Chapter 5: Electrostatics and Magnetism

Pull a load of laundry from the dryer and you’ll notice socks clinging to shirts, crackling sounds when you peel pieces apart, and maybe a small spark when you touch the metal door. That’s not magic — it’s electric charge in action.

When two different materials rub together (like clothes tumbling in a dryer), electrons transfer from one surface to the other. One material ends up with extra electrons (net negative charge); the other is missing some (net positive charge). The resulting attraction between opposite charges is what makes your socks cling to your shirts.

This chapter is the foundation for everything in electrostatics, circuits, and magnetism. Get the basics of charge — what it is, how it moves, how things become charged — and the rest builds straightforwardly on top.

What Is Electric Charge?

Electric charge is a fundamental property of matter, just like mass. There are exactly two types: positive and negative. Protons carry positive charge; electrons carry negative; neutrons carry none. Like charges repel; opposite charges attract.

Quantization of Charge

Charge comes in discrete packets. You can’t have half an electron’s worth of charge. The smallest unit of free charge is the elementary charge:

The SI unit of charge is the coulomb (C). One coulomb is an enormous amount of charge — roughly $6.25 \times 10^{18}$ electrons. In everyday static electricity (like the laundry example), you’re typically dealing with nanocoulombs ($10^{-9}$ C) or microcoulombs ($10^{-6}$ C).

Conservation of Charge

Charge is never created or destroyed — only transferred. Rub a balloon on your hair: the balloon gains $-5$ nC of charge, your hair gains $+5$ nC. Total system charge is still zero. (Just like before.)

Conductors vs. Insulators

• Conductors (metals, saltwater, plasma) have electrons that can move freely throughout the material. Touch a charged conductor and the charge spreads over the entire surface. Metals conduct because their outer electrons are delocalized in a “sea” of shared electrons.
• Insulators (rubber, glass, plastic, dry wood) hold electrons tightly in place. Charge placed on an insulator stays where you put it — it doesn’t spread. This is why you can charge one end of a plastic rod and the other end stays neutral.
• Semiconductors (silicon, germanium) sit in between. Their conductivity can be tuned by adding impurities (doping) — that’s the basis of every chip in every electronic device. Not a major MCAT topic, but you should know the category exists.

Three Methods of Charging

Charging by Friction (Triboelectric)

Rub two different materials together → electrons transfer from one to the other. Which material gains electrons depends on the triboelectric series. Classic example: rubbing a glass rod with silk pulls electrons off the glass (glass → positive, silk → negative). This is also what’s happening when you shuffle across carpet on a dry day and shock yourself on a doorknob — friction with the carpet has charged you.

Charging by Contact (Conduction)

Touch a charged object to a neutral conductor. Charge flows between them until they reach the same potential. Both objects end up with the same sign of charge. If a negatively charged rod touches a neutral metal sphere, electrons hop from rod to sphere, and now both are negative.

Charging by Induction

Bring a charged object near (but not touching) a neutral conductor. Charges in the conductor rearrange — opposite charges drift toward the nearby object, like charges drift to the far side. Now ground the conductor (connect it to Earth) while the charged object is still nearby — the repelled charges drain off into the ground. Remove the ground, then remove the charging object, and the conductor is left with a net charge opposite to the original object.

Polarization (of Insulators)

Even insulators respond to nearby charges. When a charged rod is brought near a neutral insulator, the electrons within each atom or molecule shift slightly, creating tiny induced dipoles. The material isn’t truly charged, but the near side becomes slightly attracted to the rod. This is why a charged balloon sticks to a neutral wall, why a charged comb attracts small bits of paper, and why dust collects on TV screens.

Hold two magnets in your hands. Push two north poles toward each other and you feel resistance that gets dramatically stronger as they get close. Flip one magnet around and they snap together so hard they pinch your fingers. Now imagine the same thing for electric charges instead of magnets — same intuition, but charges can be positive or negative, and the force between them follows a precise mathematical rule discovered by Charles-Augustin de Coulomb in 1785.

That rule is Coulomb’s law, and it governs almost every electrostatic problem you’ll see on the MCAT. The structure of the equation looks almost identical to Newton’s law of gravitation — which is a useful coincidence to remember, because if you know one, the other is mostly mechanical.

The Equation

The force is attractive when the charges have opposite signs (one +, one −) and repulsive when they have the same sign (both + or both −). The sign of the product $q_1 q_2$ tells you which: negative product → attraction; positive → repulsion.

The Inverse-Square Law

The single most important feature of Coulomb’s law is the $r^2$ in the denominator. Force drops off fast with distance:

• Double the distance → force drops to $\frac{1}{4}$.
• Triple the distance → force drops to $\frac{1}{9}$.
• Halve the distance → force quadruples.

Coulomb’s Law vs. Newton’s Law of Gravitation

Coulomb’s law looks almost identical to gravity:

The electric force is much stronger than gravity. The electrostatic force between a proton and electron in a hydrogen atom is about $10^{39}$ (a 1 followed by 39 zeros) times stronger than the gravitational force between them. Gravity only dominates on cosmic scales (planets, stars, galaxies) because most macroscopic objects are electrically neutral — positive and negative charges cancel out, so the residual electric force across the room is tiny. Mass never cancels — gravity always adds up.

Superposition of Forces

When more than two charges are present, the net force on any one charge is the vector sum of all the individual Coulomb forces acting on it. Calculate the force from each other charge separately, then add the force vectors. This is the principle of superposition — and it’s what lets you handle multi-charge problems with the same simple two-body equation.

