🎯⭐ INTERACTIVE LESSON

Magnetic Fields and Forces

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Magnetic Fields and Forces - Complete Interactive Lesson

Part 1: Magnetic Fields

🧲 Magnetic Fields — Sources and Properties

Part 1 of 7 — Where Do Magnetic Fields Come From?

Magnetism has been known for millennia — ancient Greeks observed lodestone attracting iron. Today we understand that all magnetic fields originate from moving electric charges. Whether it's a bar magnet, an electromagnet, or Earth itself, the underlying source is always charges in motion.

Sources of Magnetic Fields

1. Permanent Magnets

Bar magnets and horseshoe magnets produce fields due to the aligned spin of electrons within their atoms. In ferromagnetic materials (iron, nickel, cobalt), electron spins in microscopic domains align to create a net magnetic field.

2. Moving Charges

A single charge $q$ moving with velocity $\vec{v}$ creates a magnetic field:

$\vec{B} = \frac{\mu_0}{4\pi} \frac{q\vec{v} \times \hat{r}}{r^2}$

where $\mu_0 = 4\pi \times 10^{-7}$ T·m/A is the permeability of free space.

3. Electric Currents

A current-carrying wire produces a magnetic field that circles around the wire. This is the most practical source of controllable magnetic fields — the basis for electromagnets, motors, and MRI machines.

Key Insight

Electricity and magnetism are deeply connected. Moving charges create magnetic fields, and (as we'll see later) changing magnetic fields create electric fields. This unification is one of the great triumphs of physics.

Magnetic Field Lines

Magnetic field lines visualize the $\vec{B}$ field, similar to electric field lines for $\vec{E}$ .

Earth's Magnetic Field

Earth behaves like a giant bar magnet, but with an important twist:

Geographic vs. Magnetic Poles

Earth's geographic North Pole is near Earth's magnetic South pole
A compass needle's north end points toward geographic north because it's attracted to the magnetic south pole located there
The magnetic axis is tilted about 11° from the rotation axis

Field Strength

Earth's magnetic field strength at the surface:

$B_{\text{Earth}} \approx 25 - 65 \text{ μT} \approx 0.25 - 0.65 \text{ Gauss}$

Units of Magnetic Field

The SI unit of magnetic field $\vec{B}$ is the Tesla (T):

$1 \text{ T} = 1 \frac{\text{kg}}{\text{A·s}^2} = 1 \frac{\text{N}}{\text{A·m}} = 1 \frac{\text{Wb}}{\text{m}^2}$

Magnetic Fields Concept Quiz 🧲

Exit Check — Magnetic Field Basics ✅

Part 2: Force on Moving Charges

⚡ Force on Moving Charges

Part 2 of 7 — The Magnetic Force Law

Electric fields push charges along the field direction. Magnetic fields do something stranger: they push charges perpendicular to both the velocity and the field. This sideways force is the key to everything from particle accelerators to the Northern Lights.

The Magnetic Force on a Moving Charge

A charge $q$ moving with velocity $\vec{v}$ in a magnetic field experiences a force:

Part 3: Circular Motion in B Fields

🌀 Charged Particles in Magnetic Fields

Part 3 of 7 — Circular Motion, Mass Spectrometers, and Velocity Selectors

When a charged particle moves perpendicular to a uniform magnetic field, the magnetic force acts as a centripetal force, causing the particle to travel in a circle. This simple fact underlies some of the most powerful instruments in physics and chemistry.

Circular Motion in a Magnetic Field

A charge $q$ moving with speed $v$ perpendicular to a uniform field $\vec{B}$ experiences:

Part 4: Force on Current-Carrying Wires

🔌 Force on Current-Carrying Wires

Part 4 of 7 — From Single Charges to Wires

A current is simply a flow of charges. Since each moving charge experiences a magnetic force, a current-carrying wire in a magnetic field experiences a net force. This is the principle behind electric motors, loudspeakers, and galvanometers.

