28.17 - Interactive problem: helical particle motion
In this simulation you will have a chance to observe a charged particle following a
helical path through a magnetic field.
If you use the initial velocity provided by the simulation, which is perpendicular to
the field, the particle will move in a circle. You can cause its path to be a helix by
supplying a component to its initial velocity that is parallel to the field.
To do this, first make sure the magnetic field lines are viewed as pointing straight
down. Now you may drag the tip of the velocity vector arrow to set the initial speed
and direction of the particle. Once you have the initial velocity you want, change the
viewing angle by moving the slider provided for this purpose to a position near the
middle of its range. Press GO to observe the helical motion of the particle.
28.18 - Magnetic force on a current-carrying wire
A magnetic field exerts a force on a wire carrying a current. Since the moving charges
in the wire í electrons in this case í cannot escape from it, the wire as a whole will
react to the magnetic force on them, just as a large net full of helium balloons will rise
due to the balloons’ individual buoyancies.
At the right, we depict a configuration that shows how to determine the magnetic force
exerted by a uniform magnetic field on a straight, current-carrying wire. We show only a
segment of the wire, and not the entire circuit loop that allows current to flow.
The right-hand rule can be used to determine the direction of the force. Since the
current shown is conventional (positive), the thumb points in the direction of the force
when the fingers wrap from the direction of current flow to the magnetic field. (If the
current were shown as flowing electrons, the thumb would point in the direction
opposite to the current.) In this case the force points out of the screen, toward you.
To calculate the magnitude of the force exerted on a given length of wire, use the
second equation shown in Equation 1. The amount of force increases with the amount
of current, the length of the wire and the strength of the magnetic field. Since more
current means either more electrons flowing, or the electrons moving faster, this
relationship follows from the equation for force on a single charge,F = qvB sin ș (or to
state it using cross-product notation, F = qv × B). A greater length of wire will also
experience more force since it contains more moving charge.
Variables
We list below the variables that are used in the derivation, but which do not already
appear in Equation 1. The vector L appearing in the equation is the “directed length” of
the wire segment. That is, L is parallel to the segment, in the direction of the current,
and its magnitude L equals the length of the segment.
Strategy
- Use the definition of the current flowing through the segment to find that IL = qv.
- Substitute this equation into the cross-product formula for the force exerted on a
moving charge by a magnetic field to get Equation 1.
Physics principles and equations
The force exerted on a moving charged particle by a magnetic field is,
F = qv × B
Magnetic force on a wire
Uniform magnetic field exerts force on
current-carrying wire
Proportional to current, length of wire
Direction of force found with right-hand
ruleF = IL × B
F = ILB sin ș
F = force, I = current
L = directed length of wire segment
B = magnetic field
ș = angle between wire and field
amount of free charge in wire segment Q
velocity of a charge carrier in segment v
time for carrier to move through segment ǻt