For the MCAT, superposition problems usually involve only two or three charges arranged symmetrically. Use symmetry to simplify: if two charges exert equal forces at right angles, the net force is along the diagonal with magnitude $F\sqrt{2}$. Review vector addition if perpendicular force combinations feel rusty.

Common MCAT Problem Types

• Distance-change problems: “If the distance is halved, what happens to the force?” Apply inverse-square: $F$ × 4.
• Charge-change problems: “If both charges double, what happens to the force?” $F \propto q_1 q_2$, so doubling both → $F$ × 4.
• Three-charge equilibrium: “Where should a third charge be placed so the net force on it is zero?” Find the point where forces from the two fixed charges are equal and opposite.

In the previous section, you learned how to compute the force between two specific charges. But here’s the deeper question: what is a charged object doing to the space around it, even before another charge shows up?

That’s where the electric field comes in. It’s one of the most important ideas in physics, and once you internalize it, electromagnetism becomes much more intuitive.

What Is an Electric Field?

Compare to gravity. Earth creates a gravitational field around itself. Even when nothing is falling, the field is there — sitting silently in space. Drop a ball into that field and the ball feels a force. The field existed before the ball arrived.

An electric field works the same way. A charged object creates an “influence zone” around itself. Bring another charge into that zone and it feels a force. The field was there all along; the second charge just reveals it.

The Formula: Force Per Unit Charge

We define the electric field as the force a positive test charge would experience, divided by that charge:

Direction matters. The electric field points the way a positive charge would be pushed (this is a convention):

• Near a positive source charge: $E$ points away (positive charges repel each other).
• Near a negative source charge: $E$ points toward it (positives are attracted).

Field from a Point Charge

If you know Coulomb’s law ($F = kq_1 q_2/r^2$), you can derive the field formula. Force on a test charge $q_2$ near source charge $q_1$ is $F = kq_1 q_2/r^2$. Divide by $q_2$:

Notice what this formula does not contain: the test charge. The field is a property of space created by the source charge. It exists whether or not another charge is there to feel it.

Visualizing Fields: Field Lines

Electric fields are invisible, so how do we picture them? Physicists use field lines — imaginary lines drawn throughout space showing which direction a positive test charge would be pushed at every point.

The Rules of Field Lines

1. Lines point away from positive charges (where positives would be pushed away).
2. Lines point toward negative charges (where positives would be attracted).
3. Lines never cross (every point in space has only one field direction).
4. Line density = field strength. Where lines are packed close together, the field is strong.

Multiple Charges: Superposition

What if several charges create fields at the same point? They don’t interfere or cancel in any complicated way — fields add as vectors. This is the principle of superposition.

How to use it:

1. Calculate the field from charge #1 at the point of interest (magnitude and direction).
2. Calculate the field from charge #2 at the same point.
3. Add the vectors. Same direction → magnitudes add. Opposite directions → subtract. Perpendicular → use the Pythagorean theorem.

Each charge creates its field as if the others weren’t there.

The Most Important Setup: Parallel Plates

Take two large metal plates, put positive charge on one and negative charge on the other, set them facing each other. What’s the field between them?

The field is uniform — same strength and direction everywhere between the plates. Field lines are straight, parallel, and evenly spaced, running from the positive plate to the negative plate.

This setup is special because most electric fields vary with distance ($E = kq/r^2$). Between parallel plates, the field is constant. That makes the math dramatically simpler for many problems.

Why this matters: a uniform field means constant force on any charge between the plates. Constant force → constant acceleration. Constant acceleration → you can use the kinematic equations you already know. Many MCAT problems involve charged particles accelerating between plates — they solve like projectile-motion problems with the electric force playing the role of gravity.

Gauss’s Law: The Conceptual Version

The MCAT only tests Gauss’s law conceptually — no calculations. But the idea is elegant and worth understanding.

The concept: Draw any imaginary closed surface (think a balloon) around some charges. Count how many field lines poke through that surface. Gauss’s law says: that count depends only on how much charge is inside the surface.

It does not depend on:

• The shape of the surface.
• Where inside the surface the charges sit.
• Charges outside the surface.

In §5.3 you learned the rules of electric fields. Now it’s time to apply those rules to the specific charge arrangements the MCAT loves: single point charges, pairs of charges, dipoles, and parallel plates. Each has a characteristic field pattern you should recognize on sight.

The strategy is always the same: calculate $E$ from each source charge, get the direction (away from + charges, toward − charges), and add the vectors.

Electric Field from a Single Point Charge

The field radiates symmetrically outward from a positive point charge and inward toward a negative one. Field strength falls off as $1/r^2$ — same inverse-square pattern as Coulomb’s law.

Superposition with Multiple Charges

When several point charges are present, the field at any location is the vector sum of the individual fields from each charge. The recipe:

1. Pick the point where you want to know the field.
2. Calculate $E$ from each source charge using $E = kq/r^2$.
3. Determine the direction of each individual field (away from + charges, toward − charges).
4. Add the vectors.

Two Equal Positive Charges

Place two identical positive charges on a horizontal line. At the midpoint between them, the fields from each charge point in opposite directions and cancel completely. $E = 0$ at the midpoint. Above or below the midpoint, the fields partially add, producing a net upward or downward field perpendicular to the line.

