Magnetism and Moving Charges
Keep a moving charge’s source field separate from its force in an external field.
AP Physics 2 Topic 12.2 · College Board CED printed p. 83, 84 / PDF p. 90, 91 · See this topic in the course outline
A moving charge plays two roles: it creates a field, and it can experience a force from another field. Keep those roles separate, then use directions before putting numbers into a formula.
Before you start
- Right-hand vector directions
- Centripetal force mv²/r
- Electric force qE and potential difference
A moving charge creates a magnetic field
A single moving charged object produces a magnetic field as well as an electric field. At a specified observation point, its magnetic field depends on its velocity and on the distance and direction from the charge to that point. Let point from the charge to the observation point. For a positive charge, use the right-hand sense of : the field is perpendicular to both vectors. Reverse the direction for a negative charge.
With charge and distance fixed, greater speed produces a stronger magnetic field. With speed and angle fixed, increasing distance weakens it. At a fixed speed and distance it is greatest where the velocity and position vectors are perpendicular, and zero along the velocity line. A useful map for a positive charge moving right has field out of the page at points above its path and into the page below. This is a qualitative source-field model; no calculus sum over charge distributions is required.
For perpendicular vectors, use a three-finger right-hand procedure: point the index finger along the first vector, point the middle finger along the second vector, and the thumb gives their cross-product direction. For a source field, the inputs are index along v and middle along r from the charge to the observation point. For a force, the inputs are index along v and middle along the external B. Reverse the thumb direction for a negative charge. If the two inputs are parallel or antiparallel, the relevant cross product is zero.
| Question | First input (index) | Second input (middle) | Positive charge (thumb) | Negative charge |
|---|---|---|---|---|
| Source field at a point above a right-moving charge | v right → | r up ↑ | B out of page ⊙ | B into page ⊗ |
| Force in an external into-page field | v right → | B into page ⊗ | F up ↑ | F down ↓ |
Magnetic forces describe interactions between moving charges: one moving charge produces a magnetic field, and that field can exert a magnetic force on another moving charge. A stationary second charge has no magnetic force from that field, although an electric force can still act. Keep the field-producing object distinct from the object whose force you predict.
Magnetic force changes direction, not kinetic energy
Separate the source from the object that experiences the force. A magnetic field can exert a force on a moving charge; a stationary charge has no magnetic force. For the magnitude, , where θ is the angle between velocity and field. The right-hand rule gives for positive charge; reverse it for negative charge. Force is perpendicular to both velocity and field.
| θ between v and B | Force magnitude | Motion implication if B acts alone |
|---|---|---|
| 0° | 0 | Motion parallel to B continues |
| 90° | |q|vB | Velocity changes direction |
| 180° | 0 | Motion antiparallel to B continues |
AP quantitative force tasks here use 0°, 90°, or 180°. At other angles, reason qualitatively: only the component of velocity perpendicular to B contributes; the force is less than the perpendicular maximum. A magnetic force does no work on a point charge because it is perpendicular to its displacement at every instant. With only a uniform B and perpendicular velocity, it supplies centripetal force: . The radius is positive for either charge sign. This algebraic model uses constant mass at nonrelativistic speeds, well below c; it must never predict v ≥ c.
To produce the positive-charge direction when v points right and B points into the page, point the right index finger right and the middle finger away from you. The thumb points up. For an electron with the same v and B, reverse that direction: force points down. This gesture predicts force, while the v-and-r gesture in the preceding passage predicts a source field.
Add electric and magnetic forces; explain the Hall voltage
When electric and magnetic fields coexist, calculate their effects independently and add forces as vectors: . The electric force can change speed; the magnetic force by itself cannot. Oppositely directed forces can cancel for a particular velocity, giving a straight path. Equal magnitudes require in the perpendicular case; choosing directions is as necessary as matching magnitudes.
