D.3 Motion in electromagnetic fields

Start with the force

A charge in a uniform electric field experiences $F=qE$. A positive charge accelerates in the field direction; a negative charge accelerates opposite to it. In vacuum, if the field is uniform, the acceleration is constant: $a=qE/m$.

Read the trajectory

A particle initially moving perpendicular to a uniform electric field has constant velocity in the direction perpendicular to the field and constant acceleration along the field, so its path is parabolic. A particle initially at rest accelerates along a field line.

Solve the motion

Find the force and acceleration direction first, then use constant-acceleration equations. The time in the field comes from motion along the entry direction; the transverse displacement comes from the electric acceleration.

Common trap

Do not make an electron accelerate in the electric-field direction. The field direction is defined for a positive charge; an electron accelerates toward the positive plate.

Identify the magnetic force

A moving charge in a magnetic field experiences a force perpendicular to both its velocity and the field. A stationary charge, or a charge moving parallel to the field, has zero magnetic force.

Explain circular motion

When velocity is perpendicular to a uniform magnetic field, the magnetic force supplies centripetal force without changing the particle’s speed:
$$qvB=\frac{mv^2}{r}\quad\Rightarrow\quad r=\frac{mv}{|q|B}$$
The path is circular.

Use the full motion picture

A velocity component parallel to the field is unchanged, while the perpendicular component produces circular motion. Together they can form a helical path. Magnetic force does no work because it is perpendicular to instantaneous velocity, so kinetic energy stays constant.

Common trap

Do not say a magnetic field speeds up a charge or changes its kinetic energy. It changes direction, not speed, when no electric field is present.

Separate the two forces

With perpendicular uniform electric and magnetic fields, a charged particle can experience an electric force $F_E=qE$ parallel to the electric field and a magnetic force $F_B=qvB$ perpendicular to both velocity and magnetic field. For the correct geometry, these forces can point in opposite directions.

Find the undeflected speed

If the particle travels straight because the forces cancel, $F_E=F_B$:
$$|q|E=|q|vB\quad\Rightarrow\quad v=\frac{E}{B}$$
The charge magnitude cancels, so particles of different charge magnitude can be undeflected at the same selected speed.

Check the geometry

The cancellation condition depends on velocity being perpendicular to both fields and on the electric and magnetic force directions being opposite. If the particle is deflected, compare the vector forces rather than applying E/B blindly.

Common trap

Do not use the electric-field direction alone to predict the path, and do not insert the particle’s mass into v=E/B. This is a force-balance condition.

Use the magnitude equation

For a charge $q$ moving at speed $v$ through magnetic field strength $B$,
$$F=|q|vB\sin\theta$$
where $\theta$ is the angle between velocity and field. The force is zero for parallel motion and largest for perpendicular motion.

Find the direction

The magnetic force is perpendicular to both velocity and field. Use the right-hand rule for a positive charge; reverse the result for a negative charge. This force bends the path but does no work.

Connect to circular motion

For perpendicular motion, set $F=|q|vB$ equal to $mv^2/r$ to obtain $r=mv/(|q|B)$. Increasing $|q|$ or $B$ reduces the radius; increasing $m$ or $v$ increases it.

Common trap

Do not use the right-hand rule without reversing for an electron, and do not use $\theta$ as the angle between the field and the force. It is the angle between velocity and field.

Use the conductor equation

A straight conductor carrying current $I$ through a magnetic field experiences force
$$F=BIL\sin\theta$$
where $L$ is the length of conductor in the field and $\theta$ is the angle between conventional current and the field.

Read the angle limits

The force is zero when the wire is parallel to the field and maximum when it is perpendicular. Use conventional current direction for the force rule, not the electron drift direction.

Connect the models

The conductor formula is the combined magnetic force on moving charge carriers: $I=q/t$ and $L=vt$ lead from $F=qvB\sin\theta$ to $F=BIL\sin\theta$.

Common trap

Do not use electron motion as the current direction, and do not include wire length outside the region where the magnetic field exists.

Use the force-per-length equation

For two long parallel wires carrying currents $I_1$ and $I_2$, separated by distance $r$,
$$\frac{F}{L}=\frac{\mu_0I_1I_2}{2\pi r}$$
The force acts perpendicular to the wires along the line joining them.

Read attraction and repulsion

Parallel currents in the same direction attract; currents in opposite directions repel. Each wire experiences an equal-magnitude force in the opposite direction. The result follows because each wire lies in the magnetic field produced by the other.

Check the scaling

The force per unit length increases with either current and decreases inversely with separation. Keep $F/L$ units as $\mathrm{N\,m^{-1}}$, equivalently $\mathrm{kg\,s^{-2}}$.

Common trap

Do not reverse same-direction current behaviour, and do not confuse force on a finite length with force per unit length.