Worked example 1: A motor coil has N = 20 turns, current 2.0 A, area 1.5 x 10^-3 m^2, and sits in a 0.80 T magnetic field at the angle of maximum torque. Find the torque. What changes if the current direction reverses?

tau_max = NIAB = (20)(2.0)(1.5 x 10^-3)(0.80) = 4.8 x 10^-2 N m
The turns multiply the torque because each turn contributes its own force couple in the same direction.
If the current reverses, the torque direction reverses.
Reversing the current flips the magnetic moment of the loop, so the field twists it the other way.
Check: zero current or zero field would kill the torque entirely.
The device depends on the interaction between current and magnetic field, not on either piece alone.

Worked example 2: A cell membrane has a potential difference of 70 mV across a thickness of 5.0 nm. Estimate the average electric field across the membrane.

E = Delta V/d = 0.070/(5.0 x 10^-9) = 1.4 x 10^7 V/m
A small voltage across a tiny biological distance can correspond to a very large field magnitude.
Interpretation: the biology language changes, but the field bookkeeping does not.
Membrane potentials are still voltage differences and electric fields, not a separate kind of physics.
Check: if the membrane were thicker at the same voltage, the average field would be smaller.
Field strength is about potential change per distance, not about voltage alone.

Worked example 3: A spectral absorption line appears at 500 nm. Find the photon energy and explain in one sentence why the discrete line needs more than classical wave transport alone.

E = hc/lambda = (6.63 x 10^-34)(3.0 x 10^8)/(5.0 x 10^-7) = 3.98 x 10^-19 J
The wavelength of the electromagnetic wave determines the photon energy scale once the matter interaction is quantized.
E = 2.48 eV
Converting to electron-volts makes the atomic-scale energy easier to compare with electronic transitions.
Interpretation: classical EM explains the incoming wave, but the fact that only specific lines are absorbed requires discrete energy levels in matter.
That is the classical-to-quantum handoff at the edge of this topic.

Common mistake: treating membrane potentials, motor torque, and spectral lines as unrelated topics because the context changes. The field bookkeeping is shared; what changes is the material system and, at atomic scales, whether quantum structure is needed.

Electromagnetism