NCERT Solutions for Class 12 Mathematics
Class 12 Physics: Chapter 4 - Moving Charges and Magnetism
Introduction
This chapter delves into the relationship between electricity and magnetism, focusing on the magnetic effects produced by moving charges. Key concepts include the Biot-Savart law, Ampere's Circuital law, and the motion of charged particles in magnetic fields.
Topics Covered
- Magnetic Force
- Motion in a Magnetic Field
- Biot-Savart Law
- Ampere's Circuital Law
- Straight and Toroidal Solenoids
- Force between Two Parallel Currents
- Moving Coil Galvanometer
Magnetic Force
When a charged particle moves through a magnetic field, it experiences a force known as the magnetic force. This force is given by:
F = q(v × B)
where F is the magnetic force, q is the charge, v is the velocity, and B is the magnetic field.
Motion in a Magnetic Field
A charged particle moving perpendicular to a uniform magnetic field follows a circular path. The radius of this path is given by:
r = (mv) / (qB)
where m is the mass of the particle, v is the velocity, q is the charge, and B is the magnetic field.
Biot-Savart Law
The Biot-Savart law describes the magnetic field generated by a current-carrying element. It is given by:
dB = (μ₀/4π) * (I * dl × r̂) / r²
where dB is the infinitesimal magnetic field, μ₀ is the permeability of free space, I is the current, dl is the length element, r̂ is the unit vector from the element to the point where the field is calculated, and r is the distance between them.
Ampere's Circuital Law
Ampere's Circuital law relates the magnetic field along a closed loop to the electric current passing through the loop. It is stated as:
∮B · dl = μ₀I
where B is the magnetic field, dl is the differential length vector, and I is the current enclosed by the loop.
Straight and Toroidal Solenoids
Straight Solenoid
A straight solenoid is a coil of wire with many turns, generating a uniform magnetic field inside it. The magnetic field inside a solenoid is given by:
B = μ₀nI
where n is the number of turns per unit length, and I is the current.
Toroidal Solenoid
A toroidal solenoid is a solenoid bent into a circular shape, generating a magnetic field confined within its core. The magnetic field inside a toroid is given by:
B = (μ₀NI) / (2πr)
where N is the total number of turns, I is the current, and r is the radius of the toroid.
Force between Two Parallel Currents
Two parallel currents attract each other if they flow in the same direction and repel if they flow in opposite directions. The force per unit length between two parallel currents is given by:
F/L = (μ₀I₁I₂) / (2πd)
where I₁ and I₂ are the currents, and d is the distance between the wires.
Moving Coil Galvanometer
A moving coil galvanometer is a device used to measure small electric currents. It works on the principle that a current-carrying coil placed in a magnetic field experiences a torque. The deflection of the coil is proportional to the current flowing through it.
Important Terms and Meanings
Magnetic Force
Magnetic Force: The force experienced by a moving charge in a magnetic field.
Biot-Savart Law
Biot-Savart Law: A law describing the magnetic field generated by a current element.
Ampere's Circuital Law
Ampere's Circuital Law: A law relating the magnetic field around a closed loop to the current passing through it.
Solenoid
Solenoid: A coil of wire that generates a uniform magnetic field when an electric current passes through it.
Toroidal Solenoid
Toroidal Solenoid: A solenoid bent into a circular shape, confining the magnetic field within its core.
Galvanometer
Galvanometer: An instrument used to detect and measure small electric currents.
Frequently Asked Questions (FAQ)
1. What is the magnetic force?
The magnetic force is the force experienced by a moving charge in a magnetic field, given by F = q(v × B).
2. How does a charged particle move in a magnetic field?
A charged particle moving perpendicular to a uniform magnetic field follows a circular path with a radius given by r = (mv) / (qB).
3. What is the Biot-Savart law?
The Biot-Savart law describes the magnetic field generated by a current-carrying element, given by dB = (μ₀/4π) * (I * dl × r̂) / r².
4. What does Ampere's Circuital law state?
Ampere's Circuital law states that the integral of the magnetic field around a closed loop is equal to μ₀ times the current passing through the loop, given by ∮B · dl = μ₀I.
5. What is a solenoid?
A solenoid is a coil of wire that generates a uniform magnetic field when an electric current passes through it.
6. What is a toroidal solenoid?
A toroidal solenoid is a solenoid bent into a circular shape, confining the magnetic field within its core.
7. How is the magnetic field inside a straight solenoid calculated?
The magnetic field inside a straight solenoid is given by B = μ₀nI, where n is the number of turns per unit length and I is the current.
8. How is the magnetic field inside a toroidal solenoid calculated?
The magnetic field inside a toroidal solenoid is given by B = (μ₀NI) / (2πr), where N is the total number of turns, I is the current, and r is the radius of the toroid.
9. What is the force between two parallel currents?
The force per unit length between two parallel currents is given by F/L = (μ₀I₁I₂) / (2πd), where I₁ and I₂ are the currents, and d is the distance between the wires.
10. What is a moving coil galvanometer?
A moving coil galvanometer is a device used to measure small electric currents by observing the deflection of a current-carrying coil placed in a magnetic field.
11. How does the magnetic force affect a moving charge?
The magnetic force causes a moving charge to experience a perpendicular force that changes the direction of the charge's velocity, resulting in circular or helical motion.
12. What is the right-hand rule in magnetism?
The right-hand rule helps determine the direction of the magnetic force on a positive moving charge. Point your thumb in the direction of the velocity, your fingers in the direction of the magnetic field, and your palm will face the direction of the force.
13. Can magnetic fields do work on a charged particle?
No, magnetic fields cannot do work on a charged particle because the force is always perpendicular to the velocity, meaning there is no change in kinetic energy.
14. How does a current-carrying conductor in a magnetic field behave?
A current-carrying conductor in a magnetic field experiences a force given by F = I(L × B), where L is the length vector of the conductor.
15. What is the significance of μ₀ (mu-zero) in magnetism?
μ₀, the permeability of free space, is a constant that appears in equations relating to magnetic fields and forces, such as the Biot-Savart law and Ampere's Circuital law. Its value is approximately 4π × 10⁻⁷ T·m/A.
16. What is the role of a magnetic field in a cyclotron?
A cyclotron uses a magnetic field to force charged particles into a circular path, allowing them to accelerate repeatedly across a high-frequency alternating voltage until they reach high speeds.
17. How does the magnetic field inside a solenoid depend on its properties?
The magnetic field inside a solenoid depends on the current passing through it and the number of turns per unit length, as given by B = μ₀nI.
18. Why is the magnetic field inside a toroidal solenoid confined?
The magnetic field inside a toroidal solenoid is confined because the coil's circular shape ensures the magnetic field lines are contained within the core, reducing leakage and improving efficiency.
19. What is the importance of Ampere's Circuital law?
Ampere's Circuital law is crucial for calculating the magnetic field in symmetric situations, such as inside solenoids or around current-carrying wires, by relating the field to the enclosed current.
20. How does the force between two parallel currents lead to practical applications?
The force between two parallel currents forms the basis for the operation of devices like the ampere balance, which is used to define the unit of current, the ampere, based on the force between conductors.