Magnetism

Misconceptions

Key Concepts

Magnetic Field

A magnetic field B is a vector field that exerts forces on moving charges. Magnetic field lines point from north to south poles outside the magnet and form closed loops.

Units: Tesla (T) = kg/(A·s²) = Wb/m²

Magnetic Force on a Moving Charge

A charge q moving with velocity v in a magnetic field B experiences a force:

F = qvB sin(θ)

Where θ is the angle between v and B.

Key properties:

• •Force is perpendicular to both v and B
• •Force is zero when v is parallel to B
• •Use the right-hand rule: fingers point in v direction, curl toward B, thumb points in F direction (for positive charges)
• •For negative charges, force is opposite to right-hand rule

Magnetic Force on a Current-Carrying Wire

A wire of length L carrying current I in a magnetic field B experiences a force:

F = BIL sin(θ)

Where θ is the angle between the wire and B.

Circular Motion in a Magnetic Field

Since magnetic force is always perpendicular to velocity, a charged particle in a uniform magnetic field moves in a circle (or helix if there's a component parallel to B).

Radius of circular path: r = mv/(qB)

Period: T = 2πm/(qB) (independent of speed!)

Magnetic Field from a Long Straight Wire

A long straight wire carrying current I produces a magnetic field at distance r:

B = μ₀I/(2πr)

Where μ₀ = 4π × 10⁻⁷ T·m/A is the permeability of free space.

Direction: Use right-hand rule - thumb in current direction, fingers curl in direction of B.

Magnetic Field in a Solenoid

A solenoid with n turns per unit length carrying current I produces a uniform field inside:

B = μ₀nI

The field is nearly uniform inside and approximately zero outside.

Faraday's Law of Induction

The induced EMF in a loop equals the negative rate of change of magnetic flux:

EMF = -N × (ΔΦ/Δt)

Where Φ = BA cos(θ) is the magnetic flux, N is the number of turns.

Lenz's Law

The direction of induced current creates a magnetic field that opposes the change in flux that produced it.

Equations

[1] F = qvB sin(θ) (magnetic force on moving charge)
[2] F = BIL sin(θ) (force on current-carrying wire)
[3] r = mv/(qB) (radius of circular motion)
[4] T = 2πm/(qB) (period of circular motion)
[5] B = μ₀I/(2πr) (field from long wire)
[6] B = μ₀nI (field in solenoid)
[7] Φ = BA cos(θ) (magnetic flux)
[8] EMF = -N(ΔΦ/Δt) (Faraday's law)
[9] μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space)

Worked Examples

Example 1: Force on a Moving Proton

Problem: A proton (q = 1.6 × 10⁻¹⁹ C) moves at 3 × 10⁶ m/s perpendicular to a 0.5 T magnetic field. What is the magnetic force?

Solution:

1. •Use F = qvB sin(θ) with θ = 90°
2. •F = (1.6 × 10⁻¹⁹)(3 × 10⁶)(0.5)(1)
3. •F = 2.4 × 10⁻¹³ N

Example 2: Circular Motion of an Electron

Problem: An electron (m = 9.11 × 10⁻³¹ kg, q = 1.6 × 10⁻¹⁹ C) moves at 2 × 10⁷ m/s in a 0.01 T field. Find the radius of its circular path.

Solution:

1. •Use r = mv/(qB)
2. •r = (9.11 × 10⁻³¹)(2 × 10⁷) / [(1.6 × 10⁻¹⁹)(0.01)]
3. •r = 1.82 × 10⁻²³ / 1.6 × 10⁻²¹
4. •r = 0.0114 m = 1.14 cm

Example 3: Induced EMF

Problem: A 50-turn coil with area 0.02 m² is in a magnetic field that decreases from 0.8 T to 0.2 T in 0.1 s. What is the induced EMF?

Solution:

1. •ΔΦ = A × ΔB = 0.02 × (0.2 - 0.8) = -0.012 Wb
2. •EMF = -N × ΔΦ/Δt = -50 × (-0.012)/0.1
3. •EMF = 6 V

Explanation Patterns

1. •Identify the physical situation - Is it force on a charge, force on a wire, or induction?
2. •Draw the geometry - Show B field, velocity/current direction, and use right-hand rule
3. •Identify the angle - Carefully determine θ between the relevant vectors
4. •Apply the appropriate formula - F = qvB, F = BIL, or Faraday's law
5. •Check direction using right-hand rule - Especially important for force problems
6. •Verify units - Tesla = kg/(A·s²), Weber = T·m²

Common Problem Types

1. •Force on moving charges: Calculate force magnitude and direction
2. •Circular motion in B field: Find radius, period, or frequency
3. •Force on current-carrying wires: Calculate force on straight or curved wires
4. •Magnetic field calculations: Find B from wires or solenoids
5. •Faraday's law problems: Calculate induced EMF from changing flux
6. •Lenz's law: Determine direction of induced current