But what if the charges aren’t equal?

Drag the q1 and q2 sliders and watch the arrow pattern reorganize around the two charges. The green ring marks the null point where the fields cancel exactly. Make the charges equal, then unequal, then flip one sign, and notice where the null point moves (and when it leaves the picture entirely).

Two Opposite Charges (Dipole Configuration)

Place a positive charge on the left, a negative charge on the right. At the midpoint, both fields point in the same direction (from + toward −), so they add. The field at the midpoint is strong, not zero. (More on this in §5.8 — Electric Dipoles.)

The Dipole Field Pattern

A dipole is a positive and negative charge of equal magnitude separated by a small distance. The field pattern has a distinctive shape:

• Between the charges: field points from + to −.
• Along the axis (line connecting the charges): field points away from + and toward −, creating a continuous flow from positive to negative.
• Along the perpendicular bisector: field points antiparallel to the dipole moment.
• Far from the dipole: field falls off as $1/r^3$ — faster than a single point charge ($1/r^2$). Dipoles cancel each other out at distance much faster than a lone charge would.

You don’t need to calculate dipole fields on the MCAT, but you should recognize the pattern in a diagram: field lines emerge from the positive end and curve around to enter the negative end, forming closed-looking loops.

Electric Field Between Parallel Plates

Two large parallel conducting plates with equal-and-opposite charge produce the simplest possible field: uniform and constant between the plates.

This uniform field is the workhorse of MCAT electrostatics. A charged particle between the plates experiences a constant force ($F = qE$) → constant acceleration → projectile-motion-style problems. Charge between plates is geometrically the same as a ball thrown in gravity — and you solve it with the same kinematic equations.

Conductors in Electrostatic Equilibrium

Three facts about conductors at electrostatic equilibrium that often appear on the MCAT:

1. The electric field inside the conductor is exactly zero.
2. All excess charge sits on the outer surface.
3. The field just outside the surface is perpendicular to the surface.

All three follow from charges being free to move. If there were any field inside the conductor, the charges would move in response — until there wasn’t. Equilibrium = no field inside.

If you’ve studied gravitational potential energy, you already understand the core idea. Hold a ball above the ground — it has stored energy. Release it — the stored energy converts into motion as the ball falls.

Electric potential energy is the same concept, just with electric charges instead of gravity. When two charges are arranged in a particular configuration, there’s energy stored in that arrangement. Move the charges around, and you’re either storing more energy or releasing it. Same logic, different force — and conservation of energy still applies.

Building the Formula from What You Know

You learned in Section 5.2 that the force between two charges follows Coulomb’s law:

$\displaystyle F = \dfrac{kq_1 q_2}{r^2}$

Potential energy and force are closely related. In fact, if you know how force changes with distance, you can figure out potential energy (this involves calculus, which you don’t need for the MCAT). The result is:

Understanding the Sign: Positive vs. Negative Energy

The sign of U tells you something crucial about the system. This is where many students get confused, so let’s build intuition.

Like Charges: Positive Potential Energy

When both charges have the same sign (both positive, or both negative), the product $q_1 \times q_2$ is positive, so $U$ is positive.

What does positive energy mean physically? Think about it: like charges repel. To push them close together, you have to fight against that repulsion - you have to do work on the system. That work gets stored as potential energy, just like compressing a spring stores energy.

Unlike Charges: Negative Potential Energy

When the charges have opposite signs (one positive, one negative), the product $q_1 \times q_2$ is negative, so $U$ is negative.

What does negative energy mean? Unlike charges attract. They naturally “want” to come together - you don’t have to push them. In fact, you’d have to do work to pull them apart. The system is in an energy “hole” - it would take energy input to escape.

Quick Reference: What Does the Sign Mean?

How Does Energy Change as Charges Move?

The electric force is what physicists call a conservative force - the same category as gravity. This means we can track energy using potential energy, and total mechanical energy is conserved (as long as no other forces are doing work).

If you’ve studied conservation of energy, you know the key equation:

$\displaystyle K_1 + U_1 = K_2 + U_2$

(Initial kinetic energy + initial potential energy = final kinetic energy + final potential energy)

This means:

• When $U$ decreases, $K$ increases (potential energy converts to speed)
• When $U$ increases, $K$ decreases (speed converts to potential energy)

What Makes $U$ Increase or Decrease?

$U$ increases when you fight the natural tendency:

• Pushing like charges closer together (they want to repel)
• Pulling unlike charges apart (they want to attract)

$U$ decreases when you go with the natural tendency:

• Letting like charges fly apart (they naturally repel)
• Letting unlike charges come together (they naturally attract)

Force vs. Energy: Why One Has $r^2$ and the Other Has $r$

Here’s something that often confuses students: the force equation has $r^2$ in the denominator, but the energy equation only has $r$.

This means at large distances, you might barely feel any force between two charges (force has dropped to nearly zero), but there can still be significant potential energy stored in the system.

Classic Problem Type: Distance of Closest Approach

Here’s a common MCAT problem setup:

A positive charge is fired toward another positive charge (which is held fixed). How close does the moving charge get before it stops and bounces back?

Solution approach:

1. At the start: the moving charge has kinetic energy ($K_1 = \dfrac{1}{2}mv^2$) and some potential energy ($U_1$)
2. At closest approach: all the kinetic energy has converted to potential energy. The charge momentarily stops ($K_2 = 0$) before bouncing back
3. Apply conservation of energy: $K_1 + U_1 = U_2$

If the charges start very far apart, $U_1 \approx 0$, so:

$\displaystyle \dfrac{1}{2}mv^2 = \dfrac{kq_1 q_2}{r_{\text{closest}}}$

Solve for $r_{\text{closest}}$.

