The magnetic force on moving charges is an important idea in electromagnetism. It helps us understand how magnets interact with electric charges.

What is the Magnetic Force?

We can figure out the magnetic force using something called the Lorentz force law. This law tells us that when a charged particle moves in a magnetic field, it feels a force. This force is always at a right angle to both the speed of the particle and the magnetic field.

Here’s the equation that describes this relationship:

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

Let’s break down what these symbols mean:

• $\mathbf{F}$ is the magnetic force.
• $q$ is the charge of the particle (measured in coulombs, C).
• $\mathbf{v}$ is the speed of the charged particle (measured in meters per second, m/s).
• $\mathbf{B}$ is the magnetic field (measured in teslas, T).
• The $\times$ symbol shows that we are doing a cross product.

Important Points About the Force

1. Direction of the Force: You can find out which way the magnetic force points using the right-hand rule. Here’s how:

   • Point your fingers in the direction of the speed ($\mathbf{v}$).
   • Curl your fingers toward the magnetic field ($\mathbf{B}$).
   • Your thumb will then point in the direction of the force ($\mathbf{F}$).

2. Strength of the Magnetic Force: If the charged particle moves right across the magnetic field, we can calculate how strong the magnetic force is like this:

   $F = qvB\sin(\theta)$

   For the strongest force, we set $\theta = 90^\circ$ (because $\sin(90^\circ) = 1$). So the equation gets simpler:

   $F = qvB$

   In this equation:

   • $F$ is in newtons (N),
   • $q$ is in coulombs (C),
   • $v$ is in meters per second (m/s),
   • $B$ is in teslas (T).

Special Cases

• If the charge isn’t moving ($v = 0$), there is no magnetic force. This shows that only moving charges feel the magnetic field.
• If the charged particle travels along the same line as the magnetic field ($\theta = 0$ or $180^\circ$), the force is also zero, because $\sin(0) = 0$.

Charges in Wires

For wires that carry electric current, we can find the force like this:

$$\mathbf{F} = I \mathbf{L} \times \mathbf{B}$$

Where:

• $I$ is the current (measured in amperes, A).
• $\mathbf{L}$ is the length of the wire inside the magnetic field (measured in meters, m).

Magnetic Field from Moving Charges

The magnetic field ($\mathbf{B}$) created by a moving charge can be described with this equation:

$$\mathbf{B} = \frac{\mu_0}{4\pi} \frac{q \mathbf{v} \times \mathbf{r}}{r^3}$$

In this equation:

• $\mu_0 = 4\pi \times 10^{-7} \, \text{T m/A}$ is a constant that helps us measure how magnets work in space.
• $\mathbf{r}$ is the distance from the charge.

Understanding these ideas is really important. It helps us know how particles behave in magnetic fields and is the basis for many everyday things, like electric motors and generators.