In the Hall effect, moving carriers in a conductor are pushed sideways by an external magnetic-field component perpendicular to their motion. Charge accumulates on opposite sides, producing a transverse electric field and potential difference. At equilibrium the transverse electric and magnetic forces balance. Reversing B or the carrier motion reverses the side receiving a given charge; if B is parallel to the motion, no magnetic separation occurs. The voltage’s polarity depends on the carrier sign, so conventional current alone should not be mistaken for electron velocity.
Follow a proton through a uniform field
A proton (q = +1.60 × 10⁻¹⁹ C, m = 1.67 × 10⁻²⁷ kg) moves right at 2.00 × 10⁶ m/s in a 0.200 T field into the page. Find its initial force and circular-path radius.
Choose the proton as the system and a uniform, magnetic-only field. The speed is much smaller than c and velocity is perpendicular to B.
Why this step? These conditions justify the constant-mass circular-motion model.
F = |q|vB = (1.60 × 10⁻¹⁹)(2.00 × 10⁶)(0.200) = 6.40 × 10⁻¹⁴ N. Point the right index rightward with v and middle into the page with B; the thumb points up. The proton is positive, so its force is up. A negative charge with the same velocity and field would instead feel a downward force.
Why this step? The 90° case is the maximum magnitude; the charge sign sets direction.
.
Why this step? Equate the perpendicular magnetic force to the centripetal force, multiply by r, and cancel one factor of v to derive the radius expression.
r = mv/(|q|B) = (1.67 × 10⁻²⁷)(2.00 × 10⁶)/[(1.60 × 10⁻¹⁹)(0.200)] = 0.104 m.
Why this step? Equating magnetic and centripetal force eliminates one power of speed.
The proton bends upward initially and its speed remains 2.00 × 10⁶ m/s. Doubling B would halve r.
Why this step? The force is perpendicular to motion, so no magnetic work is done; the radius dependence checks the result.
Answer and check
Initial force: 6.40 × 10⁻¹⁴ N upward. Radius: 0.104 m.
Balance forces in a Hall strip
Positive carriers move right at 20.0 m/s through a strip of width 0.0100 m (vertical in the page). B = 0.500 T points into the page. Which edge becomes positive, and what transverse voltage magnitude balances the force?
For positive carriers, right × into-page gives an upward magnetic force. Positive charge therefore accumulates at the upper edge.
Why this step? The direction comes from the actual carrier velocity and sign.
The balancing electric field points downward, so the upper edge is at higher electric potential. At equilibrium E = vB = 10.0 V/m.
Why this step? Positive charges feel electric force along E; cancellation requires it opposite the magnetic force.
For a uniform transverse field, |ΔV| = Ew = (10.0)(0.0100) = 0.100 V.
Why this step? Potential decreases along E, and the strip width is the distance across that field.
Answer and check
Upper edge positive and 0.100 V higher than the lower edge. Reversing B reverses the polarity.
Make the model respond
Focused Physics 2 model
Moving charge in a uniform magnetic field
Predict first. Which changes reverse the magnetic force, and which change its magnitude?
|F| = |q|vB sin θ; r = mv⊥/(|q|B); v⊥ = v sin θ; F = q(v × B)
- Allowed quantitative angles (discrete values)
- Selected or reference point
On a narrow screen, scroll within a figure to read its labels. The table and explanation provide the same physics in text.
Positive q gives force in −y. |F| = 4.005 × 10^−13 N; r = 0.1044 m.
Read the model data and comparisons
| Quantity | Value | Unit | Meaning or limit |
|---|---|---|---|
| Signed charge | 1.602 × 10^−19 | C | |
| Fixed particle mass | 1.673 × 10^−27 | kg | Proton mass; negative q represents a hypothetical particle of this same mass |
| Speed | 5.000 × 10^6 | m/s | |
| Speed fraction of c | 0.01668 | ||
| Perpendicular speed | 5.000 × 10^6 | m/s | |
| Parallel speed (signed) | 0 | m/s | |
| Field B along +z | 0.5 | T | |
| Force magnitude | 4.005 × 10^−13 | N | |
| Force component Fy | -4.005 × 10^−13 | N | Velocity transverse component is +x |
| Transverse radius | 0.1044 | m |
The magnetic force depends on the part of velocity perpendicular to B. It does no work on the particle, because force is perpendicular to velocity, so speed stays constant. Radius uses |q| and v⊥; the sign of q selects bending direction. At θ = 0° or 180°, there is no force and no circular component. Only θ = 90° gives a planar circle; other nonzero transverse cases give a helix.
