Magnetic force on moving charges Knowledge-Based Study

Study notes

1. A magnetic field exerts a force on an electric charge only when two conditions are met simultaneously: the charge must be in motion relative to the field, and its velocity vector must not be parallel or antiparallel to the magnetic field vector.

   The Magnetic Force Equation

   The vector magnetic force $\vec{F}_B$ experienced by a particle carrying electric charge $q$ and moving with velocity $\vec{v}$ through a magnetic field $\vec{B}$ is given by:

   $$\vec{F}_B = q(\vec{v} \times \vec{B})$$

   The magnitude of this force is calculated as:

   $$F_B = |q|vB\sin\theta$$

   In this equation:

   • $q$ is the electric charge measured in coulombs (C).
   • $v$ is the speed of the particle in meters per second (m/s).
   • $B$ is the magnetic field strength measured in teslas (T).
   • $\theta$ is the angle between the velocity vector $\vec{v}$ and the magnetic field vector $\vec{B}$.

   Conditions Resulting in Zero Magnetic Force

   The magnetic force on a particle is zero ($F_B = 0$) under any of the following circumstances:

   • The particle is neutral ($q = 0$), such as a neutron.
   • The particle is stationary relative to the magnetic field ($v = 0$).
   • The particle travels parallel ($\theta = 0^\circ$) or antiparallel ($\theta = 180^\circ$) to the magnetic field lines, since $\sin 0^\circ = \sin 180^\circ = 0$.

   Because the magnetic force is determined by a vector cross product, $\vec{F}_B$ is always perpendicular to both the velocity $\vec{v}$ and the magnetic field $\vec{B}$. Consequently, the force acts perpendicular to the plane formed by $\vec{v}$ and $\vec{B}$.

   Using the Right-Hand Rule (RHR)

   To determine the direction of the magnetic force for a positive charge ($q > 0$):

   1. Point the fingers of your right hand along the direction of the velocity vector $\vec{v}$.
   2. Curl your fingers toward the direction of the magnetic field vector $\vec{B}$ through the smallest angle $\theta$.
   3. Your extended thumb points in the direction of the magnetic force $\vec{F}_B$.

   Behavior for Negative Charges

   For a negative charge ($q < 0$), such as an electron, the vector direction reverses because of the negative sign in $\vec{F}_B = q(\vec{v} \times \vec{B})$. You can find the direction by applying the right-hand rule as if the charge were positive and then reversing the resulting direction by $180^\circ$, or by using your left hand following the same finger alignments.

   A critical property of the magnetic force is that it is always perpendicular to the instantaneous velocity of the charge ($\vec{F}_B \perp \vec{v}$).

   Work and Kinetic Energy

   The rate at which work is done on the particle (instantaneous power $P$) is given by the dot product:

   $$P = \vec{F}_B \cdot \vec{v} = 0$$

   Because the magnetic force is always perpendicular to the direction of motion, the magnetic field does no work on a moving charged particle. Therefore, the kinetic energy and speed of the particle remain constant; only the direction of motion changes.

   Uniform Circular Motion

   When a charged particle enters a uniform magnetic field with its velocity strictly perpendicular to the field lines ($\theta = 90^\circ$), the magnetic force acts as a centripetal force with constant magnitude:

   $$F_B = |q|vB = \frac{mv^2}{r}$$

   Solving for the radius $r$ of the circular trajectory (the gyroradius or cyclotron radius) yields:

   $$r = \frac{mv}{|q|B}$$

   This relationship shows that particles with larger momentum ($mv$) follow wider circular paths, whereas stronger magnetic fields or larger charges produce tighter curves.