Electromagnetism

01

Motion in a uniform electric field

An electric field exerts a force according to charge sign. In a uniform field, that force has a fixed direction, so resolve the motion along and perpendicular to the field.

Electric field direction is the force direction on a positive test charge. For a particle with signed charge Q, the vector relation is F = QE. Its force magnitude is |Q|E: positive charge is forced along the field and negative charge against it. Field strength E has unit N/C, equivalent to V/m.

With unchanged charge and mass and no other significant force, Newton's second law gives constant acceleration a = QE/m in a chosen direction. The electric force exists even if the particle is initially stationary. Its direction does not depend on the particle's initial velocity.

Parallel and perpendicular entry

A positive particle already moving along E speeds up. Moving against E, it slows and can reverse. A negative particle moving along E instead slows and can reverse, because its force is opposite E; moving against E, it speeds up. State both the charge sign and the velocity direction before deciding.

With initial velocity perpendicular to E, the component perpendicular to the field remains constant while the component along the field changes. Constant motion in one direction combined with constant acceleration in the other produces a parabola. Use the constant-acceleration equations separately for the two components.

Outside the ideal field, with no further force, the electron keeps its exit velocity and follows a straight tangent. Reversing the charge sign reverses the force for the same E; a quantitative path comparison also needs the particle's mass and entry velocity.

02

Magnetic sources and field patterns

Permanent magnets and electric currents produce magnetic fields. A field pattern describes direction at each point; it is separate from the force or path of a particular object placed there.

A compass's north-seeking end indicates the local magnetic field direction. Outside a bar magnet, field lines run from north to south, returning inside it to form closed paths. Lines do not cross. Closer spacing can represent greater field strength within a consistently drawn pattern, but the lines are not material tracks.

The presence of a field does not mean every object experiences a nonzero force. For example, magnetic force on a charge depends on its velocity as well as the field. Keep the source of the field distinct from the body whose force is being considered.

Read the viewpoint and current direction

A dot means a named direction out of the page towards you, like an approaching arrowhead. A cross means into the page away from you, like the tail of an arrow. Check whether the symbol labels current, field or force. For an ordinary page view, use x right, y up and z out of the page; a separate end view must say where the observer is looking from.

Long straight wire

Field lines are concentric circles in planes perpendicular to the wire. Point your right thumb along conventional current; your curled fingers give the field direction. Current out of the page produces anticlockwise field lines in that face-on view. Immediately to the wire's right, the tangent field direction is upward.

Reversing current reverses every field direction. The lines are circular, not radial outward from the wire. A compass lies tangent to a local circle, not along the wire's current.

Flat circular coil

The field passes through the coil's centre along its axis and returns outside the coil. Viewed at a face, anticlockwise current gives a central field towards the observer; clockwise current gives it away. Curl the right-hand fingers with current around the coil and use the thumb for the central axial field.

The face and axial-section drawings describe the same arrangement from different directions. Do not draw the whole coil's field as concentric circles in the coil's plane: that is not the field pattern through its centre.

Long solenoid

A solenoid is a coil with many turns along a length. Well inside a long solenoid and away from its ends, the field is approximately uniform, with parallel, similarly spaced lines. Outside it, weaker field lines return from the north end towards the south end. The end region spreads out, so interior uniformity does not extend unchanged through the ends.

Curl the right fingers with current around the turns. The thumb gives the interior field and points towards the solenoid's north end. In the shown axial section, the interior field is right and the right end is north. Reversing current swaps the effective poles.

Map a field and distinguish its strength from flux

Move a small compass to several positions, record its north-end direction and connect smooth tangents. Keep current, source geometry and nearby magnetic objects fixed. Repeat with reversed current to compare the source contribution with Earth's or other background fields. Iron filings can suggest the pattern but do not by themselves identify its direction.

Magnetic flux density B describes the local field and is measured in teslas, T. Magnetic flux Φ describes the field through an area and is measured in webers, Wb. For this supplied uniform-field model with the area perpendicular to B, use Φ = BA.

A rough field of 0.1 T through an area of order 10^-3 m² suggests flux of order 10^-4 Wb. More precisely, a 0.0800 T perpendicular field through a 2.00 cm by 3.00 cm rectangle gives:

Doubling this area at unchanged B doubles the flux, while local B stays the same. One microweber is 10^-6 Wb. Using the force definition of the tesla, and J = N m and C = A s:

Keep the quantity and unit together: B in T and Φ in Wb are different, even when one uniform field is used to calculate both.