Force on a Straight Current-Carrying Wire

For a straight wire of length $L$ carrying current $I$ in a uniform magnetic field $\vec{B}$ :

Part 5: Ampère's Law & Solenoids

🔄 Magnetic Field from Currents

Part 5 of 7 — Biot-Savart, Ampère's Law, and Solenoids

So far we've studied the force that a magnetic field exerts on charges and wires. Now we turn to the source question: how do currents create magnetic fields? Two powerful laws — the Biot-Savart law and Ampère's law — give the answer.

Magnetic Field of a Long Straight Wire

The most important result: a long straight wire carrying current $I$ creates a magnetic field at perpendicular distance $r$ :

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

Part 6: Torque on Current Loops

⚙️ Torque on Current Loops

Part 6 of 7 — Magnetic Dipoles, Motors, and Galvanometers

A current loop in a magnetic field doesn't just experience a net force — it experiences a torque that tends to rotate it. This is the operating principle behind electric motors, galvanometers, and many other devices.

Torque on a Rectangular Current Loop

Consider a rectangular loop (sides $a$ and $b$ ) carrying current $I$ in a uniform field $\vec{B}$ .

Part 7: Synthesis & AP Review

🏆 Synthesis & AP Review

Part 7 of 7 — Right-Hand Rule Mastery, Common Mistakes, and AP FRQ Preview

You've learned the core of magnetism: fields from sources, forces on charges and wires, circular motion, and torque on loops. This final part ties everything together and prepares you for AP-level problems.

📋 Master Formula Sheet

Forces

Situation	Formula
Force on moving charge	$\vec{F} = q\vec{v} \times \vec{B}$ ,

Rules for Magnetic Field Lines

Outside a magnet: Lines point from North (N) to South (S)
Inside a magnet: Lines point from South to North (completing closed loops)
Lines never cross — the field has a unique direction at every point
Line density indicates field strength — closer lines = stronger $\vec{B}$
Magnetic field lines always form closed loops — there are no magnetic monopoles!

Bar Magnet Field Pattern

The field of a bar magnet looks similar to the electric field of a dipole:

Lines emerge from the North pole
Arc through space
Enter the South pole
Continue through the interior back to North

Critical Difference from Electric Fields

Electric field lines begin on positive charges and end on negative charges. Magnetic field lines have no beginning or end — they always close on themselves. This reflects the fact that magnetic monopoles do not exist (as far as we know).

This is quite weak compared to lab magnets (which can be 1 T or more), but it's strong enough to orient compass needles and protect us from charged particles in the solar wind.

Earth's magnetic field is generated by convection currents of molten iron in the outer core — essentially a natural dynamo. The field slowly changes over time and has even reversed polarity many times throughout geologic history.

The older CGS unit is the Gauss (G):

$1 \text{ T} = 10{,}000 \text{ G} = 10^4 \text{ G}$

Typical Magnetic Field Strengths

Source	Strength
MRI machine	1.5 – 3 T
Strong lab magnet	1 – 10 T
Refrigerator magnet	~5 mT
Earth's surface	~50 μT
Interstellar space	~0.1 nT

The Permeability Constant

$\mu_0 = 4\pi \times 10^{-7} \text{ T·m/A}$

This constant appears in all formulas for magnetic fields produced by currents, just as $\varepsilon_0$ appears in electric field formulas.

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

$|\vec{F}| = |q|vB\sin\theta$

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

The force is perpendicular to both $\vec{v}$ and $\vec{B}$ — it points "out of the plane" formed by the velocity and field vectors
No force on stationary charges — if $v = 0$ , then $F = 0$
No force when $\vec{v} \parallel \vec{B}$ — if or , then
Maximum force when $\vec{v} \perp \vec{B}$ — when ,
The magnetic force does no work — since $\vec{F} \perp \vec{v}$ , the force changes direction but not speed

The Right-Hand Rule

To find the direction of $\vec{F} = q\vec{v} \times \vec{B}$ :

For Positive Charges ( $q > 0$ )

Point your fingers in the direction of $\vec{v}$
Curl them toward $\vec{B}$

For Negative Charges ( $q < 0$ )

Use the right-hand rule as above, then reverse the direction (the force is opposite to what the right-hand rule gives).

Alternatively: use your left hand for negative charges.

Common Directions

$\vec{v}$ direction	$\vec{B}$ direction

Memory Aid

Think: "velocity, field, force" — point, curl, thumb. For a positive charge, this directly gives the force. For an electron, flip it.

