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, away from the edges, the field is a very good approximation to uniform.

For a charge in an electric field, F = qE, where F is the electric force (N), q is the electric charge (C) and E is the electric field strength (N C⁻¹ or V m⁻¹). A positive charge feels a force in the direction of the field; a negative charge feels a force opposite to the field. Newton’s second law gives a = F/m, where a is acceleration (m s⁻²) and m is mass (kg). So, if the electric field is uniform, the charged particle has constant acceleration.

For parallel plates, E = V/d, where V is the potential difference between the plates (V) and d is the plate separation (m). Combining this with Newton’s second law gives the acceleration due to the electric field: a = qE/m = 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, except the force is electrical rather than gravitational.

When the particle enters the field with velocity perpendicular to the field, the motion splits 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 because 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 vₓ, where vₓ is horizontal velocity (m s⁻¹), then the time in the field is t = X/vₓ, where t is time (s). If the initial vertical velocity is zero, the vertical deflection is sᵧ = ½a t², where sᵧ is vertical displacement (m). So, 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. That 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 is so useful: 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 current-carrying conductor experiences a magnetic force. A uniform magnetic field has constant magnitude and direction throughout 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 perpendicular to a uniform magnetic field, the magnetic force is always at right angles to the particle’s velocity. A force like that changes the direction of motion, not the speed. If the particle stays inside the field, it moves in a circle. The magnetic force acts as the centripetal force, meaning 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 is not a small-effect approximation; it comes straight from the geometry of the force. The particle is still accelerating, since the direction of its velocity changes, but its speed and kinetic energy remain constant.

For perpendicular motion, set the magnetic force equal to the centripetal force: qvB = mv²/r, where v is speed (m s⁻¹), B is magnetic field strength (T) and r is the radius of the circular path (m). Hence r = mv/qB. This is one of the most useful equations in the topic: a larger mass or speed gives a larger circle; a larger charge or magnetic field gives a tighter circle.

If the particle enters at an angle rather than exactly perpendicular to the magnetic field, split its velocity into components. The perpendicular component produces circular motion. The parallel component is unaffected. Circular motion combined with steady motion along the field gives a helix.

Determining charge-to-mass ratio

The specific charge of a particle is the ratio of its electric charge to its mass. It is written q/m and has unit C kg⁻¹. In a known magnetic field, the radius of curvature lets us identify or compare particles because r depends on m/q.

A common method accelerates charged particles from rest through a potential difference before they enter a uniform magnetic field at right angles. The electrical energy gained becomes kinetic energy: ½mv² = qV. Combining this with r = mv/qB gives q/m = 2V/B²r². The symbols in this formula have the same meanings as above. So the experiment needs measurements of accelerating potential difference, magnetic field strength and path radius.

In a fine-beam tube, electrons pass through low-pressure gas. Collisions excite gas atoms, so the glowing track makes the circular path visible. Historically, measurements of this type 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 help probe the nature of matter: the sign of curvature tells you the sign of charge, and the radius tells you 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 also quantized, including atomic energy levels and angular momentum in bound quantum systems. The experimental clue is often a pattern of discrete outcomes rather than a continuous spread.

Crossed fields and no deflection

Put a uniform electric field and a uniform magnetic field at right angles, and a charged particle may pass through in a straight line. For that to happen, the electric force and the magnetic force have to be equal in magnitude and act in opposite directions.

For a particle moving perpendicular to both fields, the electric force has magnitude Fᴱ = qE, where Fᴱ is the electric force due to the electric field (N). The magnetic force has magnitude Fᴮ = qvB, where Fᴮ is the magnetic force due to the magnetic field (N). With no deflection, Fᴱ = Fᴮ, so qE = qvB, giving v = E/B.

The charge cancels out. So, for a chosen pair of field strengths, only particles with one particular speed go straight through. Slower particles have too little magnetic force; faster particles have too much. The direction of deflection depends on the signs and the field directions, but the balance condition stays the same: 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 tidy example of using two fields against each other. An 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 first pass through a velocity selector. After that, they enter a region with magnetic field only, where they follow 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 one of the key practical answers to “what can be deduced from the path?” Using the radius in the magnetic field, together 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, its speed, the magnetic field strength, and the angle between the velocity and the field. Its magnitude is F = qvB sin θ, where θ is the angle between the particle’s velocity and the magnetic field (° or rad). Use the magnitude of the charge in the formula; deal with the sign of the charge when you work out the direction.

Don’t ignore the sine term. If the particle moves parallel to the magnetic field, θ = 0° and the magnetic force is zero. If it moves perpendicular to the magnetic field, θ = 90° and the force is maximum. Be careful not to use the angle to the normal or the angle to the page by accident; the angle is 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 in terms of conventional current: a moving positive charge is like conventional current in the direction of motion; a moving electron corresponds to conventional current in the opposite direction.

That perpendicular force is why magnetic fields are so useful for studying particles. A magnetic field does not speed a particle up, but it bends its track. The radius of the bend measures 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 is the model direction in which positive charge flows in a circuit. Because a current-carrying wire has moving charges, it feels a force in a magnetic field when the current has a component perpendicular to that field. That effect is the motor effect: electrical energy transferred into kinetic energy by a magnetic force.

For a straight conductor in a uniform magnetic field, F = BIL sin θ, where I is current (A), L is the length of conductor within the magnetic field (m), and θ is the angle between the current direction and the magnetic field (° or rad). 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 find the force direction: first finger for field, second finger for conventional current, thumb for force. I always tell students to hold 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 remember what they represent. A field line is a diagrammatic curve whose tangent gives the direction of a field at a point. They are not elastic strings. Even so, the “catapult field” picture is a useful model: the field from the current combines with the external magnetic field, giving a stronger field on one side of the wire and a weaker field 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. Reversing the current also reverses the force, so readings in both current directions help reduce zero errors and improve precision. If B, L and θ are fixed, a graph of force against current should be linear. A second test varies the length of wire in the field; the force should be proportional to that length, provided the field strength is reasonably uniform.