Formula Sheet: Magnetism

Magnetic Fields and Forces

Magnetic Field (B)

• SI Unit: Tesla (T) = kg/(A·s²) = N/(A·m)
• Gauss: 1 T = 10⁴ gauss (older unit, occasionally used)
• Magnetic field lines: Flow from North pole to South pole outside a magnet

Magnetic Force on a Moving Charge

\[F_B = qvB\sin\theta\]

• \(F_B\) = magnetic force (N)
• \(q\) = charge (C)
• \(v\) = velocity of charge (m/s)
• \(B\) = magnetic field strength (T)
• \(\theta\) = angle between velocity vector and magnetic field vector
• Maximum force: When \(\theta = 90°\) (perpendicular), \(F_B = qvB\)
• Zero force: When \(\theta = 0°\) or \(180°\) (parallel or antiparallel)
• Direction: Determined by right-hand rule

Right-Hand Rule for Magnetic Force

• For positive charges: Point fingers in direction of velocity, curl toward magnetic field direction; thumb points in direction of force
• For negative charges: Use right-hand rule, then reverse the direction
• Alternative formulation: Thumb = velocity, fingers = magnetic field, palm = force direction (for positive charges)

Magnetic Force on a Current-Carrying Wire

\[F_B = BIL\sin\theta\]

• \(F_B\) = magnetic force on wire (N)
• \(B\) = magnetic field strength (T)
• \(I\) = current in wire (A)
• \(L\) = length of wire in magnetic field (m)
• \(\theta\) = angle between current direction and magnetic field
• Maximum force: When wire is perpendicular to field (\(\theta = 90°\)), \(F_B = BIL\)
• Direction: Use right-hand rule (thumb = current direction, fingers = field, palm = force)

Charged Particle Motion in Magnetic Fields

Circular Motion of Charged Particle

• When: Particle enters magnetic field perpendicular to field lines
• Result: Uniform circular motion in plane perpendicular to \(\vec{B}\)
• Force provides: Centripetal acceleration

\[F_B = F_c\] \[qvB = \frac{mv^2}{r}\]

Radius of Circular Path

\[r = \frac{mv}{qB}\]

• \(r\) = radius of circular path (m)
• \(m\) = mass of particle (kg)
• \(v\) = speed of particle (m/s)
• \(q\) = charge of particle (C)
• \(B\) = magnetic field strength (T)
• Note: Larger mass or higher velocity → larger radius
• Note: Stronger field or greater charge → smaller radius

Period of Circular Motion

\[T = \frac{2\pi r}{v} = \frac{2\pi m}{qB}\]

• \(T\) = period (s)
• Note: Period is independent of velocity and radius
• Application: Cyclotrons use this principle

Frequency of Circular Motion

\[f = \frac{1}{T} = \frac{qB}{2\pi m}\]

• \(f\) = frequency (Hz)
• Cyclotron frequency: Also called gyrofrequency

Angular Velocity

\[\omega = \frac{2\pi}{T} = \frac{qB}{m}\]

• \(\omega\) = angular velocity (rad/s)

Magnetic Fields Produced by Currents

Magnetic Field Around a Long Straight Current-Carrying Wire

\[B = \frac{\mu_0 I}{2\pi r}\]

• \(B\) = magnetic field strength (T)
• \(\mu_0\) = permeability of free space = \(4\pi × 10^{-7}\) T·m/A
• \(I\) = current (A)
• \(r\) = perpendicular distance from wire (m)
• Field pattern: Concentric circles around wire
• Direction: Right-hand rule (thumb = current, fingers curl in field direction)

Magnetic Field at Center of a Circular Loop

\[B = \frac{\mu_0 I}{2R}\]

• \(B\) = magnetic field at center (T)
• \(I\) = current in loop (A)
• \(R\) = radius of loop (m)
• Direction: Perpendicular to plane of loop (right-hand rule: fingers curl with current, thumb points in field direction)