Energy in Uniform Electric Fields

When you have a uniform field (like between parallel plates), there’s a simpler formula for potential energy change:

$\displaystyle \Delta U = qEd$

where:

• $q$ = the charge being moved
• $E$ = the electric field strength
• $d$ = distance moved along (or against) the field direction

This is the electrical version of $\Delta U = mgh$ in gravity. Compare them:

You’ve seen voltage written on batteries: 1.5 V on a AA, 9 V on a transistor battery, 12 V on a car battery. But what does that number actually mean? And how is “voltage” different from the “potential energy” we covered in the previous section?

This is one of the trickiest distinctions in physics, and the MCAT loves to test it. The good news: there’s a single mountain-and-hiker analogy that makes it click forever. Let’s break it down.

The Big Idea: Voltage Is Like Elevation

Imagine you are hiking and you look at a topographic map. The map shows the elevation at every point on the landscape - 500 meters here, 800 meters there - regardless of whether anyone is actually standing at those spots. The elevation is a property of the location, not of any particular hiker.

Voltage (electric potential) is the electrical version of elevation.

Just as a map can tell you the height at every point on a mountain, voltage tells you the “electrical height” at every point in space around a charged object. This electrical height exists whether or not any other charge is sitting there to experience it.

Building Up from What You Know

In the previous section, you learned about electric potential energy (U) - the energy stored when two charges are positioned relative to each other. That formula was:

$\displaystyle U = \dfrac{kq_1 q_2}{r}$

(energy depends on BOTH charges)

Now we want to separate out the contribution of just ONE charge - the source charge that creates the “electrical landscape.” We define:

$\displaystyle V = \dfrac{kq}{r}$

(voltage depends on only the SOURCE charge)

The voltage $V$ tells you what the source charge has done to space. If you later bring in a second charge, you can easily find the potential energy by multiplying:

$\displaystyle U = qV$

(energy = charge × voltage)

Key Properties of Voltage

• Voltage is a scalar - it has magnitude but no direction. Unlike electric field (a vector), you don’t need to worry about components or angles.
• Voltage is created by source charges - it is a property of a location in space
• Voltage is positive near positive charges and negative near negative charges
• Voltage decreases with distance from a positive charge (since V = kq/r, bigger r means smaller V)
• Voltage is zero at infinity - we define our reference point there

The Critical Distinction: V vs. U

This is the most important concept in this section. The MCAT tests it repeatedly because students often confuse these two quantities.

The Analogy That Makes It Click

Think about a mountain:

1. The mountain’s height (elevation) exists whether or not anyone is climbing it. A peak might be 4,000 meters high - that’s a fact about the location itself.

2. A climber’s potential energy depends on both the mountain’s height AND the climber’s mass. A 100 kg climber at 4,000 m has twice the gravitational PE of a 50 kg climber at the same height.

Now translate this to electricity:

1. Voltage (V) is like the mountain’s height. It’s a property of the location, created by source charges. It exists whether or not any test charge is there.

2. Potential energy (U) depends on both the voltage AND the charge you place there. U = qV - the energy depends on the “height” (V) and the “mass” (q).

Side-by-Side Comparison

Potential Difference: What Batteries Actually Measure

When you see “9V” on a battery, what does that mean? It’s the potential difference - the difference in voltage between the two terminals.

In most physics problems, what matters isn’t the absolute voltage at a point, but the difference in voltage between two points:

$\displaystyle \Delta V = V_{\text{final}} - V_{\text{initial}}$

This difference tells you how much energy per coulomb is available to move charges from one point to another.

How Charges Move in Voltage Differences

Think about what you learned about potential energy and conservation of energy. When an object moves from high potential energy to low potential energy, that lost PE becomes kinetic energy.

The same happens with charges:

• A positive charge naturally moves from high voltage to low voltage (like a ball rolling downhill)
• A negative charge naturally moves from low voltage to high voltage (attracted toward the positive region)

Energy from Voltage Differences

The change in potential energy when a charge moves through a voltage difference is:

$\displaystyle \Delta U = q \Delta V$

If a positive charge ($q > 0$) moves from high $V$ to low $V$ ($\Delta V < 0$), then $\Delta U$ is negative - the charge loses potential energy. That energy becomes kinetic energy. The charge speeds up.

How Voltage Connects to Electric Field

You learned in Section 5.3 that electric fields point from positive charges toward negative charges. Here’s another way to think about it:

Electric fields point from high voltage to low voltage.

This makes sense: the field points in the direction a positive charge would naturally move, which is “downhill” on the voltage landscape.

The mathematical relationship in a uniform field (like between parallel plates) is:

$\displaystyle E = \dfrac{V}{d}$

where $V$ is the voltage across the plates and $d$ is the distance between them. The field strength tells you how quickly the voltage changes with distance.

The Electron-Volt: A Convenient Energy Unit

When dealing with single electrons or atoms, the joule is an inconveniently large unit. It’s like measuring your commute in millimeters.

