Unit 4: Magnetic Fields

Magnetic field fundamentals and representation

A magnetic field is a vector field that describes how magnetic forces act on moving electric charges and on currents. In electrostatics, charges create electric fields that act on charges whether they move or not; magnetism is different in a crucial way: a magnetic field does not exert a magnetic force on a charge that is stationary in the reference frame of the field source. Instead, it influences charges in motion (and equivalently, electric currents).

The magnetic field is written as:

\vec{B}

Its SI unit is the tesla (T). A useful way to interpret the tesla is via the magnetic force relationships:

1\ \text{T} = 1\ \frac{\text{N}}{\text{A m}} = 1\ \frac{\text{N s}}{\text{C m}}

Visualizing magnetic fields: field lines, strength, and common conventions

Magnetic field lines are a visualization tool: the direction of the tangent to a field line gives the direction of \vec{B} at that location, and the density of lines indicates relative magnitude (closer lines mean stronger field). For currents, field lines commonly form closed loops around the current.

A standard direction convention used constantly is “into/out of the page”:

• A dot symbol means a vector points out of the page (toward you).
• A cross symbol means a vector points into the page (away from you).

When drawing magnetic fields, focus on four ideas:

• Direction of the field: for magnets, field lines outside the magnet point from the north pole to the south pole; for currents, use the right-hand rule.
• Strength of the field: line density indicates strength; fields are generally stronger near the source and weaker farther away.
• Shape of the field: straight wires produce circular patterns; loops and solenoids produce more complex patterns with a strong axial component.
• Interactions with other fields: magnetic fields can interact with electric fields and with other magnetic fields, creating complex patterns and enabling devices such as motors and MRI systems.

Right-hand rule for magnetic fields (direction of \vec{B} from a current)

The right-hand rule is the standard mnemonic technique to determine the direction of the magnetic field produced by a current.

For a straight current-carrying wire, point your right-hand thumb in the direction of the conventional current. Your curled fingers show the direction of the magnetic field loops around the wire.

For a circular loop of current, curl your right-hand fingers in the direction of the conventional current around the loop. Your thumb points in the direction of the magnetic field through the loop along its axis (the “inside/center” direction). The field lines then curve around and return outside the loop in the opposite sense, consistent with magnetic field lines forming closed loops.

A solenoid is a coil of wire that produces a magnetic field when current passes through it. For a long (ideal) solenoid, the field inside is approximately uniform and parallel to the solenoid’s axis, and the field outside is small. The field strength increases if you increase the number of turns per length or the current.

Bar magnets, poles, and the “no magnetic monopoles” idea

A bar magnet has two poles, north and south. Outside the magnet, field lines go from north to south, and the field is strongest near the poles and decreases as you move away.

At the level of AP Physics C: E&M, you treat magnetic poles as always coming in pairs: there are no isolated magnetic charges (no magnetic monopoles in the standard course treatment). That idea is captured by Gauss’s law for magnetism:

\oint \vec{B}\cdot d\vec{A} = 0

This says the net magnetic flux through any closed surface is zero: magnetic field lines do not start or stop inside the surface; whatever goes in must come out.

In vector-calculus form (mainly conceptual for AP):

\nabla \cdot \vec{B} = 0

Superposition

Magnetic fields obey superposition: if multiple currents are present, the net field is the vector sum of each contribution. This is powerful because you can add known results (wire + loop + solenoid), but it is also a common source of mistakes because you must add fields as vectors and keep directions consistent.

Exam Focus

• Typical question patterns
  • Determine the direction of \vec{B} at a point near a current-carrying wire or loop using a right-hand rule.
  • Use \oint \vec{B}\cdot d\vec{A} = 0 qualitatively to argue that magnetic field lines form closed loops.
  • Add magnetic field vectors from multiple current sources.
• Common mistakes
  • Treating magnetic field lines like electric field lines (starting/ending on isolated “poles” as if they were charges).
  • Adding magnitudes instead of vectors when combining fields.
  • Mixing up “into/out of page” conventions and reversing directions.

