The basic idea: constant force, constant acceleration

An electric field is a region of space where an electric charge experiences an electric force. A uniform electric field has the same magnitude and direction at every point in the region being considered. Between large, oppositely charged parallel plates, and away from the edges, the field is a very good approximation to uniform.

For a charge in an electric field,

A positive charge is forced in the direction of the field; a negative charge is forced opposite to the field. Newton’s second law gives

so a charged particle has constant acceleration if the electric field is uniform.

For parallel plates,

Put this into Newton’s second law and the acceleration due to the electric field is $a = \frac{qE}{m} = \frac{qV}{md}$.

Motion parallel and perpendicular to the field

If a charged particle starts from rest, or moves along the field direction, the motion is one-dimensional. It speeds up or slows down depending on the sign of the charge and the field direction. The kinematics are the same as in mechanics; the force is electrical rather than gravitational.

When the particle enters the field with velocity perpendicular to the field, the motion separates into two independent components. The component of velocity parallel to the plates stays unchanged because there is no force in that direction. The component of velocity in the field direction changes uniformly, since the electric force is constant. The path is parabolic, just like a projectile in a uniform gravitational field.

If the field region has length $X$, where $X$ is the distance travelled through the plates parallel to the initial velocity ($m$), and the horizontal component of velocity is $v_x$, where $v_x$ is horizontal velocity ($m\,s^{-1}$), then the time in the field is

If the initial vertical velocity is zero, the vertical deflection is

For a fixed field and fixed entry speed, the deflection increases with the square of the plate length. That “square” dependence is a useful fingerprint of constant acceleration.

Energy and field comparisons

An electric field can do work on a moving charge because the electric force can have a component along the displacement. The particle’s kinetic energy may therefore change. This contrasts with magnetic fields, where the force is perpendicular to the velocity and does no work on a point charge.

A conservative field is a field in which the work done moving an object between two points depends only on the start and end points, not on the path. Electrostatic and gravitational fields are conservative. That is why conservation of energy works so well here: a charged particle accelerated through a potential difference gains kinetic energy equal to the loss in electric potential energy, provided no non-conservative forces are involved.

The gravitational force on an electron or proton in typical laboratory electric fields is usually absurdly small compared with the electric force. This shows the strength of electromagnetic interactions compared with gravity at the particle scale. In broad order of strength for fundamental interactions: strong nuclear, electromagnetic, weak nuclear, gravitational. Gravity dominates planets and stars not because it is strong between particles, but because it is always attractive and acts over astronomical amounts of mass.

The same energy idea helps estimate sizes in nuclear and atomic physics. In a head-on approach of a positively charged alpha particle towards a positively charged nucleus, the alpha particle slows as kinetic energy is stored as electric potential energy. The closest approach gives an upper estimate for the nuclear radius. It’s a neat example of forces, fields and energy doing the measuring for us.

Magnetic fields change direction, not speed

A magnetic field is a region of space where a moving charge, or a conductor carrying current, experiences a magnetic force. A uniform magnetic field has the same magnitude and direction everywhere in the region being considered. In diagrams, dots usually show a field coming out of the page, like arrow tips; crosses show a field going into the page, like arrow tails.

When a charged particle moves at right angles to a uniform magnetic field, the magnetic force always acts perpendicular to its velocity. A force at 90° changes the direction of motion, not the speed. If the particle stays inside the field, it therefore follows a circular path. The magnetic force provides the centripetal force, the resultant force directed towards the centre of a circular path.

The kinetic energy stays constant in a magnetic field because a force perpendicular to the displacement does no work. This isn’t a small-effect approximation; it comes straight from the geometry of the force. The particle is still accelerating, since its velocity is changing direction, but its speed and kinetic energy stay the same.

For motion perpendicular to the field, set magnetic force equal to centripetal force:

Hence $r = \frac{mv}{qB}$. This is one of the most useful equations in the topic: greater mass or speed gives a larger circle, while greater charge or magnetic field strength gives a tighter one.

If the particle enters at an angle rather than exactly perpendicular to the magnetic field, split its velocity into components. The perpendicular component produces the circular motion. The parallel component is unaffected. Put those together and the path is a helix.

Determining charge-to-mass ratio

The specific charge of a particle is its electric charge divided by its mass. It is written $q/m$ and has unit $C\,kg^{-1}$. In a known magnetic field, the radius of curvature can be used to identify or compare particles because $r$ depends on $m/q$.

A standard method accelerates charged particles from rest through a potential difference, then sends them into a uniform magnetic field at right angles. The electrical energy gained becomes kinetic energy: $\tfrac{1}{2}mv^2 = qV$. Combining this with $r = \frac{mv}{qB}$ gives

.

So the experiment requires measurements of accelerating potential difference, magnetic field strength and path radius.

In a fine-beam tube, electrons pass through low-pressure gas. Their collisions excite gas atoms, and the glowing track shows the circular path. Historically, measurements of this kind gave the charge-to-mass ratio of the electron before the elementary charge was known separately. More generally, curved tracks in magnetic fields are one way moving charges are used to probe the nature of matter: the direction of curvature gives the sign of charge, while the radius gives information about momentum and specific charge.

Charge itself is quantized: isolated particles are found with charges in integer multiples of the elementary charge. Other quantities met later in physics are quantized too, including atomic energy levels and angular momentum in bound quantum systems. The experimental clue is often a set of discrete outcomes rather than a continuous spread.

Crossed fields and no deflection

When uniform electric and magnetic fields are at right angles to each other, a charged particle may pass straight through with no deflection. For that to happen, the electric force and magnetic force must have equal magnitudes and act in opposite directions.

