Moving Charges And Magnetism

1. Understanding the Setup:

• The conductor is 1 cm long (i.e., 0.01 m), symmetrically placed along the x-axis centered at the origin. That means it extends from to .
• Current
• Point of observation is at

2. Magnetic Field Due to a Finite Current Segment:

We use the Biot–Savart law for a finite straight conductor:

where:

• = perpendicular distance from wire to point =
• Since the wire is symmetric about the origin, , and they are measured from the center of the wire to the ends.
• (very small)
• , so

So the approximate magnetic field is:

3. Final Answer:

Option (C) is correct.

(a) Magnetic Field B

The magnetic field due to a long straight current-carrying wire is given by Ampère’s law:

B = μ₀I / 2πd

The right-hand rule states that the direction of B at a distance d above the wire is along the positive k-direction (out of the plane). Thus:

B = μ₀I / 2πd

(b) Magnetic Force F_m

The force on a charged particle moving in a magnetic field is given by:

F_m = q(v × B)

Given that:

• v = v̅ i (along the x-axis),
• B = B̅ k (along the z-axis),

Using the cross product:

v × B = (v̅ i) × (B̅ k)

Using the vector identity i × k = −j, we get:

F_m = qvB(−j)

Thus:

F_m = −qvB̅ j

(c) Electric Field E

Since the charge moves undeviated, the net force on the particle must be zero. This means that the electric force F_e = qE must cancel out the magnetic force:

qE = −F_m

Substituting F_m = −qvB̅ j:

Dividing by q:

E = vB̅ j

Quick Tip: The required electric field is:

E = vB̅ j

Characteristics of Equipotential Surfaces

1. Key Characteristics:

• Constant Potential: The electric potential is the same at all points on an equipotential surface, meaning no work is done in moving a charge along the surface.
• Perpendicular to Electric Field: Equipotential surfaces are always perpendicular to the electric field lines. This is because the electric field is defined as the gradient of the potential, so it must be normal (perpendicular) to the equipotential surfaces.

2. Given Information:

• The electric field is uniform and directed along the -axis.
• The potential at any point is given by the equation: where:
  • is the potential at the origin,
  • is the magnitude of the electric field,
  • is the distance along the -axis.

3. Calculation of Potential at :

Since the electric field is along the -axis, the distance along the -axis is 4 m (ignoring the -coordinate of 3 m because the field is only in the -direction).

The potential at this point is given by the formula:

4. Conclusion:

Thus, the potential at the point is .

In analyzing the given assertion and reason, we need to establish their validity and relationship.

The assertion (A) states: "The deflection in a galvanometer is directly proportional to the current passing through it." This assertion is true. A galvanometer works on the principle that when an electric current passes through a coil in a magnetic field, it experiences a magnetic torque that causes it to rotate. The amount of deflection is directly related to the current flowing through it.

The reason (R) states: "The coil of a galvanometer is suspended in a uniform radial magnetic field." This reason is also true. The galvanometer's design ensures that the coil is in a uniform radial magnetic field, which maintains consistent sensitivity across the rotation of the coil.

While both the assertion and the reason are correct statements, the reason given is a part of the design characteristic of the galvanometer, ensuring uniform sensitivity. However, it does not directly explain why the deflection is proportional to the current. Instead, the proportionality of deflection to current is primarily because of the magnetic torque proportional to the current in the coil. Therefore, the correct option is:

Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).

Net Magnetic Field at Point P Due to Two Infinitely Long Conductors

Given:

• Conductor 1: Carries current along the -X axis (along XX' axis)
• Conductor 2: Carries current along the -Y axis (along YY' axis)
• Point P is at coordinates

Solution:

1. Magnetic Field Due to a Long Straight Conductor:

The magnetic field at a distance from a long straight conductor carrying current is given by the formula: where: - is the magnetic field, - is the permeability of free space ( ), - is the current in the conductor, - is the perpendicular distance from the conductor to the point where the magnetic field is being calculated.

2. Magnetic Field Due to Conductor 1 (along XX' axis):

The distance of point P from conductor 1 (along the XX' axis) is (since the conductor is along the -axis). The magnetic field at point P due to conductor 1, using the formula, is: The direction of the magnetic field can be found using the right-hand rule. Since the current flows along the -X axis, the magnetic field at point P will be directed out of the page (towards the viewer) by the right-hand rule.

3. Magnetic Field Due to Conductor 2 (along YY' axis):

The distance of point P from conductor 2 (along the YY' axis) is (since the conductor is along the -axis). The magnetic field at point P due to conductor 2 is: Again, applying the right-hand rule, since the current flows along the -Y axis, the magnetic field at point P will be directed into the page (away from the viewer).