Magnetic & electric field interactions and the Lorentz force on charges

When a charged particle is placed in an electric field, it experiences a force in the direction of the field (for positive charge) or opposite the field (for negative charge):

\vec{F} = q\vec{E}

When a charged particle moves through a magnetic field, it experiences the Lorentz magnetic force, which is perpendicular to both the velocity and the magnetic field:

\vec{F} = q\vec{v} \times \vec{B}

When both fields are present, the net force is the vector sum:

\vec{F} = q\vec{E} + q\vec{v} \times \vec{B}

These interactions are important in contexts such as particle accelerators, plasma physics, astrophysics, particle detectors, and technologies such as MRI.

Magnitude and when the magnetic force is zero

The magnitude of the magnetic part is:

F = |q|vB\sin\theta

Here \theta is the angle between \vec{v} and \vec{B}.

• If the motion is parallel or antiparallel to \vec{B}, then \sin\theta = 0 and the magnetic force is zero.
• If the motion is perpendicular to \vec{B}, then F = |q|vB.

A strong magnetic field only produces a strong force if the charge has a velocity component perpendicular to the field.

Direction: right-hand rule and the sign of the charge

For a positive charge, point your fingers along \vec{v}, curl toward \vec{B}, and your thumb gives \vec{F}.

For a negative charge, the force is opposite the direction found from the right-hand rule for a positive charge. Forgetting this sign flip is a frequent source of lost points.

Magnetic forces do no work (magnetostatics)

Because \vec{F} is perpendicular to \vec{v}, the instantaneous power from the magnetic force is zero:

P = \vec{F}\cdot \vec{v} = 0

So a magnetic field by itself cannot change a particle’s speed (kinetic energy); it only changes the direction of motion.

Worked example: force direction and magnitude

A proton moves to the right with speed v through a uniform magnetic field pointing into the page. Find the direction and magnitude of the magnetic force.

Because velocity is perpendicular to the field, the magnitude is:

F = |q|vB

For a proton, q is positive, so:

F = evB

Using the right-hand rule (fingers right for \vec{v}, curl into the page for \vec{B}), the thumb points up, so the force is upward.

Exam Focus

• Typical question patterns
  • Use \vec{F} = q\vec{v}\times\vec{B} (or the combined-field form) to find direction and/or magnitude of force.
  • Determine when magnetic force is zero (velocity parallel to \vec{B}).
  • Conceptual: explain why a magnetic field does not change a particle’s speed.
• Common mistakes
  • Forgetting to reverse direction for negative charges.
  • Using F = qvB without the \sin\theta factor.
  • Claiming magnetic force does work because the path curves (curving does not imply work if the force is perpendicular to velocity).

Magnetic force on currents: force on wires, torque on loops, and torque basics

In circuits and devices you usually treat moving charge collectively as a current I. A current-carrying wire segment in an external magnetic field experiences a force because the moving charges inside it experience Lorentz forces.

Force on a straight current segment

For a straight segment of wire of length L in a uniform magnetic field:

\vec{F} = I\vec{L} \times \vec{B}

Here \vec{L} points in the direction of conventional current with magnitude equal to the length of the segment in the field.

The magnitude is:

F = ILB\sin\theta

where \theta is the angle between the current direction and \vec{B}. Equivalently, only the perpendicular component of the magnetic field contributes:

B_{\perp} = B\sin\theta

Direction (wire-force right-hand rule): a common “three-finger/palm” version is: point your thumb along current, fingers along \vec{B}, and the palm pushes in the direction of \vec{F}.

Worked example: force on a wire segment

A wire carries current I = 3.0\ \text{A} upward through a uniform field B = 0.20\ \text{T} to the right. A length L = 0.50\ \text{m} of the wire is in the field.

Because current is perpendicular to \vec{B}, the magnitude is:

F = ILB = (3.0)(0.50)(0.20) = 0.30\ \text{N}

For direction, use \vec{L} \times \vec{B}: up cross right gives into the page.

Closed loops, motors, and why “forces cancel” can still produce rotation

If the wire is part of a closed loop in a uniform field, the net force on the loop is typically zero because forces on opposite sides cancel as vectors. However, those forces can act at different locations and form a couple, producing a nonzero torque. This is the core physics behind an electric motor: forces on different sides of the loop produce rotation even when the net translational force is zero.

Torque on a current loop and magnetic dipole moment

For a planar loop with N turns, current I, and area A, define the magnetic dipole moment:

\vec{\mu} = NIA\hat{n}

The unit vector \hat{n} is perpendicular to the plane of the loop and is set by a right-hand rule: curl fingers with conventional current; thumb gives \hat{n}.

