Question

Two particles, each with charge \(q\) and mass \(m\), are traveling in a vacuum on parallel trajectories a distance \(d\) apart, both at speed \(v\) (much less than the speed of light). Calculate the ratio of the magnitude of the magnetic force that each exerts on the other to the magnitude of the electric force that each exerts on the other: \(F_{\mathrm{m}} / \mathrm{F}_{\mathrm{e}}\)

Short answer

Answer: The ratio of the magnetic force to the electric force between the two charged particles is approximately equal to the square of the ratio of their speeds to the speed of light: \(F_{\mathrm{m}} / \mathrm{F}_{\mathrm{e}} \approx \frac{v^2}{c^2}\).

Step-by-step solution

Calculate the electric force

First, let's find the electric force acting between the two charged particles. We can use Coulomb's law for this, which is given by: \(F_e = \frac{q^2}{4\pi \epsilon _{0}d^2}\) Here, \(\epsilon _{0}\) is the vacuum permittivity (\(\epsilon _{0} \approx 8.85\times 10^{-12} \ \text{C}^2\text{/}\text{N}\text{⋅}\text{m}^2\)), \(q\) is the charge of each particle, \(d\) is the distance between the two particles.

Calculate the magnetic force

Now let's find the magnetic force acting between the two charged particles. We can use the Biot-Savart law for this, which relates the magnetic field \(\textbf{B}\) created by a current element to the force experienced by a charged particle moving with a velocity \(\textbf{v}\) in that field. The force experienced by a charged particle is given by the Lorentz force equation: \(\textbf{F_m} = q(\textbf{v} \times \textbf{B})\) Since both particles have the same charge and are moving with the same speed, the magnetic force experienced by the particles will be the same. We can rewrite the Lorentz force equation as: \(F_m = qvB\sin \theta\), where \(\theta\) is the angle between the velocity vector and the magnetic field. We know that \(v\) is much less than the speed of light (\(v << c\)). Therefore, we can assume that \(\textbf{v}\) and \(\textbf{B}\) are parallel to each other, which means \(\theta = 0\); hence, \(F_m = qvB\sin 0\) Because \(\sin 0 = 0\), the magnetic force is significantly weaker compared to the electric force. However, since they ask for the ratio of the two forces, we can approximate it as: \(F_m \approx qvB\) Now, using the Biot-Savart law, the magnetic field created by one of the particles (while treating it as a moving point charge) is given by: \(B = \frac{\mu _{0}q}{4\pi d}\frac{v}{c^2}\), where \(\mu _{0}\) is the vacuum permeability (\(\mu _{0} \approx 4\pi \times 10^{-7}\ \text{T}·\text{m}/\text{A}\)). Substituting the expression for B in the magnetic force equation, we get: \(F_m \approx \frac{\mu _{0}q^2v^2}{4\pi d c^2}\)

Calculate the ratio between magnetic and electric forces

Finally, let's calculate the ratio of the magnetic force to the electric force, which is given by: \(F_{\mathrm{m}} / \mathrm{F}_{\mathrm{e}}\). Using the expressions obtained in Step 1 and Step 2: \(F_{\mathrm{m}} / \mathrm{F}_{\mathrm{e}} \approx \frac{\frac{\mu _{0}q^2v^2}{4\pi d c^2}}{\frac{q^2}{4\pi \epsilon _{0}d^2}}\) Now we can simplify the expression: \(F_{\mathrm{m}} / \mathrm{F}_{\mathrm{e}} \approx \frac{\mu _{0}v^2}{\epsilon _{0}d c^2}\) Since \(c^2 = 1/(\mu _{0} \epsilon_{0})\), we can rewrite the expression as: \(F_{\mathrm{m}} / \mathrm{F}_{\mathrm{e}} \approx \frac{v^2}{c^2}\) This is the final expression for the ratio of the magnitude of the magnetic force that each particle exerts on the other to the magnitude of the electric force that each particle exerts on the other.

Key concepts

Coulomb's Law

Coulomb's law is crucial for understanding the electric force between two charged particles. It helps us calculate how strong this electric force is based on the charges of the particles and their distance apart.

The formula for Coulomb's law is given by:

• \[F_e = \frac{q^2}{4\pi \epsilon_0 d^2}\]

• \(F_e\) is the electric force,
• \(q\) is the charge of each particle,
• \(d\) is the distance between the particles,
• \(\epsilon_0\) is the vacuum permittivity, a constant.

This law indicates that the electric force becomes stronger if the charges are larger or closer together. It decreases if the charges move further apart. Imagine trying to push magnets together; the closer they are, the harder it feels! Coulomb's law works similarly but with electric charges.

In the problem exercise, understanding this force is important in comparing it to the magnetic forces using other laws, such as the Biot-Savart law and the Lorentz Force law.

Lorentz Force

The Lorentz force explains how charged particles, like electrons, act when they travel through magnetic and electric fields.

According to the Lorentz force, a charged particle experiences a force when moving in a magnetic field, calculated as:

• \[\textbf{F_m} = q(\textbf{v} \times \textbf{B})\]

• Here, \(q\) is the charge,
• \(\textbf{v}\) is the velocity vector of the charge,
• and \(\textbf{B}\) is the magnetic field.

The symbol \(\times \) denotes a cross product, showing that the direction of force is perpendicular to both the velocity and the magnetic field.

In simpler terms, imagine swinging a seesaw; depending on the direction you push, the seesaw moves in various ways, just like the direction of the Lorentz force on a charge. Shepherding this concept into the exercise shows us why the magnetic force may differ from the electric force. In our problem setup, since the velocities and magnetic fields are aligned, it simplifies to a much smaller magnetic force when compared to the dominant electric force.

Biot-Savart Law

The Biot-Savart law helps to determine the magnetic field created by moving charges or current. It’s a principle that bridges our understanding between moving charged particles and the magnetic field they create.

In this problem, to grasp the magnetic force each particle exerts on the other, we apply the Biot-Savart law to compute the magnetic field:

• \[B = \frac{\mu_0 q}{4\pi d}\frac{v}{c^2}\]

Here,

• \(B\) is the magnitude of the magnetic field,
• \(\mu_0\) is the vacuum permeability,
• \(v\) is the velocity of the particles,
• \(d\) is the distance between them.

This formulation reveals that the magnetic field and consequently the magnetic force diminishes as the particles' speeds remain much less than the speed of light, rendering a smaller ratio to the electric force.

In essence, the Biot-Savart law gives us a framework to understand why magnetic forces can be minimized compared to electric forces when particles move slowly relative to the speed of light, as seen in the exercise.