Magnetic Forces on Charges and Wires

Hello! In our last lesson, we established that electric currents are the source of magnetic fields, using Ampère's Law to calculate the field around a long, straight wire. We now turn to the other side of this interaction: how magnetic fields exert forces on moving charges.

This lesson focuses on the Lorentz force, the fundamental force experienced by a charge moving in a magnetic field. We will cover the force on a single moving particle and then extend that concept to derive the force on a current-carrying wire. Mastering this will be crucial for our later discussions on electric motors and generators.

By the end of this lesson, you will be able to calculate the magnitude and direction of the magnetic force on both individual moving charges and current-carrying wires.

1. The Magnetic Force on a Moving Charge

A magnetic field exerts a force, known as the Lorentz force, on any moving charged particle. A key feature of this force is that it acts only on moving charges; a stationary charge in a magnetic field experiences no force.

Given your background in vector calculus, the most precise way to define the magnetic force $\vec{F}$ on a particle with charge $q$ moving with velocity $\vec{v}$ through a magnetic field $\vec{B}$ is using the vector cross product:

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

This compact equation contains all the information about the force, including its magnitude and direction.

The magnitude of the force is given by:
$F=|q|vB\sin \theta$
where $\theta$ is the angle between the velocity vector $\vec{v}$ and the magnetic field vector $\vec{B}$. Notice two important cases:

• If the charge moves parallel or anti-parallel to the magnetic field ($\theta =0^{\circ }$ or $\theta =180^{\circ }$), $\sin \theta =0$ and the force is zero.
• The force is maximum when the charge moves perpendicular to the field ($\theta =90^{\circ }$), as $\sin 90^{\circ }=1$.

The following video provides an excellent visual introduction to this force and the rule for determining its direction.

Direction: The Right-Hand Rule

As the video explains, the direction of the force $\vec{F}$ is given by the right-hand rule because of the cross product. The resulting force vector is always perpendicular to the plane formed by $\vec{v}$ and $\vec{B}$.

Here's the convention for a positive charge $q$:

• Point your index finger in the direction of velocity ($\vec{v}$).
• Point your middle finger in the direction of the magnetic field ($\vec{B}$).
• Your thumb will point in the direction of the magnetic force ($\vec{F}$).

For a negative charge (like an electron), you can use the same rule and then simply reverse the direction of the resulting force, or use your left hand.

An Important Consequence: Magnetic Forces Do No Work

Since the magnetic force $\vec{F}$ is always perpendicular to the velocity $\vec{v}$, it can change the direction of a particle's motion, but it cannot change its speed or kinetic energy. Recall that the work done $W$ by a force is $W=\int \vec{F}\cdot d\vec{s}$, and power $P$ is $P=\vec{F}\cdot \vec{v}$. Because $\vec{F}$ is perpendicular to $\vec{v}$, their dot product is always zero.

$P=\vec{F}\cdot \vec{v}=(q(\vec{v}\times \vec{B}))\cdot \vec{v}=0$

This is a fundamental property of magnetic forces. They can steer charged particles (as in a particle accelerator) but cannot speed them up or slow them down.

Test your understanding!

An electron ($q=-1.6\times 10^{-19}$ C) travels at $2.0\times 10^6$ m/s in the positive x-direction into a uniform magnetic field of $0.5$ T pointing in the positive y-direction. What is the magnitude and direction of the force on the electron?

Show answer

First, let's calculate the magnitude. The velocity and magnetic field are perpendicular, so $\theta =90^{\circ }$ and $\sin \theta =1$.
$F=|q|vB\sin \theta =(1.6\times 10^{-19}\text{C})\times (2.0\times 10^6\text{m/s})\times (0.5\text{T})\times 1$
$F=1.6\times 10^{-13}\text{N}$

Next, let's find the direction. Using the right-hand rule:

• Index finger (velocity $\vec{v}$) points in the +x direction.
• Middle finger (magnetic field $\vec{B}$) points in the +y direction.
• Your thumb (force $\vec{F}$) points in the +z direction (out of the page).

However, the electron has a negative charge, so we must reverse this direction. The force on the electron is in the negative z-direction (into the page).

2. The Magnetic Force on a Current-Carrying Wire

We can extend the concept of the Lorentz force from a single charge to the macroscopic force on a wire carrying an electric current. A current is simply a flow of many charge carriers.

