Magnetic forces, waves and thin lenses

Distinguish magnetic effects from electric effects

An electric field can exert a force on a stationary charged particle. A magnetic field exerts the magnetic part of the force on a moving charge, with magnitude $F=|q|vB\sin\theta$. Here q is the charge, v its speed, B the magnetic-field magnitude and theta the angle between velocity and field. Parallel or antiparallel motion has zero magnetic force because the sine is zero. Perpendicular motion gives the largest magnitude for fixed charge, speed and field. A stationary charge has zero magnetic force in this model.

The magnetic force is perpendicular to the velocity and magnetic field. For a positive charge, the right-hand rule determines the direction; a negative charge reverses that direction. In a uniform magnetic field acting alone, the force changes the direction of motion but does no work on the particle, because it is perpendicular to the displacement. The speed and kinetic energy remain constant. A curved path therefore does not automatically imply that the particle is gaining kinetic energy.

Currents create magnetic fields, and a current-carrying wire placed in an external magnetic field can experience a force. For a straight wire segment, the magnitude is $F=BIL\sin\theta$, with theta between the current direction and the field. The same parallel-versus-perpendicular test applies. A compass aligns with a local magnetic-field direction, providing a way to represent field patterns. Field lines are a drawing convention for direction and relative strength, rather than physical threads attached to the magnet.

Follow a disturbance rather than transporting the whole medium

A wave carries a disturbance and transfers energy. In a mechanical wave, particles of the medium move locally while the disturbance propagates. For a transverse wave, the oscillation is perpendicular to propagation; for a longitudinal wave, it is parallel. Sound in air is longitudinal, involving compressions and rarefactions. Electromagnetic waves can travel through vacuum and do not require a material medium in the way sound does. Avoid using the word wave to imply that all wave types share the same medium or oscillation direction.

Frequency f counts cycles per second, measured in hertz. Period T is the time per cycle, so $T=1/f$. Wavelength is the distance between successive equivalent points, such as adjacent crests. For a periodic wave, speed obeys $v=f\lambda$. Use a consistent unit for wavelength before multiplying. If the speed is fixed, increasing frequency shortens wavelength. Amplitude instead measures the size of the oscillation; it is distinct from both wavelength and the speed with which the pattern travels.

When a wave passes into a different medium at a stationary boundary, its frequency remains set by the source. Its speed can change, and the wavelength changes accordingly. This is compatible with refraction, where propagation direction may also change. A numerical frequency change inferred solely from a slower speed is therefore incorrect for this standard boundary situation. Identify which quantity the source fixes and which quantity the medium determines before comparing the two sides.

Use a stated convention for a thin lens

For an ideal thin lens in air, use $1/f=1/d_o+1/d_i$. In the convention here, a real object has positive object distance d_o, a real image has positive image distance d_i on the opposite side, and a virtual image has negative d_i on the object's side. A converging lens has positive focal length f, while a diverging lens has negative f. Other sign conventions exist, so keep this one consistent throughout a calculation rather than mixing rules.

Rearrange the reciprocal equation carefully, then take the reciprocal only at the end. Lateral magnification is $m=-d_i/d_o$ in this convention. A negative value means the image is inverted; the absolute value gives its size relative to the object. A converging lens with a real object beyond the focal point produces a real inverted image. With the object inside the focal distance, it produces a virtual upright image. A real image can be projected on a screen at its position, whereas a virtual image cannot be caught there by placing a screen behind the lens.