In a uniform magnetic field, the loop experiences torque:

\vec{\tau} = \vec{\mu} \times \vec{B}

Magnitude:

\tau = \mu B\sin\theta

where \theta is the angle between \vec{\mu} and \vec{B}. The loop tends to rotate to align \vec{\mu} with \vec{B}.

Potential energy of a magnetic dipole

The torque corresponds to an energy preference:

U = -\vec{\mu}\cdot \vec{B}

Minimum energy occurs when \vec{\mu} is parallel to \vec{B}; maximum energy occurs when they are antiparallel.

Torque basics (general mechanics connection)

Independently of magnetism, the magnitude of a torque from a force applied at a lever arm is:

\tau = Fr\sin\theta

Here r is the distance from the rotation axis to the point of application, and \theta is the angle between the radius vector and the force.

A practical extension is torsional stiffness of a wire in twisting setups, defined by:

k = \frac{\tau}{\theta}

In that context, \theta is the angle of twist (in radians), and k depends on material properties and geometry (length, diameter, cross-sectional shape).

Exam Focus

• Typical question patterns
  • Compute force on a wire using \vec{F} = I\vec{L}\times\vec{B} and determine direction.
  • Analyze a rectangular loop in a uniform field: identify which sides feel force and determine the net torque.
  • Use \vec{\tau} = \vec{\mu}\times\vec{B} and U = -\vec{\mu}\cdot\vec{B} to relate orientation to stability.
• Common mistakes
  • Using electron flow direction instead of conventional current direction for \vec{L}.
  • Forgetting that loop torque depends on the angle between \vec{\mu} and \vec{B} (not between the loop plane and \vec{B} unless you convert carefully).
  • Claiming a loop in uniform \vec{B} must accelerate linearly because forces act on its sides (net force is typically zero even when torque is nonzero).

Charged particle motion in a uniform magnetic field (circular and helical paths)

Because the magnetic force is perpendicular to velocity, it naturally acts like a centripetal force when the velocity has a component perpendicular to \vec{B}.

Pure circular motion (velocity perpendicular to \vec{B})

If \vec{v} is perpendicular to \vec{B}, the magnetic force provides centripetal force:

|q|vB = \frac{mv^2}{r}

Solving for the radius:

r = \frac{mv}{|q|B}

The cyclotron angular frequency follows from v = r\omega:

\omega = \frac{|q|B}{m}

The period is:

T = \frac{2\pi m}{|q|B}

A key nonrelativistic result is that T does not depend on speed or radius, only on m, |q|, and B.

Helical motion (velocity has parallel and perpendicular components)

Decompose velocity into components relative to the field:

• perpendicular component produces circular motion
• parallel component is unaffected (constant speed along the field)

This produces a helix. The pitch (distance advanced along the field per revolution) is:

p = v_{\parallel}T

Velocity selector (crossed fields)

In a velocity selector, perpendicular electric and magnetic forces cancel for one speed. Setting magnitudes equal:

qE = qvB

gives the selected speed:

v = \frac{E}{B}

This is commonly used in mass spectrometers: select speed, then use magnetic curvature to infer m/q.

Worked example: radius of curvature

An electron (mass m, charge magnitude e) enters a uniform magnetic field B = 0.050\ \text{T} with speed v = 2.0\times 10^6\ \text{m/s} perpendicular to the field. The radius is:

r = \frac{mv}{eB}

Direction of curvature depends on the sign of charge: the electron curves opposite the direction predicted by the right-hand rule for a positive charge.

Exam Focus

• Typical question patterns
  • Use r = \frac{mv}{|q|B} to find curvature radius or infer m/q from a track.
  • Decide whether motion is circular or helical from the angle between \vec{v} and \vec{B}.
  • Use the velocity-selector relation v = \frac{E}{B} in multi-step setups.
• Common mistakes
  • Using q instead of |q| in radius formulas and getting an unphysical negative radius.
  • Treating the magnetic force as constant in a fixed direction; it changes direction continuously as the particle turns.
  • Thinking magnetic force changes speed; it changes direction only (for magnetostatic fields).