Right-Hand Rule Drill 👋

Determine the direction of the magnetic force on a positive charge in each scenario.

Force on Negative Charges

For an electron ( $q = -e$ ) or any negative charge:

$\vec{F} = q\vec{v} \times \vec{B} = (-e)\vec{v} \times \vec{B}$

The negative sign reverses the force direction compared to a positive charge.

Example

An electron moves to the right ( $+x$ ) in a magnetic field pointing up ( $+y$ ):

Right-hand rule for $+q$ : $\hat{x} \times \hat{y} = \hat{z}$ → out of page

Practical Importance

In most conductors, the current carriers are electrons (negative charges). When applying the force law to current-carrying wires, we can either:

Use $\vec{F} = q\vec{v} \times \vec{B}$ with and the actual electron velocity

Both approaches give the same force direction on the wire.

Electron Force Direction Drill 🔬

Now find the force direction on an electron ( $q = -e$ ).

Exit Quiz — Magnetic Force ✅

This force is always perpendicular to $\vec{v}$ , so it acts as a centripetal force:

$qvB = \frac{mv^2}{r}$

Solving for the radius of circular motion:

$\boxed{r = \frac{mv}{qB}}$

Faster particles → larger radius (linear in $v$ )
Heavier particles → larger radius (linear in $m$ )
Stronger field → smaller radius (inverse in $B$ )
More charge → smaller radius (inverse in $q$ )

The speed $v$ does not change — the magnetic force does no work! The particle moves in a perfect circle at constant speed.

Cyclotron Period and Frequency

The period (time for one full circle) is:

$T = \frac{2\pi r}{v} = \frac{2\pi m}{qB}$

The cyclotron frequency is:

$\boxed{f = \frac{qB}{2\pi m}}$

And the angular frequency:

$\omega = \frac{qB}{m}$

Remarkable Result

The period and frequency are independent of speed and radius! A faster particle travels a bigger circle, but it completes the circle in the same time as a slower particle. This is the principle behind the cyclotron particle accelerator.

What if $\vec{v}$ has a component along $\vec{B}$ ?

The component $v_\parallel$ (parallel to $\vec{B}$ ) is unaffected — no force
The component (perpendicular to ) produces circular motion

Mass Spectrometer

A mass spectrometer separates ions by mass using a magnetic field.

How It Works

Ionize atoms or molecules → they become charged
Accelerate through a potential difference $V$ : kinetic energy $= qV$ , so $v = \sqrt{2qV/m}$
Enter a magnetic field region → travel in semicircles
Detect where they land on a detector plate

The Separation

The radius of the semicircle is:

$r = \frac{mv}{qB}$

Substituting $v = \sqrt{2qV/m}$ :

$r = \frac{m\sqrt{2qV/m}}{qB} = \frac{\sqrt{2mV/q}}{B}$

$\boxed{r = \frac{1}{B}\sqrt{\frac{2mV}{q}}}$

Different masses land at different positions (different $r$ ), allowing isotope identification and molecular analysis.

Velocity Selector

A velocity selector uses crossed electric and magnetic fields to select particles with a specific speed.

Setup

$\vec{E}$ and $\vec{B}$ are perpendicular to each other and to the particle's velocity $\vec{v}$
Electric force: $F_E = qE$ (in one direction)
Magnetic force: $F_B = qvB$ (in the opposite direction)

Condition for Straight-Line Motion

For the particle to pass through undeflected:

$qE = qvB$

$\boxed{v = \frac{E}{B}}$

Key Features

Only particles with speed $v = E/B$ pass through — all others are deflected
The selected speed is independent of charge and mass
This is used as the first stage in many mass spectrometers to ensure all entering particles have the same speed

Circular Motion Calculation Drill 🔢

A proton ( $m = 1.67 \times 10^{-27}$ kg, $q = 1.6 \times 10^{-19}$ C) moves at $4.0 \times 10^6$ m/s perpendicular to a 0.20 T magnetic field.

Radius of the circular path (in m, to 3 significant figures)
Cyclotron period (in s, use scientific notation like 3.3e-7)
Cyclotron frequency (in Hz, use scientific notation like 3.0e6)

Exit Quiz — Particles in Magnetic Fields ✅

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

where $\vec{L}$ points in the direction of conventional current flow.

