated by those currents between the wires will both point in the same direction, in this case, into the plane of the page. These wires would repel each other.
However, if two parallel wires have currents traveling in the same direction, the magnetic fields generated by those currents between the wires will both point in opposite directions resulting in the wires attracting each other. Also notice in the right hand diagram shown below the familiar "ellipses" that we are
By using the RHR to determine the direction of the forces their respective magnetic fields exert on each other, we can see that these wires would attract each other.
The formula used to calculate these attractive or repulsive forces is:
F = Bperpendicular IL
F12 = (µoI1/2πr) I2L2
F12 = (µo/2πr) I1I2L2
F12 = (4π x 10-7/2πr) I1I2L2
F12 = (2 x 10-7/r) I1I2L2
where
F12 represents the force on wire 2 caused by its presence in the magnetic field of wire 1
I2 is the current flowing in wire 2
L2 is the length of the current segment of wire 2 in the field of wire 1
r is the distance between the wires
Velocity Selector
If you want the proton to travel in a straight path through the magnetic field instead of a circular one, a second field must be introduced into the region. That second field would have to exert an equal but opposite force on the proton. The second field would be an electric field that is perpendicular to the existing magnetic field. When these forces are balanced, the charged particle will travel straight through both fields at a constant velocity.
The equation for a velocity selector is FB = FE
qvB = qE v = E/B
A velocity selector was originally used to isolate different energy beta particles (β) emitted during neutron decay. Remember that beta rays are actually streams of electrons released from the nucleus.
n → p+ + β- + anti-electron neutrino
Initially, scientists thought that linear momentum might not be conserved during beta decay. However, the introduction of the anti-electron neutrino accounted for the missing