Magnetic fields produced by currents: the Biot–Savart law

For steady currents (magnetostatics), the fundamental “building rule” for fields from currents is the Biot–Savart law, discovered by Biot and Savart (1820). In principle it lets you integrate over a current distribution to find \vec{B} anywhere.

Biot–Savart law (differential contribution)

For a current element I d\vec{\ell} and a field point located by vector \vec{r} from the element:

d\vec{B} = \frac{\mu_0}{4\pi}\frac{I d\vec{\ell} \times \hat{r}}{r^2}

Each contribution is perpendicular to both the current element direction and the line from the element to the point.

Validity and assumptions (important conceptually): Biot–Savart is used for steady currents and does not include effects from time-varying electric fields. Many introductory applications also assume the current is steady and effectively uniform through the wire.

Standard results you should recognize

For a long straight wire at distance r:

B = \frac{\mu_0 I}{2\pi r}

For the field at the center of a circular loop of radius R:

B = \frac{\mu_0 I}{2R}

For N turns:

B = \frac{\mu_0 N I}{2R}

On the axis of a loop a distance x from the center:

B = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}

Sample problem: long straight wire (3 cm away)

A long straight wire carries a current of 5 A. Find the magnetic field at a point 3 cm away.

Use:

B = \frac{\mu_0 I}{2\pi r}

Substitute I = 5\ \text{A} and r = 0.03\ \text{m}:

B = \frac{4\pi\times 10^{-7}\cdot 5}{2\pi\cdot 0.03}

This evaluates to:

B = 3.33\times 10^{-5}\ \text{T}

Worked example: field from a straight wire (10 cm away)

A long straight wire carries I = 5.0\ \text{A}. Find the magnitude of \vec{B} at r = 0.10\ \text{m}.

B = \frac{\mu_0 I}{2\pi r}

B = \frac{\mu_0(5.0)}{2\pi(0.10)}

Using \mu_0 \approx 4\pi\times 10^{-7}:

B \approx 1.0\times 10^{-5}\ \text{T}

Exam Focus

• Typical question patterns
  • Use known Biot–Savart results (straight wire, loop center/axis) and superposition.
  • Determine field direction around a wire or through a loop using right-hand rules.
  • Multi-step: find B from a wire, then use \vec{F} = I\vec{L}\times\vec{B} to find force on another wire.
• Common mistakes
  • Mixing up r (distance from wire) with radius R of a loop.
  • Using the straight-wire formula for a finite wire without being told it is “long” (infinite approximation).
  • Getting direction wrong by pointing thumb along \vec{B} instead of along the current when using the right-hand rule.

Ampère’s law: using symmetry to find \vec{B} efficiently

While Biot–Savart is very general, Ampère’s law (discovered by Ampère, 1826) can be a faster method for highly symmetric magnetostatic situations.

Ampère’s law (integral form)

For steady currents:

\oint \vec{B}\cdot d\vec{\ell} = \mu_0 I_{\text{enc}}

The integral is taken around a closed Amperian loop, and I_{\text{enc}} is the net current passing through any surface bounded by that loop.

When Ampère’s law is useful (and when it is not)

Ampère’s law is always true for magnetostatics, but it is only useful when symmetry lets you simplify the integral (constant B on the path and known direction relative to d\vec{\ell}). It works well for:

• infinite (or very long) straight wires
• ideal long solenoids
• ideal toroids
• infinite current sheets (less common on AP, but conceptually similar)

It is not very useful for finite wires of arbitrary shape or off-axis loop points without symmetry.

Deriving the long straight wire field with Ampère’s law

Choose a circular Amperian loop of radius r centered on the wire. Symmetry gives constant tangent B, so:

\oint \vec{B}\cdot d\vec{\ell} = B(2\pi r)

Set equal to \mu_0 I:

B(2\pi r) = \mu_0 I

So:

B = \frac{\mu_0 I}{2\pi r}

Magnetic field inside an ideal solenoid

For a long solenoid with turn density n (turns per unit length) carrying current I, the inside field is approximately uniform:

B = \mu_0 n I

If the solenoid has N turns over length \ell:

B = \mu_0 \frac{N}{\ell} I

The field can be increased by increasing current or turns per length.

Magnetic field of an ideal toroid

For a toroid with N turns carrying current I, inside the core region at radius r:

B = \frac{\mu_0 N I}{2\pi r}

Outside the toroid, the field is much smaller (idealized as zero).