To make this connection more rigorous, please read the following brief derivation.

As the text shows, by considering the total charge in an infinitesimal segment of wire $d\vec{l}$, we arrive at the differential force:
$d\vec{F}=I(d\vec{l}\times \vec{B})$
where $I$ is the current and $d\vec{l}$ is a vector representing the length and direction of the wire segment.

For a straight wire of length $L$ in a uniform magnetic field $\vec{B}$, we can integrate this to get the total force:
$\vec{F}=I(\vec{L}\times \vec{B})$
where $\vec{L}$ is a vector whose magnitude is the length of the wire and whose direction is the direction of the current.

The magnitude of this force is:
$F=ILB\sin \theta$
where $\theta$ is now the angle between the direction of the current and the magnetic field.

Direction and Calculation Methods

The direction of the force on a wire is found using the same right-hand rule as before, with velocity $\vec{v}$ replaced by current direction $\vec{L}$.

The following video demonstrates this and introduces a powerful alternative method using vector components, which you may find more systematic given your mathematical background.

The vector method shown in the video is a direct application of the cross product definition:
If $\vec{L}=L_x\hat{i}+L_y\hat{j}+L_z\hat{k}$ and $\vec{B}=B_x\hat{i}+B_y\hat{j}+B_z\hat{k}$, then $\vec{F}=I(\vec{L}\times \vec{B})$ can be calculated component-wise. This avoids the potential ambiguity of orienting your hand in 3D space.

Another common mnemonic is Fleming's Left-Hand Rule, often used in the context of motors. It assigns force, field, and current to the thumb, forefinger, and middle finger, respectively. While useful, it's generally best to stick to one consistent convention based on the mathematical cross product.

3. Application: Force Between Two Parallel Wires

We can now combine what we learned in the last lesson with today's topic. Consider two parallel wires. Wire 1, carrying current $I_1$, creates a magnetic field $\overrightarrow{B_1}$. This field then exerts a force on Wire 2, which carries current $I_2$.

From the last lesson, the magnitude of the magnetic field from Wire 1 at the location of Wire 2 (a distance $r$ away) is:
$B_1=\frac{{\mu }_0{I}_1}{2\pi r}$
Now, using today's formula, the force on a length $L$ of Wire 2 is:
$F_2=I_{2}L{B}_1\sin \theta$
Since the field from Wire 1 is perpendicular to the current in Wire 2, $\theta =90^{\circ }$ and $\sin \theta =1$. Substituting $B_1$, we get the force per unit length:
$\frac{F_2}{L}=I_2{B}_1=\frac{{\mu }_0{I}_1{I}_2}{2\pi r}$

What about the direction?

• Parallel Currents (same direction): Use the right-hand rule for the field from Wire 1 (fingers curl around the wire). At Wire 2, the field $\overrightarrow{B_1}$ points into the page. Now use the right-hand rule for the force on Wire 2 (current up, field in), and you'll find the force points towards Wire 1. The wires attract.
• Anti-parallel Currents (opposite directions): The field $\overrightarrow{B_1}$ is still into the page at Wire 2. But now the current $I_2$ is down. The force rule gives a force pointing away from Wire 1. The wires repel.

This interaction was historically used to define the Ampere.

Conclusion

Today we explored the Lorentz force, the crucial link that explains how magnetic fields influence the world of moving charges.

Key Takeaways:

• A charge $q$ moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$ experiences a force $\vec{F}=q(\vec{v}\times \vec{B})$.
• A straight wire of length $L$ carrying current $I$ in a magnetic field $\vec{B}$ experiences a force $\vec{F}=I(\vec{L}\times \vec{B})$.
• The direction of the magnetic force is always perpendicular to both the velocity/current and the magnetic field, as determined by the right-hand rule.
• A fundamental consequence is that magnetic forces do no work; they can only change the direction of a charged particle's motion, not its speed.
• These principles explain why parallel currents attract and anti-parallel currents repel.

Preview of the Next Lesson:
We have now established that moving a charge through a magnetic field creates a force. In the next lesson, we will see how this very same Lorentz force is the microscopic origin of electromagnetic induction. We will explore how moving a conductor through a magnetic field (or changing the field around a conductor) can induce a voltage and drive a current—the principle that makes electrical generators possible.