$\boxed{|\vec{F}| = BIL\sin\theta}$

where $\theta$ is the angle between the wire (current direction) and $\vec{B}$ .

Wire perpendicular to $\vec{B}$ ( $\theta = 90°$ ): $F = BIL$ (maximum force)
Wire parallel to $\vec{B}$ ( $\theta = 0°$ ): $F = 0$ (no force)

Derivation from Charge Force

Current $I = nqv_dA$ where $n$ = charge density, $v_d$ = drift velocity, $A$ = cross-section area.

Total force on charges in length $L$ :

$F = (nAL)(qv_dB) = (nqv_dA)(LB) = BIL$

The macroscopic wire force follows directly from the microscopic charge force.

Direction of Force on a Wire

Use the same right-hand rule as for individual charges:

Point fingers along $\vec{I}$ (conventional current direction)
Curl toward $\vec{B}$
Thumb points in the direction of $\vec{F}$

Example

A wire carries current to the right ( $+x$ ) in a magnetic field pointing into the page ( $-z$ ):

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

$\hat{x} \times (-\hat{z}) = -(\hat{x} \times \hat{z}) = -(-\hat{y}) = +\hat{y}$

The force is upward ( $+y$ ).

This is exactly how a loudspeaker works: alternating current in a coil within a magnetic field pushes the speaker cone back and forth, producing sound waves.

Wire Force Direction Drill 🎯

Find the force direction on a current-carrying wire.

Forces Between Parallel Wires

Two parallel wires carrying currents exert magnetic forces on each other.

Same Direction Currents → Attract

Wire 1 creates a field at Wire 2. By the right-hand rule, this field causes a force on Wire 2 toward Wire 1. By Newton's third law, Wire 1 is also pulled toward Wire 2.

Parallel currents attract.

Opposite Direction Currents → Repel

If the currents flow in opposite directions, the force pushes the wires apart.

Antiparallel currents repel.

Force Per Unit Length

The magnetic field from Wire 1 at distance $d$ :

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

Force on a length $L$ of Wire 2:

$F = B_1 I_2 L = \frac{\mu_0 I_1 I_2 L}{2\pi d}$

Force per unit length:

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

Definition of the Ampere

This force is so fundamental that the ampere was historically defined as: the current that, flowing in two parallel wires 1 m apart, produces a force of $2 \times 10^{-7}$ N per meter of length.

Wire Force Calculation Drill 🔢

Use $\mu_0 = 4\pi \times 10^{-7}$ T·m/A.

A 0.50 m wire carrying 8.0 A is perpendicular to a 0.30 T field. What is the force on the wire (in N)?
Two parallel wires carry 10 A each in the same direction, separated by 5.0 cm. What is the force per unit length between them (in N/m, use scientific notation like 4.0e-4)?
Do the wires in (2) attract or repel? (Type "attract" or "repel")

Round all answers to 3 significant figures.

Exit Quiz — Forces on Wires ✅

where $\mu_0 = 4\pi \times 10^{-7}$ T·m/A.

Direction: Right-Hand Rule for Wires

Point your right thumb in the direction of the current. Your fingers curl in the direction of $\vec{B}$ .

Current flowing up → $\vec{B}$ circles counterclockwise (viewed from above)
Current flowing toward you → $\vec{B}$ circles counterclockwise

$B \propto I$ — double the current, double the field
$B \propto 1/r$ — field decreases with distance (but only as $1/r$ , not $1/r^2$ )
Field lines are concentric circles centered on the wire
Field is tangent to these circles at every point

Ampère's Law

Ampère's law is the magnetic analog of Gauss's law — it relates the magnetic field around a closed loop to the current passing through the loop:

$\boxed{\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}}$

where $I_{\text{enc}}$ is the total current enclosed by the Amperian loop.

When to Use Ampère's Law

Ampère's law is most useful when the geometry has high symmetry:

Long straight wire — use a circular Amperian loop centered on the wire
Solenoid — use a rectangular loop
Toroid — use a circular loop inside the toroid

Deriving the Wire Field

For a long straight wire, choose a circular Amperian loop of radius $r$ :

$\oint B \, dl = B(2\pi r) = \mu_0 I$

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

The same result we stated above — Ampère's law confirms it elegantly.