03

Force on a current-carrying conductor

A magnetic field can exert a force on a current-carrying wire. Find its magnitude from the active length and angle, and its direction from the field and conventional current.

For a straight conductor with length l inside a uniform external field B, carrying current magnitude I:

The angle θ is between current and B. The active length l is the part in the field, not automatically the entire wire. Use the field produced by the magnet or another source; do not treat the wire's own field as though it pushes the wire sideways by itself.

Perpendicular current gives the maximum BIl. A conductor parallel or antiparallel to B gives zero magnetic force because sin θ = 0. A current-carrying conductor can therefore be in a nonzero field without experiencing a magnetic force.

Use Fleming's left-hand rule

Hold the left thumb, first finger and second finger mutually perpendicular. The first finger is field, the second is conventional current and the thumb is force on the conductor. If current is not perpendicular to B, use its perpendicular component for the direction rule; the sine factor supplies its magnitude.

For current right and B into the page, force is up. Reversing either current or field reverses the force; reversing both restores it. For current up and B right, the force is into the page. Name the selected conductor: the magnet's opposite reaction acts on a different body.

Calculate and check the scale

A rough B of 0.1 T, I of 2 A and active length of 0.1 m suggest a perpendicular force around 0.02 N. For the supplied precise model, B = 0.0800 T into the page, I = 2.50 A right and l = 0.120 m:

Now consider the separate arrangement with the same magnitudes and an actual 30.0° angle between current and B:

In its shown view, B is up and current is 30.0° to the right of up, so the force is out of the page. An in-page arrow cannot make a 30° angle to a field directed straight into the page; those directions would be perpendicular.

Define magnetic flux density

Magnetic flux density is the force per unit current per unit length on a straight conductor placed perpendicular to the field:

One tesla gives 1 N on a 1 m perpendicular active length carrying 1 A. Without the perpendicular condition, F/(Il) would give B sin θ instead of B. A field value in teslas is not a force until the current, active length and orientation are known.

04

Measure magnetic flux density

A current balance turns a magnetic force into a measurable change in a balance reading. The balance can weigh the magnet assembly while the force of interest acts on a separately supported wire.

Place a straight active wire length l perpendicular to the approximately uniform field between magnet poles. A low-voltage circuit supplies current through the wire, with an ammeter in series and a suitable current control. The magnet assembly rests on the balance; the wire and its support do not touch or rest on that assembly.

In the shown front view, current is right and B is into the page, from the front north pole towards the rear south pole. The wire feels an upward magnetic force. The magnet feels an equal downward reaction, increasing the balance reading. These forces act on different bodies.

Subtract the current-off baseline

The display is calibrated as a mass-equivalent value. A reading increase does not mean the magnet has gained physical mass. If Δm is the increase converted to kilograms, the additional downward force on the magnet is gΔm, equal in magnitude to the upward force on the wire.

A change around 0.5 g suggests a force around 0.005 N. With current around 1 A and active length around 0.05 m, this suggests B of order 0.1 T. Check that the actual balance resolution can resolve such a difference.

For the following supplied ideal model, use l = 0.0500 m and g = 9.81 N/kg. Positive current is right and positive wire force is up:

The total 150.60 g display is not the magnetic change. Using it without subtracting the baseline would include the ordinary load supported by the balance. Reversing current to -1.20 A in this model gives 149.40 g: the wire force is now down and the magnet reaction up.

Use a gradient and investigate the intercept

At fixed field, active length and perpendicular angle, F = BlI. Plot signed wire force F = gΔm vertically against signed current I. The expected gradient is Bl, so:

Convert a graph gradient in mN/A to N/A before using it. If plotting display change directly, convert its gradient to kg/A and multiply by g. The zero intercept follows from this ideal model; an actual nonzero intercept is something to investigate, not a reason to force the fit through the origin.

A complete measurement method

1. Choose a low-voltage supply, current control, ammeter and wire within their ratings. Switch off before changing the wiring or active length. Measure the straight length that lies in the approximately uniform field.
2. Support the wire independently, set it perpendicular to B, and keep its return conductors outside the active field region. Ensure no wire, lead or support touches the weighed magnet assembly.
3. Check balance and ammeter zero, range and resolution. Record a current-off baseline and wait for a settled reading. The balance range must accommodate the assembly as well as the small magnetic change.
4. Record several actual current settings and matching balance readings, with repeats. Keep active length, angle, wire position and magnet arrangement fixed. Monitor heating and baseline drift.
5. Repeat with reversed current using the stated sign convention. Convert display differences to force, plot F against I, and infer B from the gradient divided by l.

