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

When a charged particle is placed in an environment with various fields, several forces can act on it depending on the situation. Below are the key forces that can act on a charged particle: 

1. Electrostatic Force (Coulomb's Force):

The electrostatic force is the force exerted on a charged particle due to the presence of another charge. This force is described by Coulomb's law, which states that the force between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them:

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

where:

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

This force is attractive if the charges are of opposite sign and repulsive if the charges are of the same sign.

2. Gravitational Force:

The gravitational force acts on any object with mass, including charged particles. The force is given by Newton's law of 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 two objects,
• \( r \) is the distance between the centers of the two masses.

However, the gravitational force on charged particles is extremely small compared to other forces, such as the electrostatic force, because the masses of subatomic particles are much smaller than their charges.

3. Magnetic Force:

If the charged particle is moving through a magnetic field, it experiences a magnetic force. This force is described by the Lorentz force law, which states that the magnetic force on a moving charged particle is proportional to the charge, the velocity of the particle, and the magnetic field strength:

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

where:

• \( F_B \) is the magnetic force,
• \( q \) is the charge of the particle,
• \( \vec{v} \) is the velocity of the particle,
• \( \vec{B} \) is the magnetic field vector,
• \( \times \) represents the cross product.

The direction of the magnetic force is perpendicular to both the velocity of the charged particle and the magnetic field.

4. Electromagnetic Force (Lorentz Force):

The total force acting on a charged particle in the presence of both electric and magnetic fields is called the Lorentz force. It is the combination of the electrostatic force and the magnetic force acting on the particle:

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

where:

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

The Lorentz force governs the motion of charged particles in both electric and magnetic fields.

5. Centripetal Force (In Circular Motion):

If a charged particle is moving in a circular path due to the presence of a magnetic field, the magnetic force provides the necessary centripetal force to keep the particle in that circular motion. The centripetal force is given by:

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

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:

If a charged particle is placed in an external electric field, it experiences a force given by:

\[ F_E = qE \]

where:

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

This force is responsible for the acceleration of the charged particle in the direction of the electric field if the charge is positive, and in the opposite direction if the charge is negative.

Conclusion:

In summary, the various forces that can act on a charged particle include:

• Electrostatic force (Coulomb's force),
• Gravitational force (though negligible for subatomic particles),
• Magnetic force (when the particle is moving in a magnetic field),
• Electromagnetic force (combination of electric and magnetic forces),
• Centripetal force (in circular motion),
• Electric force (in an electric field).