Magnetic Field Equations to Know for AP Physics 2

Magnetic fields play a crucial role in understanding how charged particles and currents interact. These notes cover key equations and principles, including forces on moving charges, magnetic fields from wires, and the laws governing induction and flux, all essential for AP Physics 2.

1. Magnetic force on a moving charge: F = qvB sin θ

   • The force (F) experienced by a charge (q) moving with velocity (v) in a magnetic field (B) depends on the angle (θ) between the velocity and the magnetic field.
   • Maximum force occurs when θ = 90°, meaning the charge moves perpendicular to the magnetic field.
   • The direction of the force can be determined using the right-hand rule.

2. Magnetic force on a current-carrying wire: F = ILB sin θ

   • The force (F) on a wire carrying current (I) in a magnetic field (B) also depends on the angle (θ) between the wire and the magnetic field.
   • The force is maximized when the wire is perpendicular to the magnetic field (θ = 90°).
   • The direction of the force can be found using the right-hand rule, similar to the force on a moving charge.

3. Magnetic field due to a long straight wire: B = (μ₀I) / (2πr)

   • The magnetic field (B) created by a long straight wire carrying current (I) decreases with distance (r) from the wire.
   • The constant μ₀ is the permeability of free space, a fundamental physical constant.
   • The magnetic field forms concentric circles around the wire, with direction determined by the right-hand rule.

4. Magnetic field at the center of a circular loop: B = (μ₀I) / (2R)

   • The magnetic field (B) at the center of a circular loop of radius (R) carrying current (I) is directly proportional to the current and inversely proportional to the radius.
   • This equation shows that a tighter loop (smaller R) produces a stronger magnetic field.
   • The direction of the magnetic field follows the right-hand rule, curling around the loop.

5. Magnetic field inside a solenoid: B = μ₀nI

   • The magnetic field (B) inside a long solenoid is uniform and directly proportional to the current (I) and the number of turns per unit length (n).
   • The field lines inside the solenoid are parallel and closely spaced, indicating a strong and uniform magnetic field.
   • The direction of the magnetic field can be determined using the right-hand rule.

6. Ampère's Law: ∮B·dl = μ₀I

   • Ampère's Law relates the integrated magnetic field (B) around a closed loop to the total current (I) passing through that loop.
   • This law is useful for calculating magnetic fields in symmetrical situations, such as solenoids and toroids.
   • The constant μ₀ is the permeability of free space, linking magnetic fields to electric currents.

7. Magnetic flux: Φ = BA cos θ

   • Magnetic flux (Φ) measures the quantity of magnetic field (B) passing through a surface area (A) and depends on the angle (θ) between the field and the normal to the surface.
   • Maximum flux occurs when the magnetic field is perpendicular to the surface (θ = 0°).
   • Changes in magnetic flux can induce electromotive force (EMF) in a circuit.

8. Faraday's Law of Induction: ε = -dΦ/dt

   • Faraday's Law states that the induced electromotive force (ε) in a circuit is proportional to the rate of change of magnetic flux (Φ) through the circuit.
   • The negative sign indicates the direction of induced EMF opposes the change in flux, as per Lenz's Law.
   • This principle is fundamental in the operation of generators and transformers.

9. Lenz's Law: The induced current opposes the change in magnetic flux

   • Lenz's Law states that the direction of induced current will always be such that it opposes the change in magnetic flux that produced it.
   • This law ensures the conservation of energy and is a consequence of Faraday's Law.
   • It can be visualized by considering the effect of a changing magnetic field on a loop of wire.

10. Motional EMF: ε = Blv

• Motional EMF (ε) is generated when a conductor of length (l) moves with velocity (v) through a magnetic field (B).
• The induced EMF is directly proportional to the strength of the magnetic field, the length of the conductor, and its velocity.
• This principle is used in applications such as electric generators and railguns.