Lead forces or contact with the magnet can change the balance reading without measuring only the intended active segment. Field nonuniformity and fringing make the effective field along that segment less simple; using an overstated active length would underestimate B for the same force/current reading. Current reversal helps distinguish a current-dependent force from a fixed offset, but does not remove every systematic error.

Subtraction also affects precision. If rounding alone places each of the two display readings within 0.005 g, their difference can be within 0.010 g. Relative to the 0.60 g change, this alone is about 1.7%, before current, length, alignment and other uncertainties. These are stated rounding bounds, not universal balance specifications.

Keep the original readings, including suspected anomalies. Investigate their cause and explain any selected fitting subset. The measurement-record method shows how to preserve actual readings separately from generated examples.

05

Magnetic force on a charged particle

Magnetic force depends on a particle's velocity as well as its charge. Determine the force for positive-charge motion first, then reverse that force for a negative particle moving the same way.

For charge Q moving at speed v through magnetic flux density B, with angle θ between velocity and field, the force magnitude is:

The form F = BQv sin θ uses Q as the charge magnitude when calculating a magnitude. If Q is signed, handle the direction separately rather than reporting a negative magnitude. At perpendicular entry, F = B|Q|v. A stationary charge, a neutral particle, or motion exactly parallel or antiparallel to B has zero magnetic force.

Keep velocity and charge sign separate

Use x right, y up and z out of the page. For a positive charge, its motion has the conventional-current direction, so Fleming's left-hand rule gives the force. For an electron with the same stated velocity, reverse that force once. Do not reverse the velocity as well and then reverse the force a second time.

• Velocity right, B into the page: positive-charge force is up; electron force is down.
• Velocity up, B right: positive-charge force is into the page; electron force is out of the page, towards the observer.

Reversing only velocity, only B or only charge sign reverses the force for a nonzero-force case. Reversing both velocity and B restores the original force for the same charge. The force is perpendicular to both velocity and B; it does not point along either.

Calculate a magnitude

An electron with |Q| = 1.60 × 10^-19 C moves right at 2.00 × 10⁷ m/s through a 1.00 mT field into the page. Convert 1.00 mT to 1.00 × 10^-3 T:

At a different angle, retain sin θ. The angle is between the actual velocity vector and B, not between force and either of them.

Why the magnetic force does no work

The magnetic force is perpendicular to the particle's instantaneous velocity and therefore to its small displacement at that instant. It does no work on the particle. With magnetic force alone, speed and kinetic energy remain constant while velocity direction can change.

A uniform B does not mean a fixed force direction: as the velocity turns, the perpendicular force turns too. For perpendicular entry this leads to the circular path. Any velocity component parallel to B remains unaffected by the magnetic force, so not every entry direction gives a circle in the page.

06

Circular motion in a magnetic field

With perpendicular entry into a uniform magnetic field, a charged particle keeps its speed while its velocity turns. The magnetic force supplies the inward resultant required for circular motion.

For constant speed v on a circle of radius r, inward acceleration has magnitude v²/r. With magnetic force as the only significant force, its perpendicular-entry magnitude B|Q|v therefore gives:

The magnetic force is the actual inward force; do not add a second force called centripetal force. At each point, velocity is tangent to the circle and magnetic force points towards its centre. The force does no work, so this model has constant speed, not constant horizontal velocity.

Compare the electric and magnetic paths

In the uniform electric-field example, force has a fixed upward direction, horizontal velocity stays constant and the growing vertical component increases total speed. The path is a parabola within that field.

In the magnetic-only example, force continually turns with velocity and stays perpendicular to it. The path is circular at constant speed, while both horizontal and vertical velocity components change. Use the field type and initial direction to choose the motion model; a charged particle does not always follow one standard curve.

07

Select a beam speed

Electric and magnetic forces can oppose one another for particles travelling in a chosen direction. Only the matching incident speed makes their magnitudes equal and leaves the charged beam undeviated.

Let the incident velocity be right, E downward and B into the page. These directions are mutually perpendicular. A positive charge feels electric force down and magnetic force up. An electron has both force directions reversed: electric force up and magnetic force down.

Establish that the forces oppose before equating their magnitudes. For a nonzero charge travelling in the stated direction:

The charge magnitude cancels, and mass does not enter this ideal force-balance condition. Neutral particles do not satisfy this nonzero-charge selection argument. Reversing the incident direction without changing either field makes the two forces reinforce rather than cancel.

