Question

What are the various forces acting on the charged particle?

Hint:

The total force on a charged particle moving in an electric and magnetic field is the vector sum of the electric and magnetic forces.

Answer 1

A charged particle within a field-rich environment can be subjected to multiple forces based on the prevailing conditions. The primary forces acting on a charged particle are delineated below:

1. Electrostatic Force (Coulomb's Force):

This force arises from the interaction between two electric charges. Coulomb's law quantifies this force, establishing it as directly proportional to the product of the charges' magnitudes and inversely proportional to the square of the separation distance:

\[ F = k_e \frac{|q_1 q_2|}{r^2} \]

wherein:

• \( F \) denotes the magnitude of the electrostatic force,
• \( k_e \) is Coulomb's constant (\( 8.99 \times 10^9 \, \text{N m}^2/\text{C}^2 \)),
• \( q_1 \) and \( q_2 \) represent the magnitudes of the charges,
• \( r \) signifies the distance between the charges.

The force is attractive for charges of opposite signs and repulsive for like charges.

2. Gravitational Force:

All objects possessing mass, including charged particles, are subject to gravitational force, as described by Newton's law of universal gravitation:

\[ F_g = \frac{G m_1 m_2}{r^2} \]

where:

• \( F_g \) is the gravitational force,
• \( G \) is the gravitational constant (\( 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)),
• \( m_1 \) and \( m_2 \) are the masses of the interacting objects,
• \( r \) is the distance between their centers.

It is noteworthy that the gravitational force exerted on charged particles is exceedingly weak in comparison to other forces, such as the electrostatic force, due to the diminutive masses of subatomic particles relative to their charges.

3. Magnetic Force:

A charged particle in motion within a magnetic field experiences a magnetic force. This interaction is governed by the Lorentz force law, which indicates that the magnetic force is proportional to the particle's charge, velocity, and the magnetic field strength:

\[ F_B = q \vec{v} \times \vec{B} \]

with:

• \( F_B \) representing the magnetic force,
• \( q \) being the charge of the particle,
• \( \vec{v} \) denoting the particle's velocity vector,
• \( \vec{B} \) signifying the magnetic field vector,
• \( \times \) denoting the cross product.

The resultant magnetic force is orthogonal to both the particle's velocity and the magnetic field.

4. Electromagnetic Force (Lorentz Force):

The combined force experienced by a charged particle subjected to both electric and magnetic fields is termed the Lorentz force. It is the summation of the electrostatic and magnetic forces:

\[ \vec{F} = q(\vec{E} + \vec{v} \times \vec{B}) \]

where:

• \( \vec{F} \) is the net force on the charged particle,
• \( \vec{E} \) is the electric field vector,
• \( \vec{v} \) is the velocity vector of the charged particle,
• \( \vec{B} \) is the magnetic field vector,
• \( q \) represents the charge of the particle.

The Lorentz force dictates the behavior of charged particles under the influence of electric and magnetic fields.

5. Centripetal Force (In Circular Motion):

When a charged particle follows a circular trajectory due to a magnetic field, the magnetic force serves as the centripetal force maintaining this motion. The centripetal force is calculated as:

\[ F_c = \frac{m v^2}{r} \]

where:

• \( F_c \) is the centripetal force,
• \( m \) is the mass of the charged particle,
• \( v \) is the speed of the particle,
• \( r \) is the radius of the circular path.

6. Electric Force in an Electric Field:

A charged particle situated within an external electric field experiences an electric force defined by:

\[ F_E = qE \]

where:

• \( F_E \) denotes the electric force,
• \( q \) is the charge of the particle,
• \( E \) is the electric field strength.

This force drives the acceleration of a positive charge in the direction of the electric field and a negative charge in the opposite direction.

Conclusion:

In summary, the forces that can impact a charged particle include:

• Electrostatic force (Coulomb's force),
• Gravitational force (typically negligible for subatomic particles),
• Magnetic force (acting on moving particles in a magnetic field),
• Electromagnetic force (the resultant of electric and magnetic forces),
• Centripetal force (required for circular motion),
• Electric force (experienced in an electric field).