Magnetic Field Inside a Solenoid

\[B = \mu_0 nI\]

• \(B\) = magnetic field inside solenoid (T)
• \(n\) = number of turns per unit length = N/L (turns/m)
• \(N\) = total number of turns
• \(L\) = length of solenoid (m)
• \(I\) = current (A)
• Field characteristics: Uniform inside, nearly zero outside
• Direction: Right-hand rule (fingers curl with current, thumb points toward North pole)

Alternative Solenoid Formula

\[B = \frac{\mu_0 NI}{L}\]

• Same as above, just written with total turns and length separately

Electromagnetic Induction

Magnetic Flux

\[\Phi_B = BA\cos\theta\]

• \(\Phi_B\) = magnetic flux (Wb = weber = T·m²)
• \(B\) = magnetic field strength (T)
• \(A\) = area (m²)
• \(\theta\) = angle between magnetic field and normal (perpendicular) to area
• Maximum flux: When field is perpendicular to surface (\(\theta = 0°\)), \(\Phi_B = BA\)
• Zero flux: When field is parallel to surface (\(\theta = 90°\))

Faraday's Law of Electromagnetic Induction

\[\varepsilon = -\frac{\Delta\Phi_B}{\Delta t}\]

• \(\varepsilon\) = induced EMF (electromotive force) (V)
• \(\Delta\Phi_B\) = change in magnetic flux (Wb)
• \(\Delta t\) = time interval (s)
• Negative sign: Represents Lenz's Law (direction of induced current opposes the change)
• EMF induced when: Magnetic flux through circuit changes

Faraday's Law for Multiple Loops

\[\varepsilon = -N\frac{\Delta\Phi_B}{\Delta t}\]

• \(N\) = number of loops or turns
• Note: EMF is proportional to number of loops

Lenz's Law

• Principle: The direction of induced current creates a magnetic field that opposes the change in flux that produced it
• Application: If flux is increasing, induced field opposes the increase
• Application: If flux is decreasing, induced field tries to maintain the flux
• Conservation: Result of energy conservation

Motional EMF

\[\varepsilon = BLv\]

• \(\varepsilon\) = induced EMF (V)
• \(B\) = magnetic field strength (T)
• \(L\) = length of conductor (m)
• \(v\) = velocity of conductor perpendicular to field (m/s)
• Condition: Conductor moves perpendicular to both its length and the magnetic field
• Application: Moving rod in magnetic field, generators

Induced Current

\[I = \frac{\varepsilon}{R}\]

• \(I\) = induced current (A)
• \(\varepsilon\) = induced EMF (V)
• \(R\) = resistance of circuit (Ω)
• Note: Ohm's law applies to induced EMF

Torque on Current Loop in Magnetic Field

Torque on Rectangular Loop

\[\tau = NBIA\sin\theta\]

• \(\tau\) = torque (N·m)
• \(N\) = number of loops
• \(B\) = magnetic field strength (T)
• \(I\) = current in loop (A)
• \(A\) = area of loop (m²)
• \(\theta\) = angle between magnetic field and normal to loop
• Maximum torque: When \(\theta = 90°\) (loop plane parallel to field)
• Zero torque: When \(\theta = 0°\) or \(180°\) (loop perpendicular to field)
• Application: Electric motors

Magnetic Dipole Moment

\[\mu = NIA\]

• \(\mu\) = magnetic dipole moment (A·m²)
• Torque formula using dipole moment: \(\tau = \mu B\sin\theta\)

Ampère's Law

\[\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed}\]

• Verbal description: Line integral of magnetic field around closed loop equals \(\mu_0\) times current enclosed by loop
• \(I_{enclosed}\) = net current passing through loop
• Application: Finding magnetic fields with high symmetry (straight wires, solenoids, toroids)
• MCAT note: Conceptual understanding more important than detailed calculations