Solenoids

A solenoid is a tightly wound coil of wire. It produces a nearly uniform magnetic field inside and nearly zero field outside.

Field Inside an Ideal Solenoid

$\boxed{B = \mu_0 n I}$

where:

$n = N/L$ is the number of turns per unit length
$N$ = total number of turns
$L$ = length of the solenoid
$I$ = current

Derivation (Ampère's Law)

Choose a rectangular Amperian loop with one side inside and one outside the solenoid:

Inside side (length $l$ ): $B \cdot l$ (field is uniform and parallel)
Outside side: 0 (field is negligible)
Two short sides: 0 (perpendicular to $\vec{B}$ )

Current enclosed: $I_{\text{enc}} = nlI$ (there are $nl$ turns through the loop)

$Bl = \mu_0 nlI$

$B = \mu_0 nI \checkmark$

Key Properties

Field inside is uniform (constant everywhere inside)
Field outside is approximately zero
Field strength depends on turns per length ( $n$ ) and current ( $I$ ), not the total length
A solenoid is essentially a magnetic dipole — its field pattern outside resembles a bar magnet

Magnetic Field Calculation Drill 🔢

Use $\mu_0 = 4\pi \times 10^{-7}$ T·m/A.

A long wire carries 15 A. What is $B$ at a distance of 3.0 cm from the wire? (in T, use scientific notation like 1.0e-4)
A solenoid is 0.40 m long, has 800 turns, and carries 2.0 A. What is $B$ inside? (in T, to 3 significant figures)
How far from a 20 A wire is the field equal to $1.0 \times 10^{-4}$ T? (in m, to 3 significant figures)

Concept Check — Fields from Currents

Exit Check — Fields from Currents ✅

Sides parallel to $\vec{B}$ : No force ( $\theta = 0°$ )
Sides perpendicular to $\vec{B}$ : Force $F = BIa$ on each, but in opposite directions

These opposite forces create a couple — a net torque with zero net force.

The perpendicular distance between the two force lines (the lever arm) depends on the angle $\phi$ between the loop's normal and $\vec{B}$ :

$\tau = F \cdot b\sin\phi = BIa \cdot b\sin\phi = BIA\sin\phi$

where $A = ab$ is the area of the loop.

For a coil with $N$ turns:

$\boxed{\tau = NIAB\sin\phi}$

where $\phi$ is the angle between the magnetic dipole moment $\vec{\mu}$ (normal to the loop) and $\vec{B}$ .

Magnetic Dipole Moment

The magnetic dipole moment of a current loop is:

$\boxed{\vec{\mu} = NI\vec{A}}$

where:

$N$ = number of turns
$I$ = current
$\vec{A}$ = area vector (perpendicular to the loop, direction from right-hand rule: curl fingers in direction of current, thumb = $\vec{A}$ )

The magnitude is:

$\mu = NIA$

Torque in Terms of $\vec{\mu}$

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

$|\tau| = \mu B \sin\phi$

Equilibrium Positions

$\phi = 0°$ : $\vec{\mu} \parallel \vec{B}$ → → (lowest energy)

Potential Energy

$U = -\vec{\mu} \cdot \vec{B} = -\mu B \cos\phi$

Minimum energy at $\phi = 0°$ : $U = -\mu B$
Maximum energy at $\phi = 180°$ : $U = +\mu B$

Applications

Electric Motor

A motor uses the torque on a current loop to create continuous rotation:

Current flows through a coil in a magnetic field → torque rotates the coil
A commutator (split ring) reverses the current direction every half turn
This ensures the torque always acts in the same rotational direction
The coil spins continuously!

Without the commutator, the coil would just oscillate back and forth around the equilibrium position ( $\vec{\mu} \parallel \vec{B}$ ).

Galvanometer

A galvanometer measures small currents:

Current flows through a coil suspended in a magnetic field
The magnetic torque $\tau = NIAB\sin\phi$ is opposed by a spring
The coil rotates until $\tau_{\text{magnetic}} = \tau_{\text{spring}}$

The deflection angle is proportional to the current, making it a useful measuring device.