The electron-volt (eV) is a more practical unit at atomic scales:

$1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}$

An electron-volt is the energy an electron gains when it accelerates through a voltage difference of 1 volt. Since the electron has charge $q = 1.6 \times 10^{-19}$ C:

$\displaystyle \text{Energy} = qV = (1.6 \times 10^{-19} \text{ C})(1 \text{ V}) = 1.6 \times 10^{-19} \text{ J} = 1 \text{ eV}$

You will see this unit again in atomic physics and nuclear physics.

Topographic maps use contour lines to connect points of equal elevation. Walk along a contour line and you stay at the same height — no climbing, no descending, no work against gravity.

Equipotential lines are the electrical version. They connect points of equal voltage. Move a charge along an equipotential and no work is done, because the charge stays at the same “electrical altitude.” Move it across equipotentials, and work happens. The whole topographic-map analogy from the previous section keeps paying off here.

What Are Equipotential Lines?

An equipotential line (in 2D) or equipotential surface (in 3D) connects all points that share the same electric potential V. Key properties:

• No work is done moving a charge along an equipotential. Since W = -qΔV and ΔV = 0 along an equipotential, W = 0. The charge neither gains nor loses energy.
• Equipotential lines are always perpendicular to electric field lines. This is a fundamental geometric relationship. The field points in the direction of steepest voltage drop, which is always at right angles to lines of constant voltage.
• Equipotential lines never cross. A point in space has one and only one value of V, so two different equipotential lines cannot intersect.

Equipotential Patterns for Common Configurations

Point Charge

The equipotential surfaces around a single point charge are concentric spheres (or circles in 2D). Each sphere has a constant V = kq/r. The spheres are more closely spaced near the charge (where V changes rapidly) and more widely spaced far away.

The field lines radiate outward (for +) or inward (for -), crossing the spherical equipotentials at right angles.

Parallel Plates

Between parallel plates, the equipotential surfaces are flat planes parallel to the plates (straight lines in 2D). They are evenly spaced because the field is uniform - the voltage drops by the same amount for each equal step across the gap.

If the positive plate is at +100 V and the negative plate is at 0 V with 10 cm separation, the equipotential lines at 2 cm intervals would be at +80 V, +60 V, +40 V, +20 V, evenly spaced across the gap.

Dipole

The equipotential pattern around a dipole is more complex. The equipotential surfaces are distorted, but the perpendicular bisector of the dipole (the line halfway between the two charges, perpendicular to the dipole axis) is a special equipotential: V = 0 along this entire surface.

The Perpendicular Rule in Practice

The perpendicular relationship between field lines and equipotentials is one of the most useful shortcuts on the MCAT:

• If you know the equipotential pattern, you can sketch the field lines (draw them perpendicular to the equipotentials, pointing from high V to low V).
• If you know the field lines, you can sketch the equipotentials (draw them perpendicular to the field lines).
• Where equipotential lines are closely spaced, the field is strong (voltage changes rapidly over a short distance).
• Where equipotential lines are widely spaced, the field is weak.

Conductors as Equipotential Surfaces

The entire surface of a conductor in electrostatic equilibrium is an equipotential surface. The interior is also at the same potential. This follows from the fact that E = 0 inside a conductor - if there is no field, there is no voltage change, so the entire conductor sits at a single potential.

Think about a battery. Positive end on top, negative end on the bottom. Now imagine shrinking that battery down to the size of a molecule — you’d have something very close to what physicists call a dipole.

“Dipole” literally means “two poles” — like a battery has two ends, or a bar magnet has a north and a south pole. In electricity, a dipole is any object with a positive side and a negative side separated by some distance.

This idea matters enormously in biology. Water molecules are dipoles. The amino acids in your proteins are dipoles. The phospholipids in every cell membrane are dipoles. Understanding how dipoles behave explains why salt dissolves in water, why oil and water don’t mix, why proteins fold the way they do, and how your nerve cells send signals.

Breaking Down the Concept: What Makes a Dipole?

Let’s build this up step by step.

You already know from Section 5.1 that objects can have positive or negative charge. Now imagine you have two small charged objects:

• One has a charge of +q (some amount of positive charge)
• The other has a charge of -q (the same amount, but negative)

If you hold these two charges apart at some distance d, congratulations - you’ve created a dipole.

The Dipole Moment: How “Strong” Is the Dipole?

Not all dipoles are created equal. A dipole with large charges far apart is “stronger” than one with tiny charges close together. We capture this with a quantity called the dipole moment.

The dipole moment tells you two things:

1. How much charge separation there is (bigger charges and bigger distances both make it larger)
2. Which direction the dipole points (from negative toward positive, by convention)

Why does the formula make sense? If you double the charge (bigger q), the dipole is twice as “strong.” If you double the separation distance (bigger d), the dipole is also twice as strong. The dipole moment captures both effects by multiplying them together.

What Happens When You Put a Dipole in an Electric Field?

This is where dipoles get interesting. Remember from Section 5.3 that an electric field pushes positive charges in one direction and negative charges in the opposite direction.

Now think about what happens to our dipole (with its positive end and negative end) when we place it in an electric field:

• The positive end gets pushed in the direction of the field
• The negative end gets pushed against the field direction

If the dipole is sitting at an angle to the field, these two pushes create a twist - what physicists call a torque. The dipole rotates until it lines up with the field.

The Torque Formula

How strong is this twisting force? It depends on three things:

1. How strong the dipole is (the dipole moment p)
2. How strong the electric field is (E)
3. What angle the dipole makes with the field (θ)

This makes intuitive sense. When the dipole is already aligned with the field, there’s no reason for it to rotate - the positive end is already pointing the way the field wants to push it. But when the dipole is sideways to the field, it experiences maximum twist.

