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Particles in Magnetic Fields
Storyboard
Electric charges moving in a magnetic field are deflected perpendicular to the direction in which they move and in which the magnetic field points.
The force acting on the particle depends on the charge, the velocity and the magnetic field is called the Lorentz force.
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Parallel currents
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When two currents are allowed to flow in a parallel manner, we observe an attractive force between the wires.
It's worth recalling that currents consist of electrons in motion, and electrons naturally repel each other due to their negative charges. However, when these charges are in motion, this repulsive force turns into an attractive force, resulting in the observed attraction between the negatively charged conductors.
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Opposite parallel currents
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When two currents are allowed to flow in a parallel but opposite direction, we observe a repulsive force between the wires.
Comparing this experiment to the one where the flow is parallel but in the same direction, the key difference lies in the presence of relative velocity in the latter case.
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Parallel currents, field is not electric
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If a metal plate is placed between both conductors, no noticeable effect is observed:
Hence, we conclude that the generated field does not correspond to a traditional electric field.
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Current effect on a compass
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When a compass is exposed to an electric current, the following observations can be made:
In summary, the compass needle:
• does not rotate if there is no electric current present
• rotates when there is a flow of electric current
• if the direction of the current flow is reversed, the rotation of the needle also reverses.
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Detection of the generated magnetic field
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When you explore the space around a wire with a compass, you'll notice that the current induces the presence of a magnetic field:
This is why parallel wires can either attract or repel each other depending on the direction of the current. The key insight here is that:
Current generates a magnetic field, and this magnetic field exerts a force on moving charges.
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Lorenz Law
Equation
The force
| $ \vec{F} = q ( \vec{E} + \vec{v} \times \vec{B} )$ |
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Magnitude of the magnetic component of the Lorentz force
Equation
The magnetic component of the Lorentz force is
| $ \vec{F} = q \vec{v} \times \vec{B} $ |
so with
its magnitude will be
| $ F = q v B \sin \theta $ |
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Circular motion in magnetic field
Equation
The magnetic component of the Lorentz force
| $ F = q v B \sin \theta $ |
It is always perpendicular to the direction of movement leading to the particle moving in a circle (the speed is tangential to it and thus always orthogonal to the radius). The radius will have to be such that the magnetic force is equal to the centrifugal force so it will have to
| $ m \displaystyle\frac{ v ^2}{ r }= q v B $ |
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Radius of the orbit in the magnetic field
Equation
Being the movement of an electric charge in a circular magnetic field satisfying the equality between the magnetic and centrifugal forces
| $ m \displaystyle\frac{ v ^2}{ r }= q v B $ |
it will have that the radius of the orbit will be
| $ r =\displaystyle\frac{ m v }{ q B }$ |
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Cyclotron frequency
Equation
Being the movement of an electric charge in a circular magnetic field satisfying the equality between the magnetic and centrifugal forces
| $ m \displaystyle\frac{ v ^2}{ r }= q v B $ |
so you can calculate the angular frequency
| $ v = r \omega $ |
with which the charge rotates around the center of the orbit
| $ \omega =\displaystyle\frac{ q B }{ m }$ |
which allows to determine the relationship between the load and mass empirically.
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Video
Video: Particle Dynamics