Magnetism and Motion of Charged Particles (4C) - MCAT Chemical and Physical Foundations of Biological Systems

$\omega = \frac{|q|B}{m}$. Cyclotron frequency emerges from equating magnetic force to centripetal force, independent of velocity for nonrelativistic cases.

Circles around wire; thumb $I$, fingers give $\vec{B}$. The right-hand rule for a straight wire follows from Ampere's law, with magnetic field lines encircling the current.

Induced current opposes the change in $\Phi_B$. Lenz's law, a consequence of energy conservation, dictates that induced current creates a field opposing the flux change.

$\mathcal{E} = -\frac{d\Phi_B}{dt}$. Faraday's law states that induced emf equals the negative rate of change of magnetic flux through the loop.

$B = \mu_0 n I$. Inside a solenoid, the uniform field results from the additive contributions of tightly wound current loops, proportional to turn density and current.

$B = \frac{\mu_0 I}{2\pi r}$. The field strength decreases inversely with distance, derived from Ampere's circuital law for an infinite straight wire.

$\Phi_B = BA\cos\theta$. Magnetic flux quantifies the component of the magnetic field passing through the surface, given by the dot product of field and area vectors.

$\vec{F}\perp\vec{v}$ (magnetic force does no work). The magnetic force is always perpendicular to velocity due to the cross product, ensuring zero work done as $\vec{F}\cdot\vec{v}=0$.

$\vec{F} = I,\vec{L}\times\vec{B}$. The vector force on a wire segment results from the cross product of the length vector in the direction of current and the magnetic field.

$F = ILB\sin\theta$. The force on a current-carrying wire integrates the Lorentz force over charge carriers, depending on current, length, field, and angle.

$1,\text{T} = \frac{\text{N}}{\text{A}\cdot\text{m}}$. The tesla is defined from the force on a current-carrying wire, expressed in base SI units as newtons per ampere-meter.

It is opposite to $\vec{v}\times\vec{B}$. A negative charge reverses the force direction due to the sign of $q$ in the Lorentz force equation.

Along $\vec{v}\times\vec{B}$ (right-hand rule). The direction follows from the cross product in the Lorentz force, determined by the right-hand rule for positive charges.

$\theta = 0^\circ$ or $180^\circ$ (,$\vec{v}\parallel\vec{B}$,). Magnetic force vanishes when velocity is parallel or antiparallel to the magnetic field because $\sin\theta=0$ at these angles.

$\vec{F} = q,\vec{v}\times\vec{B}$. The vector form of the Lorentz magnetic force arises from the cross product of velocity and magnetic field vectors, scaled by the charge.

$F = |q|vB\sin\theta$. The magnitude of the magnetic force on a moving charge is derived from the Lorentz force law, incorporating the charge, velocity, magnetic field strength, and the sine of the angle between velocity and field vectors.

$T = \frac{2\pi m}{|q|B}$. The period is the time for one circular orbit, derived from angular frequency as $T=2\pi/\omega$.

$\text{pitch} = v_{\parallel}T = \frac{2\pi m v_{\parallel}}{|q|B}$. The pitch represents axial advance per cyclotron period, combining parallel velocity with the orbital period in the magnetic field.

$r = \frac{mv}{|q|B}$. The radius balances centripetal force with magnetic force for perpendicular motion, yielding $r = \frac{mv}{|q|B}$.

Upward (force toward top of page). For positive charge with velocity rightward and field into the page, the Lorentz force directs upward via the right-hand rule, causing initial upward deflection.

Deflects to the left. An electron's negative charge reverses the force direction, leading to leftward deflection when velocity is upward and field is out of the page.

$v = \frac{E}{B}$. In crossed fields, undeflected motion occurs when electric and magnetic forces balance, so $qE = qvB$.