Force Between Current-Carrying Wires

Force per Unit Length Between Two Parallel Wires

\[\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}\]

• \(F/L\) = force per unit length (N/m)
• \(I_1, I_2\) = currents in the two wires (A)
• \(d\) = distance between wires (m)
• Parallel currents (same direction): Attractive force
• Antiparallel currents (opposite directions): Repulsive force
• Mechanism: Each wire produces magnetic field that exerts force on the other

Mass Spectrometry

Velocity Selector

\[v = \frac{E}{B}\]

• \(v\) = velocity of particles that pass through undeflected (m/s)
• \(E\) = electric field strength (V/m or N/C)
• \(B\) = magnetic field strength (T)
• Principle: Electric and magnetic forces balance: \(qE = qvB\)
• Application: Selects particles with specific velocity regardless of charge or mass

Mass-to-Charge Ratio in Mass Spectrometer

\[\frac{m}{q} = \frac{rB}{v} = \frac{rB^2}{E}\]

• \(m/q\) = mass-to-charge ratio (kg/C)
• \(r\) = radius of circular path in detector (m)
• \(B\) = magnetic field in detector (T)
• \(v\) = velocity from velocity selector (m/s)
• \(E\) = electric field from velocity selector (V/m)
• Application: Separating isotopes, identifying molecular fragments

Magnetic Materials

Types of Magnetic Materials

• Ferromagnetic: Strongly attracted to magnets (iron, nickel, cobalt); can be permanently magnetized
• Paramagnetic: Weakly attracted to magnets (aluminum, platinum); not permanently magnetized
• Diamagnetic: Weakly repelled by magnets (water, copper, carbon); effect very small

Magnetic Permeability

\[B = \mu H\]

• \(B\) = magnetic field (T)
• \(\mu\) = magnetic permeability of material (T·m/A)
• \(H\) = magnetic field intensity (A/m)
• Relative permeability: \(\mu_r = \mu/\mu_0\)
• Ferromagnetic materials: \(\mu_r >> 1\)
• Paramagnetic materials: \(\mu_r\) slightly > 1
• Diamagnetic materials: \(\mu_r\) slightly <>

Important Relationships and Concepts

Lorentz Force (Combined Electric and Magnetic)

\[\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})\]

• Total force: Sum of electric force and magnetic force
• Electric component: \(F_E = qE\) (always in direction of field for positive charge)
• Magnetic component: \(F_B = qvB\sin\theta\) (perpendicular to both v and B)

Hall Effect

• Phenomenon: Voltage develops across conductor carrying current in magnetic field
• Mechanism: Magnetic force pushes charge carriers to one side
• Application: Determining charge carrier type (positive or negative)

\[V_H = \frac{IB}{nqt}\]

• \(V_H\) = Hall voltage (V)
• \(I\) = current (A)
• \(B\) = magnetic field (T)
• \(n\) = charge carrier density (carriers/m³)
• \(q\) = charge per carrier (C)
• \(t\) = thickness of conductor (m)

Energy Considerations

• Magnetic force does no work: Force always perpendicular to velocity
• Kinetic energy unchanged: Speed of particle in pure magnetic field remains constant
• Only direction changes: Magnetic force changes direction but not speed

Right-Hand Rules Summary

• Force on moving positive charge: Fingers = velocity, curl toward B, thumb = force
• Force on current-carrying wire: Thumb = current, fingers = B, palm = force
• Field from straight wire: Thumb = current, fingers curl = field direction
• Field from loop/solenoid: Fingers curl = current, thumb = field direction (North pole)

Key Constants

• Permeability of free space: \(\mu_0 = 4\pi × 10^{-7}\) T·m/A ≈ \(1.26 × 10^{-6}\) T·m/A
• Elementary charge: \(e = 1.6 × 10^{-19}\) C
• Electron mass: \(m_e = 9.11 × 10^{-31}\) kg
• Proton mass: \(m_p = 1.67 × 10^{-27}\) kg