Module 7: Electrostatics and Magnetism

Learning Objectives

By the end of this section, you should be able to:

Understand the nature, behavior, and conservation of electric charge.
Distinguish between conductors and insulators in terms of charge mobility.
Explain the mechanisms of charge transfer: conduction, induction, and polarization.
Derive and interpret Coulomb’s Law, both mathematically and conceptually.
Relate electrostatic forces to other forces (e.g., gravitational), using scaling relationships.
Apply vector reasoning to solve force problems involving multiple charges.

Module 7 Overview: Electrostatics and Magnetism

Electric Charge and Coulomb’s Law
Electric Fields and Electric Potential
Magnetic Fields and the Lorentz Force
Magnetic Field Generated by Currents (Ampere’s Law, Right-Hand Rules)
Summary and Key Takeaways

Electric Charge and Coulomb’s Law

What Is Electric Charge?

Electric charge is one of the most fundamental properties of matter. While mass gives rise to gravitational interactions, charge gives rise to electromagnetic interactions, which are much stronger and more varied in effect.

All matter is made of atoms, which in turn consist of:

Protons (positive charge)
Electrons (negative charge)
Neutrons (no charge)

These charges interact via the electrostatic force, which is described by Coulomb’s Law.

Basic Properties of Charge

Two Types:

Positive charge
- Defined as the type carried by protons.
Negative charge
- Carried by electrons.

Important: These are labels, not absolute truths – electricity could have been defined the other way around.

Quantization of Charge:

All observable charge comes in integer multiplies of the elementary charge: e = 1.60 x 10^-19 C
A proton has +e; an electron has -e. No known particle has less.

Conservation of Charge

In any closed physical system, the total electric charge remains constant.
Charges may rearrange or transfer, but the net charge doesn’t change.

Additive Property

Charge is additive: the net charge of a system is the algebraic sum of all individual charges.

Conductors vs. Insulators

Understanding how charge moves is critical:

Material Type	Behavior
Conductors	Allow electrons to move freely (e.g., metal, salt water)
Insulators	Hold electrons tightly; charge remains localized (e.g., rubber, wood)

MCAT Application: A charged metal sphere distributes charge over its surface; an insulator retains charge at the point of contact.

Charge Transfer Mechanisms

Conduction: Direct transfer of electrons via contact.

Charge flows from one object to another until potential difference is neutralized.

Induction: Charge rearrangement without contact.

A nearby charged object induces a redistribution of charge in a conductor.
The object remains electrically neutral overall, but becomes polarized.

Polarization: Electron shift within atoms/molecules without full transfer.

Relevant in biological molecules, dielectrics, and van der Waals forces.

Coulomb’s Law: The Electrostatic Force

Coulomb’s Law mathematically describes the magnitude and direction of the force between two points charges.

$$
F = k_e \frac{|q_1 q_2|}{r^2}
$$

Where:

$$
F = \text{Electrostatic force between the charges (Newtons, N)}
$$

$$
k_e = \text{Coulomb’s constant } \left(8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2\right)
$$

$$
q_1, q_2 = \text{Magnitudes of the two charges (Coulombs, C)}
$$

$$
r = \text{Distance between the charges (meters, m)}
$$

Understanding Coulomb’s Law Conceptually

Magnitude Relationship

Force increases with larger charges.
Force decreases rapidly with distance (inversely proportional to the square of the distance).
Inverse-square law: Doubling distance reduces force to 1/4

Sign and Direction

If q1 and q2 have:
- Opposite signs → attractive force
- Same sign → repulsive force
Force is directed along the line connecting the charges.

Physical Intuition: Comparison to Gravity

$$
F_{\text{gravity}} = G \frac{m_1 m_2}{r^2} \quad \text{vs.} \quad F_{\text{electric}} = k \frac{q_1 q_2}{r^2}
$$

Both follow inverse-square laws.
Gravitational force is always attractive.
Electrostatic force can be attractive or repulsive.
Electrostatic forces are ~10³⁶ times stronger than gravity at atomic scales.

Vector Representation

Force is a vector quantity. If multiple charges are involved:

Use vector addition to find net force.
Common on the MCAT: charges in equilateral triangle, square, or 1D line.

Tip: Use symmetry to simplify direction.

MCAT Conceptual Triggers

Be alert for setups asking: “Which way does the force point?”
Emphasize magnitude vs. direction:
- Scalar question? Use magnitude formula only
- Vector question? Use signs and geometry
Expect comparisons between electric and gravitational forces.

Electric Field and Electric Potential

What Is an Electric Field?

An electric field is a region in space around a charged object where other charges experience an electric force. It is a vector field, meaning it has both magnitude and direction at every point.

Think of an electric field as a force field – if you place a small positive test charge in the region, the field tells you what force it would feel and in what direction.

Electric Field (General Definition)

$$
E = \frac{F}{q}
$$

Where:

$$
E = \text{Electric field (Newtons per Coulomb, N/C)}
$$

$$
F = \text{Electrostatic force experienced by a test charge (N)}
$$

$$
q = \text{Test charge placed in the field (C)}
$$

This definition simply says: the electric field at a point is the force per unit charge at that location. It’s a measure of how intense the electric influence is.