Assumptions and limits. Uniform B along +z, no electric field, no collisions, fixed mass equal to the proton mass. q may have either sign but negative q is a hypothetical same-mass particle, not an electron. Quantitative force display is restricted to angles 0°, 90°, 180° by the AP Physics 2 boundary; other angles are qualitative. Nonrelativistic calculation restricted to v ≤ 0.1c; the UI maximum is about 0.067c. Zero q, B, or transverse speed gives zero force and no circle. No motion is animated. Projection and radius are schematic.
Try it from memory
Write your answer and reason before opening the worked answer. Your self-assessment is a study decision; it is not an automatically graded score.
Question 1
A positive charge moves right. Compare its magnetic field at equally distant points directly above it and directly ahead of it.
Worked answer and reason
Above: out of the page and maximal for that speed and distance. Ahead: zero because r and v are parallel.
The source-field right-hand rule uses v and the position vector to the observation point.
Question 2
An electron and a proton have the same velocity rightward in a field into the page. Compare force magnitude and direction.
Worked answer and reason
Equal magnitude because their |q| values are equal. Proton force is up; electron force is down.
Apply the right-hand rule to positive charge, then reverse the direction for a negative charge.
Question 3
A particle’s path curves in a magnetic-only region. Has the field increased its kinetic energy?
Worked answer and reason
No. Its velocity direction changes, but magnetic force is perpendicular to motion and does no work, so speed and kinetic energy are constant.
Curving motion does not imply a change in speed.
Question 4
A Hall voltage is initially present. B is then turned parallel to the carrier velocity. What happens in the ideal steady state?
Worked answer and reason
The sideways magnetic force vanishes; the charge separation relaxes and the Hall voltage becomes zero.
The Hall effect requires a perpendicular magnetic-field component.
JavaScript enables saved drafts and self-assessment. You can always work on paper.
Catch the tempting mistake
- A negative charge has a negative force magnitude or radius.
- Use |q| in magnitudes and radius. Its sign reverses the force direction.
- Magnetic force accelerates a particle by making it faster.
- Acceleration can change direction only. A magnetic-only force does no work.
- The magnetic field created by a charge points along the force on that charge.
- The source field and force are different vectors governed by different right-hand-rule inputs.
Put it to work
Give the physical reason as well as the calculation. The worked solutions are on a separate page.
Scientific calculator workspace
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Scientific calculator
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For the teacher
- Require a labeled v, B, and F diagram before calculations.
- Quantitative moving-charge force questions use 0°, 90°, and 180°; discuss other angles qualitatively.
- The magnetic model is nonrelativistic and must keep v below c.
Source objectives and scope
- 12.2.A — Describe the magnetic field produced by moving charged objects.
- 12.2.B — Describe the force exerted on moving charged objects by a magnetic field.
Quantitative treatment of the magnitude of the magnetic force exerted by a magnetic field on a moving charge is limited to angles of 0, 90, and 180 degrees between the velocity and the magnetic field. Qualitative analysis of other angles is permitted.
Suggested skills are teaching guidance. The course applies all science practices across its objectives.