4. Resultant Magnetic Field:

The magnetic fields and are perpendicular to each other (one is out of the page and the other is into the page), so we can calculate the net magnetic field using the Pythagorean theorem: Substituting the values of and :

5. Direction of the Net Magnetic Field:

The direction of the net magnetic field can be found using the right-hand rule applied to the two perpendicular fields. The resultant magnetic field will be directed at an angle from the -axis (where is the angle between the net magnetic field and the -axis). Using trigonometry: Thus, the angle is given by:

Conclusion:

The magnitude of the net magnetic field at point P is: The direction of the net magnetic field is at an angle from the -axis, where:

1. Kinetic Energy and Radius of the Path:

The kinetic energy of a charged particle with mass and velocity is given by:

In the presence of a uniform magnetic field, the charged particle moves in a circular path with a radius , given by:

Where:

• is the radius of the circular path,
• is the mass of the particle,
• is the velocity of the particle,
• is the charge of the particle, and
• is the magnetic field strength.

2. Effect of Losing Half the Kinetic Energy:

After passing through the sheet of lead, the particle loses half of its kinetic energy, so the new kinetic energy is:

Since kinetic energy is proportional to the square of the velocity, the new velocity of the particle is:

Now, the new radius of the particle’s path is given by the equation:

Substituting into this equation, we get:

Conclusion for the Radius:

• The radius of the path decreases by a factor of . So, the new radius .

3. Time Period of Revolution:

The time period of revolution of the particle in a magnetic field is given by:

Since the velocity of the particle decreases by a factor of , the new velocity is . Therefore, the new time period is:

Substituting into the equation, we get:

Conclusion for the Time Period:

• The time period of revolution increases by a factor of . So, the new time period .

Final Summary:

• The radius of the path decreases by a factor of .
• The time period of revolution increases by a factor of .

The work done by a force is given by: Where is the force and is the displacement. The magnetic force on a moving charge is given by: Since the magnetic force is always perpendicular to the velocity of the particle, the work done by the magnetic force is zero because: Therefore, the magnetic force does no work on a moving charge, as it does not change the kinetic energy of the particle.

The source of the force acting on a current-carrying conductor placed in a magnetic field is the magnetic interaction between the moving charges (current) in the conductor and the external magnetic field. The force on a current-carrying conductor in a magnetic field is given by: Where:
is the force,
is the current,
is the length of the conductor in the magnetic field,
is the magnetic field strength,
is the angle between the magnetic field and the conductor.

For two long, straight, parallel conductors carrying steady currents and , the force per unit length between them is given by Ampère's law: Where:
is the permeability of free space,
is the distance between the two conductors.

This is the expression for the force between two parallel conductors carrying steady currents, and it defines Ampère's law.

The magnetic force and the electric force must balance each other to maintain constant velocity for the particle. Therefore: Substituting the expressions for electric and magnetic forces: Using the formula for the magnetic field produced by a current-carrying conductor at a distance : Thus, equating the forces: Solving for : Thus, the value of is:

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:

where:

• is the magnitude of the electrostatic force between the charges,
• is Coulomb's constant ( ),
• and are the charges,
• 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:

where:

• is the gravitational force,
• is the gravitational constant ( ),
• and are the masses of the two objects,
• 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:

where:

• is the magnetic force,
• is the charge of the particle,
• is the velocity of the particle,
• is the magnetic field vector,
• 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:

where:

• is the total force on the charged particle,
• is the electric field,
• is the velocity of the charged particle,
• is the magnetic field,
• 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:

where:

• is the centripetal force,
• is the mass of the charged particle,
• is the speed of the particle,
• 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:

where:

• is the electric force,
• is the charge of the particle,
• 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).

Forces on a Charged Particle in Electric and Magnetic Fields

Given:

• The particle is negatively charged and moving with a constant velocity .
• The particle is near a long straight conductor along the XX' axis carrying current towards the -X axis.
• The particle remains at a constant distance from the conductor.

1. Magnetic Field due to the Current-Carrying Conductor:

The magnetic field at a distance from a long straight conductor carrying current is given by: where: - is the magnetic field, - is the permeability of free space ( ), - is the current, - is the distance from the conductor to the point where the magnetic field is calculated.

2. Magnetic Force on the Particle:

The magnetic force on a moving charge is given by the Lorentz force law: where: - is the charge of the particle (negative charge), - is the velocity of the particle, and - is the magnetic field. Since the magnetic field is directed into the page and the particle is moving along the -X axis, applying the right-hand rule for , we find that the magnetic force will be directed **upwards**, along the positive -axis. The magnitude of the magnetic force is:

3. Electric Force on the Particle:

The particle experiences an electric force due to the uniform electric field . The electric force is given by: Since the particle is negatively charged, the electric force will be in the **opposite direction** to the electric field. If the electric field is directed along the positive -axis, the electric force will be directed along the **negative -axis**. The magnitude of the electric force is:

4. Condition for Constant Velocity:

For the particle to move with a constant velocity, the net force on the particle must be zero. Therefore, the magnetic and electric forces must balance each other. Thus: Substituting the expressions for and : Canceling from both sides:

Conclusion:

The relationship between the velocity , the current , the distance , the electric field , and the charge is given by: This equation shows that the particle will continue to move with a constant velocity as long as the electric force balances the magnetic force.