Energy Stored in a Dipole’s Orientation

When you rotate a dipole against the field (fighting the twist), you’re doing work on it - storing energy. When you let it rotate back into alignment, that stored energy is released.

This is the same idea as gravitational potential energy: lift a ball above the ground and you store energy; let it fall and that energy converts to motion.

Understanding the Energy Formula

Let’s make sense of this formula by checking extreme cases:

When θ = 0° (dipole aligned with field):

• cos 0° = 1
• U = -pE(1) = -pE (the most negative value possible)
• This is the lowest energy state - the dipole is “relaxed”

When θ = 90° (dipole perpendicular to field):

• cos 90° = 0
• U = -pE(0) = 0
• This is a middle energy state

When θ = 180° (dipole pointing opposite to field):

• cos 180° = -1
• U = -pE(-1) = +pE (the most positive value possible)
• This is the highest energy state - the dipole is “fighting” the field

Summary Table: How Angle Affects a Dipole

Why Dipoles Matter in Biology

Now that you understand what dipoles are and how they behave, let’s see why this matters for the MCAT and for understanding life itself.

Water: The Most Important Dipole

Water ($\text{H}_2\text{O}$) is a dipole, and this single fact explains an enormous amount of biology and chemistry.

Here’s why water is a dipole: Oxygen atoms attract electrons more strongly than hydrogen atoms do (oxygen is more “electronegative” - see bonding and electronegativity for more). In a water molecule, the shared electrons spend more time near the oxygen, giving it a partial negative charge (δ-). The hydrogen atoms, having lost some electron density, end up with partial positive charges (δ+).

The result: one end of the water molecule is slightly negative, the other end is slightly positive. That’s a dipole.

Dipoles in Proteins and Membranes

The same principle extends throughout biology:

• Amino acids (the building blocks of proteins) often have polar side chains - parts of the molecule that act as dipoles. When a protein folds, these dipoles interact with each other and with water, helping determine the protein’s final shape.

• Cell membranes are made of phospholipids, which have a polar “head” (a dipole that interacts with water) and nonpolar “tails” (not dipoles, so they avoid water). This is why membranes form the way they do - the polar heads face the watery environment while the nonpolar tails tuck inside.

Stick a compass on a table and the needle swings to point north. Hold a strong magnet nearby and the needle snaps toward the magnet instead. The compass is responding to magnetic fields — invisible vector fields that exert forces on moving charges and magnetic materials.

Magnetic fields differ from electric fields in one fundamental way: while electric fields can be produced by stationary charges, magnetic fields are fundamentally tied to motion — moving charges, electric currents, or the intrinsic “spin” of electrons. No motion, no magnetism. (The MRI machine in a hospital generates its enormous magnetic field by running massive electric currents through superconducting coils.)

Magnetic Field Lines

Magnetic field lines describe the direction and strength of the magnetic field, following rules similar to electric field lines:

• Outside a bar magnet, field lines emerge from the north pole and curve around to enter the south pole.
• Inside the magnet, lines run from south to north (they form continuous closed loops - they never start or stop).
• Field lines never cross.
• The density of field lines represents field strength. Lines are densest near the poles.

Sources of Magnetic Fields

Bar Magnets

A bar magnet produces the classic dipole field pattern. Every magnet has both a north and south pole. Break a magnet in half and you get two smaller magnets, each with its own north and south pole. There’s no such thing as an isolated magnetic pole (magnetic monopole).

Current-Carrying Wires

A straight wire carrying current produces circular magnetic field lines that loop around the wire. The direction follows the right-hand rule.

The field strength around a straight wire decreases with distance from the wire (proportional to $1/r$, not $1/r^2$). You won’t need this formula for the MCAT, but you should know qualitatively that the field is strongest close to the wire.

Current Loops and Solenoids

Bend a current-carrying wire into a loop and the magnetic field looks like that of a tiny bar magnet. Stack many loops together into a coil (solenoid) and you get a strong, nearly uniform field inside. This is how electromagnets work - and it’s the basis of MRI machines, electric motors, and speakers.

Paramagnetism vs. Diamagnetism

The AAMC specifically lists these two forms of magnetism. Both describe how materials respond to an external magnetic field.

Paramagnetism: Materials with unpaired electrons (like iron, aluminum, and oxygen). The unpaired electrons act like tiny magnets that weakly align with an external field, causing the material to be slightly attracted. When the external field is removed, the alignment is lost. Think of paramagnetic atoms as compass needles that align with the field but have no memory.

Diamagnetism: Materials with all electrons paired (like copper, gold, water, and most organic molecules). An external field induces a tiny opposing magnetic moment in each atom, causing the material to be very weakly repelled. Diamagnetism is present in all materials but is extremely weak and usually masked by paramagnetism or ferromagnetism when those are present.

Earth’s Magnetic Field

The Earth itself acts like a giant bar magnet. The geographic north pole is near the magnetic south pole (which is why the north-seeking end of a compass points toward geographic north - it’s attracted to the magnetic south pole). The Earth’s field protects us from charged particles in the solar wind by deflecting them toward the poles, producing auroras.

Here’s something strange: put a charged particle in a magnetic field and nothing happens. The particle just sits there. No force, no motion, no nothing.

