Question

Under what conditions is the force acting on a charge moving through a uniform magnetic field minimum?

Answer 1

The force acting on a charge moving through a uniform magnetic field is minimum when the velocity of the charge is parallel or anti-parallel to the direction of the magnetic field. In other words, the angle between the velocity vector and the magnetic field vector is $0^\circ$ or $180^\circ$.

Answer 2

The magnetic force $\vec{F}$ acting on a charge $q$ moving with velocity $\vec{v}$ in a uniform magnetic field $\vec{B}$ is given by the Lorentz force formula:

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

The magnitude of this force is given by:

$$F = |q| v B \sin\theta$$

where:

• $|q|$ is the magnitude of the charge.
• $v$ is the magnitude of the velocity of the charge.
• $B$ is the magnitude of the magnetic field.
• $\theta$ is the angle between the velocity vector $\vec{v}$ and the magnetic field vector $\vec{B}$.

For the force $F$ to be minimum, assuming the charge $q$, its velocity $v$ (since it's moving), and the magnetic field $B$ are non-zero, the value of $\sin\theta$ must be minimum.

The minimum possible value of $|\sin\theta|$ is 0.

This occurs under two conditions for the angle $\theta$:

1. When $\theta = 0^\circ$: The velocity vector $\vec{v}$ is parallel to the magnetic field vector $\vec{B}$. In this case, $\sin(0^\circ) = 0$, so $F = 0$.
2. When $\theta = 180^\circ$: The velocity vector $\vec{v}$ is anti-parallel to the magnetic field vector $\vec{B}$. In this case, $\sin(180^\circ) = 0$, so $F = 0$.

Therefore, the minimum force acting on a charge moving through a uniform magnetic field is zero, and this occurs when the charge moves parallel or anti-parallel to the direction of the magnetic field.

The conditions for the force acting on a charge moving through a uniform magnetic field to be minimum are:

When the velocity vector of the charge is parallel or anti-parallel to the magnetic field vector. This means the angle between the velocity and the magnetic field is $0^\circ$ or $180^\circ$.