Magnetic Field Formulas to Know for Principles of Physics II

Magnetic fields play a vital role in understanding electromagnetism. These formulas explain how currents create magnetic fields, the forces on moving charges, and the principles behind devices like motors and generators. Mastering these concepts is key in Physics II.

1. Biot-Savart Law

   • Describes the magnetic field generated by a small segment of current-carrying wire.
   • The magnetic field (B) is proportional to the current (I) and inversely proportional to the square of the distance (r) from the wire segment.
   • The direction of the magnetic field is determined by the right-hand rule.

2. Ampère's Law

   • Relates the integrated magnetic field around a closed loop to the electric current passing through the loop.
   • Mathematically expressed as ∮B·dl = μ₀I_enc, where μ₀ is the permeability of free space.
   • Useful for calculating magnetic fields in symmetrical situations, such as solenoids and toroids.

3. Magnetic force on a moving charge (Lorentz Force)

   • The force (F) on a charge (q) moving with velocity (v) in a magnetic field (B) is given by F = q(v × B).
   • The direction of the force is perpendicular to both the velocity of the charge and the magnetic field.
   • This principle is fundamental in understanding the motion of charged particles in magnetic fields.

4. Magnetic force on a current-carrying wire

   • A current-carrying wire in a magnetic field experiences a force given by F = I(L × B), where L is the length vector of the wire.
   • The direction of the force can be determined using the right-hand rule.
   • This principle is applied in electric motors and other electromagnetic devices.

5. Magnetic field of a long straight wire

   • The magnetic field (B) around a long straight wire carrying current (I) is given by B = (μ₀I)/(2πr), where r is the distance from the wire.
   • The field forms concentric circles around the wire, with the direction determined by the right-hand rule.
   • This formula is essential for understanding the behavior of magnetic fields in practical applications.

6. Magnetic field at the center of a circular current loop

   • The magnetic field (B) at the center of a circular loop carrying current (I) is given by B = (μ₀I)/(2R), where R is the radius of the loop.
   • The field direction is perpendicular to the plane of the loop, following the right-hand rule.
   • This concept is crucial for designing electromagnets and inductors.

7. Magnetic field of a solenoid

   • A solenoid generates a uniform magnetic field (B) inside it, given by B = μ₀(nI), where n is the number of turns per unit length.
   • The field lines inside the solenoid are parallel and closely spaced, indicating a strong and uniform field.
   • This principle is widely used in applications like transformers and magnetic field generation.

8. Magnetic flux

   • Magnetic flux (Φ) is defined as the product of the magnetic field (B) and the area (A) through which it passes, Φ = B·A·cos(θ), where θ is the angle between B and the normal to the surface.
   • It quantifies the total magnetic field passing through a given area.
   • Understanding magnetic flux is essential for applications in electromagnetism and induction.

9. Faraday's Law of Induction

   • States that a change in magnetic flux through a circuit induces an electromotive force (emf) in the circuit.
   • Mathematically expressed as ε = -dΦ/dt, where ε is the induced emf and dΦ/dt is the rate of change of magnetic flux.
   • This principle is fundamental in the operation of generators and transformers.

10. Lenz's Law

    • States that the direction of the induced current (or emf) will oppose the change in magnetic flux that produced it.
    • This law is a consequence of the conservation of energy and ensures that induced currents act to counteract changes in the magnetic field.
    • Lenz's Law is crucial for understanding the behavior of inductors and the operation of electric circuits.