Start that particle moving, though, and a force suddenly appears — and it acts in a bizarre direction. Not along the field. Not along the motion. Sideways to both.

This counterintuitive behavior is the essence of magnetic forces on charges. Once you’ve got the right-hand rule down and you’ve seen a few worked examples, it stops feeling weird and starts feeling consistent. Let’s build it up step by step.

The Key Insight: Motion Required

Compare electric and magnetic forces:

| Type | Does a stationary charge feel a force? |
|------|---------------------------------------|
| Electric force | Yes - just place it in an electric field |
| Magnetic force | No - the charge must be moving |

Why the difference? At a deep level, electric and magnetic fields are two aspects of the same phenomenon (electromagnetism), but for the MCAT, just remember: magnetic forces only act on moving charges.

The Magnetic Force Formula

This formula should remind you of the torque on a dipole (τ = pE sin θ). The sin θ factor keeps appearing because it captures “how much of the motion/orientation is perpendicular to the field.”

The Weirdest Part: The Force Is Sideways

Here’s what trips up most students: the magnetic force doesn’t push the charge in the direction of the field, and it doesn’t push along the direction of motion. It pushes perpendicular to both.

Finding the Direction: The Right-Hand Rule

So the force is perpendicular to both velocity and field. But which perpendicular direction? There are two options (up or down, left or right, etc.). The right-hand rule tells you which.

Step-by-Step Instructions

1. Hold up your right hand (this is important - not your left!)
2. Point your fingers in the direction the charge is moving (velocity v)
3. Curl your fingers toward the direction of the magnetic field (B)
4. Your thumb now points in the direction of the force (F) - but ONLY for positive charges

For negative charges (like electrons): Do the right-hand rule, then flip the direction. The force on a negative charge is opposite to what the right-hand rule gives.

Practice Example

A proton moves to the right in a magnetic field pointing into the page.

1. Point fingers right (velocity direction)
2. Curl fingers into the page (field direction)
3. Thumb points… up!

The force on the proton is upward.

Launch the proton in the simulation below and compare the red force arrow with your prediction. Then switch to an electron, or flip the field into the page, and watch the curve reverse direction. Adjust speed and field strength and note what each one does to the size of the circle.

Circular Motion in a Magnetic Field

Here’s a beautiful consequence of the sideways force: if a charged particle enters a magnetic field moving perpendicular to the field, it moves in a circle.

Why? Think about it:

• The force is always perpendicular to the velocity (sideways)
• A force perpendicular to motion doesn’t speed up or slow down the object - it just turns it
• The force keeps turning the particle, always perpendicular
• What shape has constant turning? A circle!

This is exactly like circular motion from centripetal force. The magnetic force provides the centripetal force that keeps the particle moving in a circle.

Finding the Radius

Since the magnetic force IS the centripetal force, we can set them equal:

Magnetic force = Centripetal force

$\displaystyle qvB = \dfrac{mv^2}{r}$

Solve for $r$:

What does this formula tell us?

• Heavier particles (larger $m$) → bigger circles
• Faster particles (larger $v$) → bigger circles
• More charge (larger $q$) → tighter circles
• Stronger field (larger $B$) → tighter circles

This is how mass spectrometers work: particles with different masses curve differently, allowing them to be separated and identified.

The “No Work” Rule

This is a crucial conceptual point that the MCAT loves to test.

Remember from work and energy that work is only done when a force has a component along the direction of motion:

$\displaystyle W = Fd \cos\theta$

For magnetic forces, the force is always perpendicular to the velocity. That means $\theta = 90°$, and $\cos 90° = 0$. So:

$\displaystyle W = Fd \times 0 = 0$

Velocity Selector: Combining Electric and Magnetic Forces

What if you want only particles traveling at a specific speed to pass through? Use a velocity selector.

Here’s the setup: create both an electric field AND a magnetic field in the same region, oriented perpendicular to each other. A charged particle enters and feels two forces:

1. Electric force: $F_E = qE$ (pushes in one direction, let’s say up)
2. Magnetic force: $F_B = qvB$ (pushes in the opposite direction, down)

For particles at exactly the right speed, these forces cancel:

$\displaystyle qE = qvB$

Solving for $v$:

$\displaystyle v = \dfrac{E}{B}$

Particles traveling at this speed feel no net force and go straight through. Particles traveling faster or slower get deflected.

Current in a wire is just a large number of moving charges traveling together through a conductor. Since a magnetic field exerts a force on each moving charge, it also exerts a force on the wire as a whole.

This is the principle that makes electric motors spin, speakers vibrate, and MRI machines work. Every time you turn on a fan, hear sound from headphones, or watch an electric car accelerate, the magnetic-force-on-a-current-carrying-wire equation is doing its work. The physics is a straightforward extension of the single-charge Lorentz force.

Force on a Straight Current-Carrying Wire

This equation comes directly from F = qvB sin θ, applied to all the moving charges in the wire at once. The product of charge and velocity for all the charges becomes current times length (IL replaces qv).

Direction: Right-Hand Rule (FBI)

The direction of the force on the wire follows the same right-hand rule as for a single moving charge:

1. Point your fingers in the direction of conventional current (I).
2. Curl your fingers toward the magnetic field (B).
3. Your thumb points in the direction of the force (F).

The Motor Principle

An electric motor is simply a current-carrying loop of wire placed in a magnetic field. The magnetic force pushes one side of the loop up and the other side down, creating a torque that spins the loop. A commutator reverses the current direction every half-turn so the torque always pushes in the same rotational direction.

