Shortcut Methods and Tricks for Numerical Problems on Ampere’s Law:

Magnetic field of a solenoid:

• Use the formula: $$B = \mu_0 nI,$$ where (B) is the magnetic field, (\mu_0) is the permeability of free space ((4\pi \times 10^{-7} N/A^2)), (n) is the number of turns per unit length, and (I) is the current.

Magnetic force on a moving charge:

• Use the formula: $$F = qvB\sin\theta,$$ where (F) is the magnetic force, (q) is the charge of the particle, (v) is the velocity of the particle, (B) is the magnetic field, and (\theta) is the angle between the velocity and the magnetic field.

Magnetic moment of a current loop:

• Use the formula: $$\mu = IA,$$ where (\mu) is the magnetic moment, (I) is the current, and (A) is the area of the loop.

Magnetic field due to a moving charge:

• Use the formula: $$B = \frac{\mu_0}{4\pi}\frac{qv\sin\theta}{r},$$ where (B) is the magnetic field, (\mu_0) is the permeability of free space, (q) is the charge of the moving particle, (v) is the velocity of the particle, (\theta) is the angle between the velocity and the line joining the charge and the observation point, and (r) is the distance between the charge and the observation point.

Torque on a current loop in a magnetic field:

• Use the formula: $$\tau = \mu B\sin\theta,$$ where (\tau) is the torque, (\mu) is the magnetic moment of the loop, (B) is the magnetic field, and (\theta) is the angle between the magnetic moment and the magnetic field.

Applications of Ampere’s law:

• Understand the basic principles behind the working of solenoids, electromagnets, magnetic levitation trains, magnetic resonance imaging, and particle accelerators. Use real-world examples and analogies to grasp the concepts better.