Key Difference

Motor: commutator allows continuous rotation
Galvanometer: spring limits rotation to a measurable deflection

Torque Concept Quiz ⚙️

Torque Calculation Drill 🔢

A rectangular coil has 50 turns, dimensions 8.0 cm × 5.0 cm, carries 2.0 A, and sits in a 0.30 T uniform magnetic field.

Magnetic dipole moment $\mu$ (in A·m²)
Maximum torque on the coil (in N·m)
Torque when $\vec{\mu}$ makes a 30° angle with $\vec{B}$ (in N·m)

Round all answers to 3 significant figures.

Exit Check — Torque and Dipoles ✅

Fields from Currents

Source	Formula
Long straight wire	$B = \mu_0 I/(2\pi r)$
Solenoid (inside)	$B = \mu_0 nI$
Ampère's law	$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$

Quantity	Formula
Radius	$r = mv/(qB)$
Period	$T = 2\pi m/(qB)$
Cyclotron frequency	$f = qB/(2\pi m)$
Velocity selector	$v = E/B$

Quantity	Formula
Dipole moment	$\mu = NIA$
Torque	$\tau = NIAB\sin\phi$
Potential energy	$U = -\mu B\cos\phi$

$\mu_0 = 4\pi \times 10^{-7} \text{ T·m/A}$ $e = 1.6 \times 10^{-19} \text{ C}$ $m_p = 1.67 \times 10^{-27} \text{ kg}$ $m_e = 9.11 \times 10^{-31} \text{ kg}$

⚠️ Common Mistakes on the AP Exam

Mistake 1: Wrong Hand

Using the left hand for the right-hand rule (for positive charges or conventional current). Left hand is only for negative charges.

Mistake 2: Confusing $1/r$ and $1/r^2$

Coulomb/Electric field: $F \propto 1/r^2$ , $E \propto 1/r^2$
Wire magnetic field: $B \propto 1/r$ (NOT $1/r^2$ !)

Mistake 3: Thinking Magnetic Force Does Work

The magnetic force is always perpendicular to velocity: $\vec{F} \perp \vec{v}$ . It changes but not . It does and cannot change kinetic energy.

Mistake 4: Forgetting sinθ

$F = qvB$ only when $\vec{v} \perp \vec{B}$ . The general formula is . If , .

Mistake 5: Period Depends on Speed

The cyclotron period $T = 2\pi m/(qB)$ is independent of speed. Students often think faster particles take less time — they don't. They travel a bigger circle at a proportionally higher speed.

Mistake 6: Parallel Wires Direction

Same-direction currents attract (not repel). This is counterintuitive — think of it as: the wires' fields reinforce between them, creating a pressure that pushes them together.

Right-Hand Rule Mastery Challenge 👋

Mixed scenarios — identify the correct direction or quantity.

📝 AP FRQ Preview

A common AP Physics 2 free-response question type:

Setup

A proton is launched horizontally into a region of uniform magnetic field pointing into the page.

Typical Parts

(a) Draw the path of the proton. Explain why it curves.

Answer: The proton follows a circular arc. The magnetic force $F = qvB$ acts perpendicular to the velocity, providing centripetal acceleration.

(b) Determine the radius of the circular path.

Answer: $r = mv/(qB)$ . Substitute given values.

(c) If the proton is replaced by an alpha particle ( $q = 2e$ , $m = 4m_p$ ) with the same speed, how does the radius change?

Answer: $r_{\alpha}/r_p = (4m_p \cdot qB)/(2e \cdot m_p B) \cdot (v/v) = 4m_p/(2m_p) \cdot (e/(e)) = 2$ . The alpha particle has twice the radius.

(d) Does the magnetic field do work on the proton? Justify.

Answer: No. The magnetic force is always perpendicular to velocity ( $\vec{F} \perp \vec{v}$ ), so . Speed and KE remain constant.

