Magnetic Fields I
I. Vector Math Review
A. Multiply By A Scalar
EXAMPLE:
Determine the force experienced by each of the charges if the electric field has a strength of 10 N/C and points in the +x direction as shown:
1. Magnitude, length, of the vector is the magnitude of vector times the absolute value of the charge on the object.
2. Direction of
a) q > 0 then and point in ______direction.
b) q < 0 then and point in ______direction.
B. Vector Multiplication “Cross Product” -
1. Magnitude –
2. Direction –
Vector C is always ______to both vector
______and vector ______.
3. Right Hand Rule – RHR
Step 1 – Put down your pencil if you are right-handed.
Step 2 – Sit on your left hand.
Step 3 – Place fingers along .
Step 4 – Rotate fingers into through the angle q.
Step 5 – Thump point in the direction of .
EXAMPLE: Evaluate the vector expression , where q = 2 C and vectors and are shown below:
II. Magnetism
A. Source - ______
B. Operational Definition -
The EXTRA Force that a CHARGED OBJECT acquires when it is set in Motion is called the Magnetic Force.
C. Equation – “Lorentz Force Law For Magnets”
D. Units –
Note: A 1.00 T field is an extremely large magnetic field so physicists and engineers usually work in Gauss. The conversion factor is
Thus, you must convert all magnetic fields from Gauss to Tesla before using the Lorentz Force Law for Magnets.
EXAMPLE: Wien Filter “Cross Field Analyzer”
Our problem involves a very useful device that allows use to select charged particles of a particular velocity (i.e. it’s a velocity selector)!
A 5.00 C charged object is placed in an electric field of 3.00 N/C in the +y direction and a magnetic field of 2.00 T in the +z direction as shown below
a) What is the net force on the object when the object is stationary?
b) What is the net force when the object has a velocity of 3.00 m/s in the +x direction?
c) Determine the velocity required by the object to travel undeflected through the aperture.
III. Permanent Magnets
A. Over 2000 years ago in Turkey, people discovered that rocks of iron ore would
apply a force on each other. These rocks were called magnets due to their discovery near the town of magnesia.
B. Several interesting facts concerning these rocks were discovered:
1. If a rod of iron touched these rocks, then it would become a magnet and begin interacting with these rocks. The rod was said to be have been magnetized.
2. If the magnetized rod was placed on a leaf and floated on water, the rod would always align itself so that it pointed approximately in the North/South direction on the Earth.
We now know that this is because the Earth is a big magnet due to currents inside the Earth. Thus, people were using compasses for navigation long before we had any fundamental understanding of the cause of magnetic fields. For instance, "Columbus sailed the ocean the ocean blue in 1492," but the theory of magnetism dates from the 1800's.
3. After marking the side of the rod that pointed in the northern direction as north and the side that pointed in the southern direction as south, people discovered that
a) the north side (pole) of the magnet repelled the north pole and attracted the south pole of another magnet.
b) the south pole of a magnet repelled the south pole of another magnet.
Question: From the discussion above, what type of magnetic pole is near the Earth's North Pole?
4. Cutting a magnet in half will produce two complete magnets. This discovery is probably far more intriguing today than it was long ago since it implies that the cause of magnetism is not a point source (like a point charge) but an extended source (a current loop). The search for a point source of magnetism (magnetic monopole) is an active field of physics research. We will discuss this further when we get to Gauss' Law for Magnets.
Summary: A considerable amount of empirical information about magnetism was discovered using permanent magnets. This allowed for the construction of some useful devices using permanent magnets long before any theory of magnetism existed. However, the connection between the phenomena of electricity (discovered in Greece) and magnetism (discovered in Turkey) prevented people from developing motors, generators, stronger magnets (electromagnets), and electrical power. It was the development of the battery in the late 1700's that would begin the age of electricity. Thus, many of the conventions concerning magnets (using N/S instead of +/-) are historical in nature.
IV. Magnetic Field and Magnetic Field Lines
A. The concept of the magnetic field can be developed in a manner similar to the way we developed the electric field. The magnitude of the magnetic field at a particular point in space is found by using a moving charged particle and determining the maximum force exerted on the particle for a given speed.
