When a charged particle, such as an electron or a proton, moves through a uniform magnetic field, it experiences a force known as the Lorentz force. The Lorentz force (\( \mathbf{F} \)) acting on a charged particle (\( q \)) moving with velocity (\( \mathbf{v} \)) in a magnetic field (\( \mathbf{B} \)) is given by the following equation:
\[ \mathbf{F} = q \cdot (\mathbf{v} \times \mathbf{B}) \]
Here, \( \times \) represents the vector cross product. The direction of the force is perpendicular to both the velocity of the charged particle and the magnetic field according to the right-hand rule.
The equation can also be written in component form for each direction:
\[ F_x = q \cdot (v_y \cdot B_z – v_z \cdot B_y) \]
\[ F_y = q \cdot (v_z \cdot B_x – v_x \cdot B_z) \]
\[ F_z = q \cdot (v_x \cdot B_y – v_y \cdot B_x) \]
The Lorentz force causes the charged particle to move in a circular or helical path, depending on the initial conditions.
If the charged particle is initially moving parallel or antiparallel to the magnetic field lines, it will experience no force because the cross product of parallel vectors is zero. If the initial velocity has a component perpendicular to the magnetic field, the particle will move in a circular path with a radius determined by the magnitude of its velocity and the strength of the magnetic field.
The equation for the radius (\( r \)) of the circular motion is given by:
\[ r = \frac{m \cdot v}{|q| \cdot B} \]
where \( m \) is the mass of the charged particle, \( v \) is its velocity, \( |q| \) is the magnitude of its charge, and \( B \) is the magnitude of the magnetic field.
The frequency (\( f \)) of the circular motion, which is the number of revolutions per unit time, is given by:
\[ f = \frac{|q| \cdot B}{2 \pi \cdot m} \]
In summary, the motion of a charged particle in a uniform magnetic field is characterized by circular or helical paths, depending on the initial conditions, and the radius and frequency of this motion are determined by the particle’s charge, mass, velocity, and the strength of the magnetic field.
Here is an app to calculate the key parameters of the gyro-motion.