For a particle moving perpendicular to both fields, the electric force has magnitude

The magnetic force has magnitude

With no deflection, $F^{E}=F^{B}$, so $qE=qvB$, giving $v=E/B$.

The charge cancels out. So, for a given pair of field strengths, only particles with one particular speed travel in a straight line. Slower particles do not have enough magnetic force; faster particles have too much. The direction of deflection depends on the signs and field directions, but the balance condition remains $v=E/B$.

A velocity selector is an arrangement of perpendicular electric and magnetic fields that transmits charged particles of one selected speed without deflection. It’s a clear example of using two fields against each other: the electric field can change kinetic energy in general, but in the selector the two forces cancel, so the selected particle’s velocity is unchanged while it is inside.

Mass spectrometry as an application

A mass spectrometer is an instrument that separates charged particles according to mass-to-charge ratio by using electric and magnetic fields. In a Bainbridge-type arrangement, ions pass through a velocity selector first. They then enter a region with magnetic field only, where they move in circular paths. For ions with the same charge and selected speed, heavier ions curve less and hit the detector at a different position.

This gives a practical answer to the question, “what can be deduced from the path?” Using the radius in the magnetic field, along with the known selected speed and field strength, the mass-to-charge ratio can be found. That is why magnetic deflection is useful in isotope analysis, planetary probes and modern analytical instruments.

Magnitude of the magnetic force

The magnetic force on a moving charge depends on the charge, the speed of the charge, the magnetic field strength, and the angle between the velocity and the field. For the magnitude, use

Here, the formula uses the magnitude of the charge; deal with the sign of the charge when you work out the direction.

The sine term is easy to overlook. If the particle moves parallel to the magnetic field, $\theta = 0^\circ$ and the magnetic force is zero. If it moves perpendicular to the magnetic field, $\theta = 90^\circ$ and the force is maximum. Don’t accidentally use the angle to the normal or the angle to the page; the angle needed is the one between velocity and field.

Direction of the magnetic force

The magnetic force acts perpendicular to both the velocity and the magnetic field. For a positive charge, use the usual right-hand vector rule for $v$ crossed with $B$. For a negative charge, reverse the direction. In school diagrams, it’s often quicker to think using conventional current: a moving positive charge behaves like conventional current in the direction of motion, while a moving electron corresponds to conventional current in the opposite direction.

That perpendicular force is why magnetic fields are so useful when studying particles. A magnetic field doesn’t speed a particle up; it bends its track. The radius of the bend gives a measurement of the particle’s momentum per unit charge. In cloud chambers and bubble chambers, the field representation is indirect: the field is invisible, but the curved particle trail makes the field’s effect measurable. This is a useful nature-of-science point — field lines, equations and tracks are models or representations, not the field itself.

The motor effect

A conventional current uses the model direction in which positive charge flows around a circuit. In a current-carrying wire, charges are moving, so a wire placed in a magnetic field feels a force if the current has a component perpendicular to the field. That force is the motor effect: electrical energy is transferred into kinetic energy by a magnetic force.

For a straight conductor in a uniform magnetic field,

The force is greatest when the wire is at right angles to the field. It is zero when the current is parallel to the field.

Use Fleming’s left-hand rule to get the direction of the force: first finger for field, second finger for conventional current, thumb for force. I always tell students to make the three directions genuinely at right angles — a half-hearted hand shape gives half-hearted answers.

Field-line diagrams help here, as long as you treat them as diagrams. A field line is a diagrammatic curve whose tangent gives the direction of a field at a point. Field lines are not elastic strings. Even so, the “catapult field” picture works well as a model: the field from the current combines with the external magnetic field, making the field stronger on one side of the wire and weaker on the other, so the wire is pushed sideways.

In a laboratory investigation, a straight lead is placed between magnets resting on a sensitive balance. If the current is reversed, the force reverses too, so readings in both current directions help reduce zero errors and improve precision. A graph of force against current should be linear if $B$, $L$ and $\theta$ are fixed. Another test changes the length of wire in the field; the force should be proportional to that length, provided the field strength is reasonably uniform.

Magnetic field around a straight wire

A long, straight wire carrying current produces magnetic field lines that form circles around the wire. Use the right-hand grip rule to get the direction: point your right thumb in the direction of conventional current, and your curled fingers show the direction of the magnetic field.

For a long straight wire,

The constant $\mu_0$ has value $4\pi \times 10^{-7}\ \mathrm{N\,A^{-2}}$. The relationship is $1/r$, not inverse-square, because the field from a long wire spreads around cylindrical geometry rather than out from a point source into a sphere.

Force between two parallel currents

Each wire lies in the magnetic field made by the other wire. Combine the magnetic field of one long wire with the force on the other current-carrying wire, and you get

The two wires exert equal and opposite forces on each other, as Newton’s third law requires. Currents in the same direction attract. Currents in opposite directions repel. This is still magnetism, not a separate new force: it is the magnetic force on moving charges in one wire due to moving charges in the other.

A simple demonstration uses two long, flexible foil strips carrying large currents. When the currents run in the same direction, the strips pull together; when they run in opposite directions, the strips push apart. In more quantitative work, graphs can test the proportionality. Increasing either current increases the force per unit length, while increasing separation reduces it. Plotting force-related quantities against reciprocal separation is a useful way to check the $1/r$ dependence.

The old SI definition of the ampere used the force between parallel wires. The modern definition is tied to the fixed value of the elementary charge, but the wire-force experiment is still a neat reminder that electric current and magnetism are inseparable. Electric, magnetic and gravitational fields are all represented using field strength, field lines and force laws, but the source geometry controls the distance dependence: point sources give inverse-square laws, long straight sources give $1/r$ laws, and ideal infinite sheets give uniform fields.