The magnetic field vector lies along the direction in which the moving charged particle experiences no additional force due to its motion (no force if we remove electric fields, gravity, etc.) with the direction of the vector given by the right hand rule.
B. We can build a graphical picture of the magnetic field (magnetic field lines) in a manner similar to our work with electric fields. However, there are some important differences due to the nature of the vector cross product that you need to understand.
Similarities To Electric Field Diagrams:
1. Magnetic field lines leave the north pole (like electric field lines from a positive charge) and terminate on the south pole of a magnet (like electric field lines on a negative charge). Thus, the direction of the magnetic field at any point is the direction pointed to by a compass (assuming that the Earth's magnetic field is negligible).
2. Magnetic field lines can never cross (the magnetic field has a single value at every point is space),
3. The density of magnetic field lines is proportional to the strength of the magnetic field.
The easiest way to map the magnetic field is to use iron fillings. These act like little magnets and align with the field. A compass can then be used to determine the direction of the arrow. Also, the strength of the magnetic field is obtained since more iron filings will be attracted to regions of higher magnetic field.
Differences:
1. A graph of the magnetic field lines doesn't completely specify the force on a charged particle. This is because the force also depends on the velocity of the particle. Stationary particles experience no magnetic force at all!
2. A magnetic field line DOES NOT point in the direction of the force applied on a moving charged particle. If fact, a charged particle moving the direction of the magnetic field line experiences NO magnetic force. The non-zero force experienced by any moving charge will always be perpindicular to the magnetic field lines.
3. We find that the graphs of the magnet field lines always show circulation (rotation) and never flow! More evidence that their is no point source of magnetism.
EXAMPLE: The following can not be a graph of the magnetic field because it has flow and no circulation.
EXAMPLE: The magnetic field diagram for a magnetic dipole (the simplest magnetic source known).
Current Loop Permanent Magnet
V. Magnetic Force On A Current Carrying Wire
A. Theory
The magnetic force on a differential amount of moving charge, dq, in the wire is
Using the definition of average velocity, , we have
We now use
Substituting this result and using the definition of current, we have
The total force on the wire is then found by summing up the force on each element of the wire.
Since the elements of the wire are in series, they have the same current (ie I constant) and we can pull I out of the sum.
where I is the current in the wire
is the magnetic field vector
is the displacement vector point in the direction of I.
For simple problems, the magnitude of B is the same at each point on the wire and the angle between the wire and B is also contant. In this case we have the simplified result:
B. Problem Solving Strategy
Step 1: Break up the wire into segments in which the angle between and and the direction of the resulting cross product DOES NOT CHANGE.
Step 2: You can disregard all wire segments in which I {ie. } is parallel to as
Step 3: For each remaining wire segment, find by
A. Finding
B. Find the force on this wire segment using
Step 4: Sum segment forces using vector addition!
EXAMPLE: What is the force exerted by a 5.00 T magnetic field in the +y-direction upon the 25.0 meter long wire shown below when 2.00 A of current is flowing in the wire.
Solution:
VI. Charged Particle in a Perpendicular Uniform Magnetic Field
A. The particle is undergoing ______
______.
B. The magnetic force is the ______.
C. The magnetic force
1. does ______work.
2. provides ______torque.
3. causes ______acceleration.
4. Thus ______momentum is conserved but
______momentum is NOT conserved.
D. Equation Of Motion
"Magnetic Rigidity"
Thus, all particles with the same momentum per charge ratio will travel in a circle of the same
radius, R, for a given magnetic field.
VII. Mass Spectrometers
An instrument that often uses the special properties of magnets.
The apertures only allow particles that travel in a circle with the correct radius R to enter the detector.
For the magnet, we have
CASE I: Chemistry Mass Spectrometer- and q are constant and the same for all particles
Dr. Astin in his Chemistry Nobel Prize design used a Wien filter to select the velocity of the particles inserted into his magnetic spectrometer.
Case II: Energy and q are the same for all particles traveling at non- relativistic speeds.
EXAMPLE:
A. A proton is traveling at 5.00x105 m/s in a 100 Gauss magnetic field. What is the radius of the circle?
B. What is the magnetic field required to bend singly charged helium nuclei along the same path?