Introductory Physics Volume Two

82 Magnetic Fields 4.2
We can determine the dimensions of the magnetic
field from the force equation:
F = qvB −→ B = F qv
The unit of magnetic field is the Tesla, abbreviated as
just T.
Newton
Tesla =
Coulomb · meter/second = Kilogram
Coulomb · second
§ 4.2 Magnetic Force on a Current
If a current carrying wire is placed in a magnetic field it will experience
a magnetic force. The current in the wire is composed of many
moving charges. Each of these charges experiences a magnetic force.
The force on the wire is the sum of the forces on all the moving charges
in the wire.
Theorem: Magnetic Force on a Current
Suppose that you have a wire that is carrying a current I. A small
section of wire of length ⃗ dl, where the vector points in the direction
of the current, will experience a force
⃗ dF = I ⃗ dl × ⃗ B
The force on a longer section of wire can be found by integrating
over the length of the section.
∫
⃗F = I dl ⃗ × B ⃗
If the magnetic field is uniform over the region containing the wire
then the following result can be proved.
Theorem: Force on a Current in a Uniform Field
Let ⃗ ∆l be a vector that points from the beginning to the end of a
section of wire carrying a current I. If that section is in a uniform
magnetic field, the force on that section is
⃗F = I ⃗ ∆l × ⃗ B
⊲ Problem 4.1
Consider a semicircular piece of wire or radius R in the first two quadrants
of the x-y plane. The wire carries of current I in the counterclockwise
direction.