The torque on a rectangular current loop in a magnetic field is:

$\displaystyle \tau = nIAB \sin\theta$

where $n$ = number of turns, $I$ = current, $A$ = area of the loop, $B$ = magnetic field, and $\theta$ = angle between the field and the normal to the loop. This is the same $\sin\theta$ pattern as torque on a dipole.

Forces Between Parallel Wires

Two parallel wires carrying current exert forces on each other through their magnetic fields. Each wire creates a magnetic field, and the other wire sits in that field and experiences a force.

Same-direction currents attract. The magnetic field from wire 1 pushes wire 2 toward wire 1, and vice versa.

Opposite-direction currents repel. The magnetic field from wire 1 pushes wire 2 away, and vice versa.

This might seem counterintuitive (same-direction currents attract, while same-sign charges repel), but you can verify it with the right-hand rule. First, find the direction of the magnetic field from wire 1 at the location of wire 2. Then find the force on wire 2 using F = BIL.

A Note on Units

The tesla (T) is the SI unit of magnetic field. It can be expressed in several equivalent ways:

$1 \text{ T} = 1 \text{ kg/(A·s}^2\text{)} = 1 \text{ N/(A·m)} = 1 \text{ V·s/m}^2$

The most useful equivalence for problem-solving is 1 T = 1 N/(A·m), which you can verify by checking that $F = BIL$ gives newtons when $B$ is in tesla, $I$ is in amps, and $L$ is in meters.

In 1831, Michael Faraday discovered something that quite literally changed the world: a changing magnetic field can create an electric current — without any battery or power source. Move a magnet through a coil of wire and a current flows. Stop moving the magnet and the current stops. Move it the other way and the current reverses.

This is electromagnetic induction, and it’s the principle behind every electrical generator, every transformer in every power line, every induction cooktop, every wireless phone charger, and every MRI machine. Your wall outlets exist because Faraday figured this out.

If you’ve studied electric fields and magnetic fields, you know that these fields can exert forces on charges. Induction goes a step further: a changing magnetic field can actually create an electric field, which then pushes charges around a circuit.

Magnetic Flux

Before we can talk about induction, we need the concept of magnetic flux - the total amount of magnetic field passing through a surface.

Magnetic flux ($\Phi$) through a surface is:

$\displaystyle \Phi = BA \cos\theta$

where $B$ = magnetic field (T), $A$ = area of the loop (m²), and $\theta$ = angle between $B$ and the normal (perpendicular) to the surface. The unit of flux is the weber (Wb), where 1 Wb = 1 T·m².

Flux is maximum when the field is perpendicular to the surface ($\theta = 0°$, $\cos\theta = 1$) and zero when the field is parallel to the surface ($\theta = 90°$, $\cos\theta = 0$).

Faraday’s Law (Conceptual)

Faraday’s law states that a changing magnetic flux through a loop induces an EMF (voltage) in the loop. The faster the flux changes, the larger the induced EMF.

Three ways to change flux (and induce an EMF):

1. Change $B$ - move a magnet closer to or farther from the loop, or turn an electromagnet on/off.
2. Change $A$ - expand or contract the loop (like pulling a wire through a field).
3. Change $\theta$ - rotate the loop in the field (this is how generators work).

Lenz’s Law

Lenz’s law tells you the direction of the induced current: the induced current flows in a direction that opposes the change in flux that caused it.

Applying Lenz’s Law Step by Step

1. Determine the direction of the external magnetic field through the loop.
2. Determine whether the flux is increasing or decreasing.
3. The induced current will create a magnetic field that opposes the change:
   • If flux is increasing, the induced field opposes the external field (points opposite).
   • If flux is decreasing, the induced field supports the external field (points same direction).
4. Use the right-hand rule to find the current direction that produces the needed induced field.

Example

A bar magnet with its north pole pointing down is dropped toward a horizontal loop of wire. The downward magnetic flux through the loop is increasing. By Lenz’s law, the induced current must create an upward magnetic field to oppose the increase. Using the right-hand rule, this means the current flows counterclockwise when viewed from above.

Applications

Generators

A generator is the reverse of a motor. A motor takes current and produces rotation. A generator takes rotation and produces current. Spinning a wire loop in a magnetic field continuously changes the flux (by changing θ), inducing an alternating EMF. That’s the basis of AC power generation at every power plant.

Transformers

A transformer uses electromagnetic induction to change the voltage of AC power. Two coils (primary and secondary) are wound around a shared iron core. Alternating current in the primary coil creates a changing magnetic field, which induces an EMF in the secondary coil.

The voltage ratio depends on the number of turns:

$\displaystyle \dfrac{V_2}{V_1} = \dfrac{N_2}{N_1}$

A step-up transformer ($N_2 > N_1$) increases voltage. A step-down transformer ($N_2 < N_1$) decreases voltage. Energy is conserved (ideally): if voltage goes up, current goes down proportionally.

Eddy Currents

When a conducting material (not just a wire loop) is exposed to a changing magnetic field, induced currents swirl through the bulk of the material. These are called eddy currents. They oppose the change in flux (Lenz’s law) and dissipate energy as heat. Eddy currents are why:

• A metal pendulum swinging between the poles of a magnet slows down rapidly.
• Induction cooktops heat metal pots without a flame.
• Electromagnetic brakes work without friction pads.