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When a charged particle, like an electron or a proton, moves through a magnetic field, it experiences a force known as the Lorentz force. Mathematically, this force is described by the equation:
\[\begin{equation}\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})\end{equation}\]
where \(q\) represents the charge of the particle, \(\mathbf{v}\) is the velocity vector, and \(\mathbf{B}\) is the magnetic field vector. In our scenario, since the electric field (\(\mathbf{E}\)) is null, the equation simplifies to
\[\begin{equation}\mathbf{F} = q(\mathbf{v} \times \mathbf{B})\end{equation}\]
This force acts perpendicular to both the direction of the particle's movement and the magnetic field, causing the particle to deviate from its straight-line path.

A uniform magnetic field is one where the magnetic field strength and direction are constant throughout the region of interest. In such a field, a charged particle will consistently experience the Lorentz force in the same magnitude and direction, ignoring any other forces acting upon it. This consistent application of force is crucial in determining the nature of the particle's motion and ensures that the particle's speed remains unchanged regardless of the path it takes through the field.

Helical motion occurs when a charged particle has a velocity component that is not solely perpendicular to the magnetic field direction. This leads to a combined motion with a circular path in the plane perpendicular to the magnetic field, and a linear motion along the direction of the field.
This combination creates a spiral path resembling the shape of a helix, hence the term 'helical motion'. The radius of the helix's circular cross section is related to the perpendicular component of the velocity, while the pitch, which is the distance between turns of the helix, is dependent on the velocity parallel to the field.

Circular motion is a special case of helical motion where the velocity of the charged particle is entirely perpendicular to the magnetic field direction. In this instance, since there is no velocity component parallel to the field, the motion is restricted to a plane perpendicular to the field lines.
The force acting on the particle, being perpendicular to the velocity, will change the direction of the velocity vector but not its magnitude, resulting in the particle describing a perfect circle. This circular path is characterized by a constant radius, which is determined by the particle's mass, charge, speed, and the strength of the magnetic field.