Calculate the selected speed

A field E of a few thousand N/C divided by B of a few 10^-4 T suggests a selected speed of order 10⁷ m/s. Use the supplied E = 4000 N/C and B = 0.200 mT = 2.00 × 10^-4 T:

This B is different from the 1.00 mT field in the magnetic semicircle example. For an electron with |Q| = 1.60 × 10^-19 C at the selected speed, both opposing forces have magnitude 6.40 × 10^-16 N.

The selected beam remains straight when gravity, collisions, fringing and interactions between beam particles are negligible. A selector does not automatically accelerate every incident particle to E/B; it identifies the speed for an undeviated path in this geometry.

Decide the initial deflection of other speeds

For the same electron charge and fields, electric force stays at 6.40 × 10^-16 N upward. Magnetic force depends on incident speed:

• At 1.50 × 10⁷ m/s, magnetic force is 4.80 × 10^-16 N down. The initial resultant is 1.60 × 10^-16 N up.
• At 2.50 × 10⁷ m/s, magnetic force is 8.00 × 10^-16 N down. The initial resultant is 1.60 × 10^-16 N down.

Positive charges entering with these same velocities have the opposite initial force directions. These statements describe the initial resultants. Once a particle deflects, its velocity changes and so does its magnetic force; a complete off-speed trajectory is not generally a constant-acceleration parabola.

Revision

Electromagnetism: revision summary

Name the source field, the selected body and the direction of conventional current or particle velocity. Calculate a force magnitude and a direction, then choose the motion or measurement model.

Uniform electric motion

A positive charge is forced along E and a negative charge against E, including when initially stationary. With perpendicular entry and no other significant force, the field-perpendicular velocity component is constant while the field-parallel component changes. The path is parabolic and total speed can change. Outside the ideal field, continue along the exit tangent.

Field patterns and directions

• Long straight wire: concentric field circles; right thumb follows current and curled fingers give B. Out-of-page current gives anticlockwise field in that view.
• Flat circular coil: central axial field with external return paths. Anticlockwise current seen at a face gives central field towards that observer.
• Long solenoid: approximately uniform interior away from ends, with weaker returning field outside. The grip-rule thumb gives the interior direction and points towards its north end.

Field lines are not particle tracks. A dot means the labelled quantity points out of the page and a cross means into it. Distinguish the views before comparing a face diagram with an axial section.

Conductor force and the current balance

The angle is between conventional current and the external B. Fleming's left-hand rule uses first finger for field, second for conventional current and thumb for conductor force; use the current's perpendicular component for a non-right angle. Parallel or antiparallel current gives zero force. Reversing one of I or B reverses the force; reversing both restores it.

In the shown current balance, an upward wire force produces a downward reaction on the separately weighed magnet. Convert its baseline-subtracted display change from grams to kilograms and use F = gΔm for the stated upward wire-force reference. At fixed perpendicular length, the F-against-I gradient is Bl, so divide by l to obtain B.

Keep supports separate, return wires outside the active field and geometry fixed. Check current-off drift, ranges, resolution, heating, fringing and contact forces. A changed balance display is not a changed physical mass; actual nonzero intercepts and anomalies need investigation.

Magnetic motion and velocity selection

For positive-charge motion, use the conventional-current force direction; reverse it once for a negative charge with the same velocity. A stationary charge or velocity parallel to B gives zero magnetic force. Magnetic force does no work: it changes velocity direction while speed remains constant. It is the inward force, not an extra force in addition to a centripetal force.

For the electron semicircle in x > 0, the exit lies 2r below entry and its velocity is left. Radius and entry-to-exit separation are different. Outside the ideal field, the path is a straight tangent.

This requires the intended incident direction, mutually perpendicular velocity and fields, and nonzero charge. The selected speed is independent of mass and charge magnitude. Off-speed force comparisons give the initial deflection, not automatically a full path or a process that brings every particle to the selected speed.

Quantity and unit reference

The supplied electron constants are e = 1.60 × 10^-19 C and m_e = 9.11 × 10^-31 kg. Convert 1 mT = 10^-3 T, 1 µT = 10^-6 T, 1 µWb = 10^-6 Wb, 1 ns = 10^-9 s and 1 g = 10^-3 kg before using SI equations.

For the supplied uniform perpendicular-area example, Φ = BA: local B and the flux through an area are distinct quantities. The units are 1 T = 1 kg s^-2 A^-1 and 1 Wb = 1 kg m² s^-2 A^-1.