Sample problem: Ampère’s law near a wire (corrected)

A long straight wire carries a current of 10 A. What is the magnetic field at a distance of 5 cm from the wire?

Choose a circular Amperian loop of radius r = 0.05\ \text{m}. Then:

B(2\pi r) = \mu_0 I

So:

B = \frac{\mu_0 I}{2\pi r}

Substitute I = 10\ \text{A}:

B = \frac{4\pi\times 10^{-7}\cdot 10}{2\pi\cdot 0.05}

This evaluates to:

B = 4.0\times 10^{-5}\ \text{T}

Worked example: solenoid field

A solenoid has N = 800 turns over length \ell = 0.40\ \text{m} and carries I = 0.60\ \text{A}.

First compute:

n = \frac{N}{\ell} = 2000\ \text{m}^{-1}

Then:

B = \mu_0 n I = 1200\mu_0

Using \mu_0 \approx 4\pi\times 10^{-7}:

B \approx 4.8\pi\times 10^{-4}\ \text{T}

Exam Focus

• Typical question patterns
  • Choose an appropriate Amperian loop and use symmetry to compute B for a wire, solenoid, or toroid.
  • Conceptual: explain why B is approximately uniform inside a long solenoid and small outside.
  • Decide when to use Ampère’s law versus Biot–Savart.
• Common mistakes
  • Picking an Amperian loop that does not match the symmetry and then assuming B is constant when it is not.
  • Confusing N (total turns) with n (turns per length) in solenoid problems.
  • Forgetting that toroid B depends on radius as 1/r.

Forces between current-carrying wires and the meaning of current-produced fields

Two parallel currents provide a clean link between “currents create fields” and “fields exert forces.” One wire creates a magnetic field; the other wire experiences a force from that field.

Field from wire 1 and force on wire 2

For two long parallel wires separated by distance d carrying currents I_1 and I_2:

Field from wire 1 at wire 2:

B_1 = \frac{\mu_0 I_1}{2\pi d}

Force per unit length on wire 2 (standard geometry has \sin\theta = 1):

\frac{F}{L} = I_2 B_1

So:

\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d}

Equivalently, the force magnitude on a length L is:

F = \frac{\mu_0 I_1 I_2 L}{2\pi d}

Attraction vs repulsion

• Currents in the same direction attract.
• Currents in opposite directions repel.

It is safest to be able to re-derive the direction quickly using right-hand rules rather than relying only on memorization.

Worked example: force per unit length

Two long wires are d = 0.040\ \text{m} apart carrying I_1 = 10\ \text{A} and I_2 = 6.0\ \text{A} in the same direction.

\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi d} = \frac{\mu_0(10)(6.0)}{2\pi(0.040)}

Using \mu_0 \approx 4\pi\times 10^{-7} gives:

\frac{F}{L} \approx 3.0\times 10^{-4}\ \text{N/m}

Direction: attractive because currents are in the same direction.

Exam Focus

• Typical question patterns
  • Compute F/L between two long parallel wires.
  • Direction reasoning: determine whether wires attract or repel.
  • Multi-step: compute B from one wire then use force on the other.
• Common mistakes
  • Forgetting that d is in the denominator (and is the separation distance, not the wire radius).
  • Getting attraction/repulsion backward due to a right-hand rule slip.
  • Treating the force as proportional to 1/d^2 (it is 1/d for long wires).

Magnetic flux (in magnetostatics) and how it connects to field geometry

Even before induction is studied formally, magnetic flux is a useful way to quantify “how much” magnetic field passes through an area.

Definition of magnetic flux

\Phi_B = \int \vec{B}\cdot d\vec{A}

For a flat area with uniform \vec{B}:

\Phi_B = BA\cos\theta

where \theta is the angle between \vec{B} and the surface normal (area vector).

Connecting flux to Gauss’s law for magnetism

For any closed surface:

\oint \vec{B}\cdot d\vec{A} = 0

This is the statement that net magnetic flux through a closed surface is zero.

Worked example: flux through a tilted loop

A circular loop of area A is in a uniform magnetic field of magnitude B. The loop’s normal makes angle \theta with \vec{B}.

\Phi_B = BA\cos\theta

Extreme cases: maximum flux at \theta = 0 and zero flux at \theta = \frac{\pi}{2}.