Science references for this teaching
- College Board AP Physics 2 CED, Unit 12 (supplied pinned source) — Official Unit 12 LO/EK scope and quantitative moving-charge angle boundary; raw 12.1.B4 normalized as 12.1.B.4
- OpenStax College Physics 2e: magnetic field lines — Closed-loop field convention and compass alignment
- OpenStax College Physics 2e: force on moving charge — Magnetic force magnitude and right-hand rule
- OpenStax College Physics 2e: Hall effect — Transverse force balance and Hall polarity
- OpenStax College Physics 2e: magnetic fields produced by currents — Long-wire field model and current-loop direction
- OpenStax College Physics 2e: force on a current-carrying conductor — Wire force and in-field length
- OpenStax College Physics 2e: Faraday and Lenz laws — Flux change and Lenz direction
- OpenStax College Physics 2e: motional emf — Sliding rod and mechanical-to-electrical energy accounting
Glossary and deeper reading · 14 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Electric charge
Electric charge is a conserved property that determines electric interactions. It can be positive or negative and is measured in coulombs; ordinary net charges arise from an imbalance of charged particles. For example, transferring electrons between initially neutral objects leaves one negatively charged and the other positively charged.
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Vector
A vector is a quantity with a magnitude and a direction, represented by an arrow or by components along chosen axes. Vectors add by adding corresponding components rather than simply adding their magnitudes. For example, an eastward displacement followed by a northward displacement produces a resultant pointing northeast. The zero vector has zero magnitude and no unique direction.
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Centripetal force
Centripetal force names the inward net-force component required for motion along a circular path. It is supplied by actual interactions such as tension, friction, gravity, or normal forces rather than being an additional type of force. For example, gravity supplies the centripetal force on a satellite in a circular orbit.
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Electric force
Electric force is the interaction of an electric field with a charged object. For point charges at rest, like signs repel and opposite signs attract, with direction along their separation. For example, an electron and proton exert equal-magnitude opposite forces on one another despite having different masses.
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Potential difference
Potential difference is the change in electric potential between two locations. It represents electric potential-energy change per unit charge and is measured in volts. For example, the potential difference across a resistor describes the energy transferred per coulomb passing through it.
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Magnetic field
A magnetic field assigns a vector direction and strength to each location and predicts interactions with moving charges, currents, and magnetic materials. Its strength is measured in teslas, and the force on a moving charge need not point along the field. For example, a compass aligns with the local field while a moving charge can be deflected perpendicular to it.
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Electric field
An electric field assigns the electric force per unit positive test charge to each location. The field is produced by the sources and is a vector, while the force on a negative charge points opposite the field. For example, the electric field between ideal parallel capacitor plates can accelerate an electron opposite the field arrows.
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Velocity
Velocity describes how position changes with time and includes direction. Instantaneous velocity describes one moment, while average velocity is displacement divided by elapsed time. For example, a car turning a corner can change velocity while maintaining the same speed.
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Cross product
A cross product combines two vectors to form a vector perpendicular to both. Its magnitude depends on both magnitudes and the sine of their included angle, while reversing input order reverses its direction. For example, a force whose line of action passes through the pivot gives zero torque cross product about that pivot.
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Force
A force describes an interaction that can change an object’s momentum. It is a vector measured in newtons, so both size and direction matter when forces are added. For example, a rope can pull a cart while the floor exerts friction in another direction.
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Magnitude
Magnitude is the nonnegative size of a vector quantity. It does not specify direction, and it differs from a signed component. For example, a westward velocity can have a negative x component while its speed, the velocity magnitude, stays positive.
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Displacement
Displacement is the vector change in position from the start of an interval to its end. It depends on those endpoints rather than the length or shape of the path traveled. For example, walking around a closed track and returning to the start gives zero displacement but a nonzero distance traveled.
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Hall effect
The Hall effect is the transverse charge separation produced when moving carriers in a conductor experience a magnetic force. The separation creates an electric field and voltage that can balance the sideways magnetic force. For example, reversing the applied magnetic field reverses the side on which a given carrier sign accumulates.
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Electrical conductor
An electrical conductor contains charge carriers that can move through the material. In electrostatic equilibrium, mobile charges redistribute until the electric field within the conducting material is zero. For example, excess charge on an isolated metal object resides on its surface after the charges settle.