Pro Tips for FRQs

Always state the relevant equation before plugging in numbers
Explain directions using the right-hand rule explicitly
Justify "no work" by citing $\vec{F} \perp \vec{v}$

AP-Style Calculation Challenge 🎯

An electron ( $m = 9.11 \times 10^{-31}$ kg, $q = 1.6 \times 10^{-19}$ C) is accelerated from rest through a potential difference of 500 V, then enters a uniform 0.010 T magnetic field perpendicular to its velocity.

Speed of the electron after acceleration (in m/s, use scientific notation like 1.3e7)
Radius of the circular path in the magnetic field (in m, to 3 significant figures)
Cyclotron period (in s, use scientific notation like 3.6e-9)

Final Mastery Quiz — Magnetic Fields and Forces 🏆

Your thumb points in the direction of

\vec{F}

Electron (

-q

): reverse → force is into the page (

-z

)

Or use the conventional current direction (opposite to electron flow) with

q = +e

The result is a helix (spiral) along the field direction

stable equilibrium

$\phi = 180°$ :

\vec{\mu}

antiparallel to

\vec{B}

→

\tau = 0

→ unstable equilibrium (highest energy)

$\phi = 90°$ : maximum torque

Greater current → greater deflection → read on a scale

\vec{v} \parallel \vec{B}

W = \vec{F} \cdot \Delta\vec{r} = 0

When comparing particles, use ratios to avoid numerical errors

$+x$ (right)	$+y$ (up)	$+z$ (out of page)
$+x$ (right)	$+z$ (out of page)	$-y$ (down)
$+y$ (up)	$+x$ (right)	$-z$ (into page)
$+z$ (out of page)	$+x$ (right)	$+y$ (up)

Magnetic Fields and Forces

Magnetic Fields and Forces - Complete Interactive Lesson

Part 1: Magnetic Fields

🧲 Magnetic Fields — Sources and Properties

Sources of Magnetic Fields

1. Permanent Magnets

2. Moving Charges

3. Electric Currents

Key Insight

Magnetic Field Lines

Earth's Magnetic Field

Geographic vs. Magnetic Poles

Field Strength

Units of Magnetic Field

Part 2: Force on Moving Charges

⚡ Force on Moving Charges

The Magnetic Force on a Moving Charge

Part 3: Circular Motion in B Fields

🌀 Charged Particles in Magnetic Fields

Circular Motion in a Magnetic Field

Part 4: Force on Current-Carrying Wires

🔌 Force on Current-Carrying Wires

Force on a Straight Current-Carrying Wire

Part 5: Ampère's Law & Solenoids

🔄 Magnetic Field from Currents

Magnetic Field of a Long Straight Wire

Part 6: Torque on Current Loops

⚙️ Torque on Current Loops

Torque on a Rectangular Current Loop

Part 7: Synthesis & AP Review

🏆 Synthesis & AP Review

📋 Master Formula Sheet

Forces

Rules for Magnetic Field Lines

Bar Magnet Field Pattern

Critical Difference from Electric Fields

Source

Conversion

Typical Magnetic Field Strengths

The Permeability Constant

Key Properties

The Right-Hand Rule

For Positive Charges (q>0q > 0q>0)

For Negative Charges (q<0q < 0q<0)

Common Directions

Memory Aid

Force on Negative Charges

Example

Practical Importance

Key Observations

Cyclotron Period and Frequency

Remarkable Result

What if v⃗\vec{v}v has a component along B⃗\vec{B}B?

Mass Spectrometer

How It Works

The Separation

Velocity Selector

Setup

Condition for Straight-Line Motion

Key Features

Special Cases

Derivation from Charge Force

Direction of Force on a Wire

Example

Forces Between Parallel Wires

Same Direction Currents → Attract

Opposite Direction Currents → Repel

Force Per Unit Length

Definition of the Ampere

Direction: Right-Hand Rule for Wires

Key Features

Ampère's Law

When to Use Ampère's Law

Deriving the Wire Field

Solenoids

Field Inside an Ideal Solenoid

Derivation (Ampère's Law)

Key Properties

Forces on the Sides

The Torque

For Positive Charges ( $q > 0$ )

For Negative Charges ( $q < 0$ )

What if $\vec{v}$ has a component along $\vec{B}$ ?

Torque in Terms of $\vec{\mu}$

Mistake 2: Confusing $1/r$ and $1/r^2$