Exam Focus

• Typical question patterns
  • Compute flux \Phi_B for a uniform field through a flat loop.
  • Use Gauss’s law for magnetism to reason that net flux through a closed surface is zero.
  • Interpret geometry: identify the orientation giving maximum flux.
• Common mistakes
  • Using \sin\theta instead of \cos\theta because of mixing up the angle between field and surface vs field and normal.
  • Treating flux as a vector (it is a scalar; the vector is the area vector inside the dot product).
  • Forgetting that Gauss’s law for magnetism applies to closed surfaces.

Magnetic dipoles, Earth’s field, and magnetic materials (conceptual extensions)

Many problems model magnets and current loops as magnetic dipoles. A current loop is the clearest quantitative dipole model in this course.

What is a magnetic dipole?

A magnetic dipole is any system that produces a field resembling that of a small current loop far away from the source. For a loop:

\vec{\mu} = NIA\hat{n}

This dipole moment determines how strongly the object interacts with external magnetic fields.

Earth as a (rough) magnetic dipole

Earth’s magnetic field is approximately dipole-like on large scales. A compass needle aligns because it experiences torque:

\vec{\tau} = \vec{\mu}\times\vec{B}

In typical AP setups, Earth’s field is treated as uniform over small regions.

Magnetic materials: diamagnetism, paramagnetism, ferromagnetism

Microscopically, magnetism is tied to moving charges (electron orbital motion and spin, treated qualitatively here).

• Diamagnetic: induced moment opposes an applied field (weak).
• Paramagnetic: moments tend to align with an applied field (weak attraction).
• Ferromagnetic: strong, persistent magnetization via domain alignment (strong attraction; can form permanent magnets).

Unless a problem specifies a different permeability, AP calculations typically use vacuum permeability \mu_0.

Common misconception: “magnets attract all metals”

Only certain materials (notably ferromagnets such as iron, cobalt, nickel, and many alloys) are strongly attracted. Metals like copper and aluminum are not ferromagnetic; they can still experience effects in changing magnetic fields via induction, but that is a different mechanism.

Exam Focus

• Typical question patterns
  • Treat a loop as a dipole: use \vec{\mu}, torque, and energy relations.
  • Conceptual questions about alignment and stability (minimum energy when \vec{\mu} aligns with \vec{B}).
  • Qualitative questions distinguishing magnetic material behavior.
• Common mistakes
  • Confusing the direction of \vec{\mu} (set by current right-hand rule) with the direction of current itself.
  • Saying a dipole moves in a uniform field due to torque (uniform field gives torque but often no net force).
  • Overgeneralizing that any metal is strongly attracted to magnets.

Putting it together: multi-concept reasoning and common AP-style synthesis

AP problems often test whether you can connect: fields from currents, forces from fields, and the resulting motion. Direction reasoning (right-hand rules) and careful vector/magnitude translations are as important as algebra.

Cross product relationships: vector form vs magnitude form

A consistent pattern is that magnetism uses cross products heavily, so you should expect forces/torques to be perpendicular to the “input” directions.

Synthesis example: wire creates field, field exerts force

A common two-part structure is:

1. Use a current-carrying wire to compute B at another location.
2. Use \vec{F} = I\vec{L}\times\vec{B} or F/L = IB to compute the force on another current.

This logic drives many “two-wire” and “wire near loop” questions.

Synthesis example: velocity selector plus circular motion

A mass spectrometer style chain often looks like:

1. A velocity selector sets speed:

v = \frac{E}{B}

2. A region with magnetic field bends the path:

r = \frac{mv}{|q|B}

3. Combine to solve for m/q in terms of measured radius and known fields.

Exam Focus

• Typical question patterns
  • Multi-step problems chaining: compute B from currents, then compute force/torque/motion.
  • Mixed conceptual and quantitative: justify a direction using right-hand rules, then compute a magnitude.
  • Identify symmetry and choose between Ampère’s law and Biot–Savart.
• Common mistakes
  • Switching between magnitude and vector forms without tracking angles (dropping \sin\theta or using the wrong \theta).
  • Losing sign information (electron vs proton) in direction reasoning.
  • Using a familiar formula that does not match the physical situation (Ampère’s law without symmetry, straight-wire result when the wire is not effectively long).