Charged Particle in a Magnetic Field

When a charged particle moves in a magnetic field, the quantity that remains unchanged is Kinetic energy.

1. The Nature of Magnetic Force

The magnetic force ($\mathbf{F}$) acting on a charge $q$ moving with velocity $\mathbf{v}$ in a field $\mathbf{B}$ is:

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

By the definition of a cross product, the force $\mathbf{F}$ is always perpendicular to the velocity $\mathbf{v}$.

2. Work and Kinetic Energy

Since the force is perpendicular to the direction of motion at every instant, the work done ($W$) by the magnetic field on the particle is zero. Using the dot product for power ($P$):

$$P = \mathbf{F} \cdot \mathbf{v} = 0$$

According to the Work-Energy Theorem, the change in kinetic energy ($\Delta K$) is equal to the work done:

$$W = \Delta K = 0$$

Because no work is done, the speed and the kinetic energy ($K = \frac{1}{2}mv^2$) of the particle remain constant.

3. Why the other options change:

Direction of velocity: The force acts as a centripetal force, constantly bending the path of the particle.

Momentum direction: Since momentum ($\mathbf{p} = m\mathbf{v}$) is a vector, it changes whenever the direction of velocity changes.

Angular momentum: Generally changes depending on the chosen point of reference, as the position vector $\mathbf{r}$ and momentum vector $\mathbf{p}$ are constantly rotating.

Conclusion: The magnetic field changes the direction of motion (velocity and momentum) but never the magnitude of motion (speed and kinetic energy).