D.3 Motion in electromagnetic fields

UNIT SYLLABUS

SL/HL 6 hours

Put a moving charge in a magnetic field and something remarkable happens: the force $F = qvB\sin\theta$ acts perpendicular to the velocity, so it changes direction but never speed — the charge spirals or circles at constant kinetic energy, with radius $r = mv/qB$. This single fact powers mass spectrometers, cyclotrons and the aurora. A current-carrying wire feels the same physics as $F = BIL\sin\theta$ (the motor effect), and two parallel currents attract or repel with a force per length that historically defined the ampere. Crossed electric and magnetic fields complete the toolkit: balance them and you have a velocity selector.

Guiding Questions

? How do electric and magnetic fields change the motion of charged particles?
? Why does a magnetic force never do work on a moving charge?

A positive charge entering a uniform magnetic field (into the page) is deflected into a circle: the magnetic force always points to the centre, changing direction but not speed.

✓ What the IB expects you to master

Analyse the motion of a charged particle in a uniform electric field (parabolic, projectile-like).
Analyse the motion of a charged particle in a uniform magnetic field (circular at constant speed) using $r = mv/qB$ .
Analyse motion in perpendicular (crossed) electric and magnetic fields, including the velocity selector condition $v = E/B$ .
Use $F = qvB\sin\theta$ with a hand rule for the force direction on a moving charge.
Use $F = BIL\sin\theta$ for the force on a current-carrying conductor (the motor effect).
Apply the force per unit length between parallel wires, $\frac{F}{L} = \mu_0\frac{I_1I_2}{2\pi r}$ — like currents attract.

1 Key Formulas

Force on a moving charge

F = qvB\sin\theta

Force on a conductor

F = BIL\sin\theta

Radius in a magnetic field

r = \frac{mv}{qB}

Parallel wires (per length)

\frac{F}{L} = \mu_{0}\frac{I_{1}I_{2}}{2\pi r}

Velocity selector

v = \frac{E}{B}

2 Exam Preparation & Topic Explanations

Comparing the three field motions

The IB loves the comparison: charge in a uniform E field → parabola (constant force, projectile mathematics); charge in a uniform B field → circle (perpendicular force, constant speed); charge in crossed fields → straight line at exactly $v = E/B$ , deflection otherwise.

Be fluent in one hand rule and state which you use. For negative charges, reverse the result.

Pro Exam Strategy

Magnetic force on a stationary charge is zero — $v$ in $qvB$ matters.
Faster particles in a B field make LARGER circles ( $r \propto v$ ) but the period $T = 2\pi m/qB$ is speed-independent — the cyclotron's secret.
Mass spectrometer chain: accelerate ( $qV = \frac{1}{2}mv^2$ ) → select velocity ( $v = E/B$ ) → measure radius ( $r = mv/qB$ ).
Sketching: show the force arrow perpendicular to v, pointing to the circle's centre.

3 MCQ Practice

Q1. A magnetic force never changes the kinetic energy of a charged particle because:

The force is too small
The force is always perpendicular to the velocity, so it does no work
Magnetic fields do not exert forces on charges
Kinetic energy is conserved in all fields

Q2. An electron and a proton enter the same magnetic field with the same velocity. Compared with the proton, the electron's circular path has:

A much smaller radius, curving the opposite way
A much larger radius, curving the opposite way
The same radius, curving the same way
A much smaller radius, curving the same way

Q3. Two long parallel wires carry currents in the same direction. They:

Repel each other
Attract each other
Exert no force on each other
Rotate to become perpendicular

4 Short Answer Questions

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D.2 Electric and magnetic fields

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