Electric Field Due to a Point Charge

For a single point charge Q, the field at a distance r from the charge is:

$$
\vec{E} = k_e \cdot \frac{Q}{r^2} \, \hat{r}
$$

Where:

$$
\vec{E} = \text{Electric field vector (N/C)}
$$

$$
k_e = \text{Coulomb’s constant } \left(8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2\right)
$$

$$
Q = \text{Source charge (C)}
$$

$$
r = \text{Distance from the charge (m)}
$$

$$
\hat{r} = \text{Unit vector pointing from the source charge to the location of the field}
$$

Directionality:

Positive source charge: field lines radiate outward
Negative source charge: field lines converge inward

Field Line Visuals and Interpretation

Electric field lines are a visual representation of the direction and strength of electric fields. Lines point away from positive charges and toward negative charges, indicating the direction a positive test charge would move. The density of the lines reflects field strength, closer lines mean stronger fields.

Density of field lines = field strength (closer together = stronger field)
Field lines never cross
Start on positive charges, end on negative ones
The tangent to a field line gives the force direction on a positive test charge

Electric Potential (V)

While electric field tells us about force, electric potential tells us about energy.

Definition:

$$
V = \frac{U}{q} = \frac{k_e Q}{r}
$$

Where:

$$
V = \text{Electric potential (Volts, V = J/C)}
$$

$$
U = \text{Electric potential energy (Joules, J)}
$$

$$
q = \text{Test charge (C)}
$$

$$
k_e = \text{Coulomb’s constant } (8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2)
$$

$$
Q = \text{Source charge (C)}
$$

$$
r = \text{Distance from the source charge (m)}
$$

V tells you the potential energy per unit charge at a given point in space.

Potential is a scalar quantity. It has magnitude but no direction.

Electric Potential Energy (U)

$$
U = qV = \frac{k_e q Q}{r}
$$

Where:

$$
U = \text{Electric potential energy (Joules, J)}
$$

$$
q = \text{Test charge (C)}
$$

$$
V = \text{Electric potential at the position of } q \, (\text{Volts, V})
$$

$$
Q = \text{Source charge (C)}
$$

$$
r = \text{Distance between the charges (m)}
$$

$$
k_e = \text{Coulomb’s constant } (8.99 \times 10^9 \, \text{N} \cdot \text{m}^2/\text{C}^2)
$$

U: Potential energy of test charge q in the field of source charge Q
If charges are o the same sign, U > 0 results in repulsion
If charges are of opposite sign, U < 0 results in attraction

Conceptual Analogy: Field vs. Potential

Quantity	Type	Analogy
Electric field (E)	Vector	Slope of a hill (push/force)
Electric potential (V)	Scalar	Elevation of the hill (energy level)
Potential energy (U)	Scalar	Actual energy a charge has at that point

Field lines: show where and how a charge feels a force
Potential contours: show where a charge gains or loses energy

Positive charges move from high to low potential.

Negative charges move from low to high potential (opposite to E).

Work-Energy Relationship

$$
W = -\Delta U = q \, \Delta V
$$

Where:

$$
W = \text{Work done by the electric field (Joules, J)}
$$

$$
\Delta U = \text{Change in electric potential energy (J)}
$$

$$
q = \text{Charge moving through the field (C)}
$$

$$
\Delta V = \text{Change in electric potential (Volts, V)}
$$

Work done by the field decreases potential energy.
Work done against the field increases potential energy.

Summary Table

Concept	Formula	Notes
Electric field (general)	$$ E = \frac{F}{q} $$	Force per unit charge
Field from point charge	$$ E = \frac{k Q}{r^2} $$	Direction depends on source sign
Electric potential (V)	$$ V = \frac{k Q}{r} $$	Scalar energy per unit charge
Potential energy (U)	$$ U = qV = \frac{k q Q}{r} $$	Stored interaction energy
Work by field	$$ W = -\Delta U = q \, \Delta V $$	Positive when moving “downhill”

MCAT Reasoning Tips

A positive charge placed in a positive field moves away from the source – its potential decreases, kinetic energy increases.
Be comfortable translating between energy, potential, field, and force depending on what the question gives you.
Expect diagrams that test your ability to:
- Trace direction of motion
- Predict changes in U, V, or E
- Distinguish between potential difference and field strength

Magnetic Fields and the Lorentz Force

What Is a Magnetic Field?

A magnetic field is a region in space where a moving charge or a current experiences a force. Unlike electric fields, which originate from static charges, magnetic fields are created by motion, specifically, by the movement of charge.

This of magnetic fields as the consequence of electric fields observed in motion. They are not “separate forces” but another manifestation of electromagnetic interaction.

Unlike electric fields, which originate from static charges, magnetic fields are produced by moving charges. These fields represent a fundamental aspect of electromagnetic interaction, describing how motion gives rise to influence over other moving charges.

The Earth has a magnetic field due to the movement of molten metal in its core.
Wires carrying electric current generate magnetic fields around them.
Magnetic fields are inherently loop-like, they have no beginning or end.

What Is the Lorentz Force?

The Lorentz force is the total electromagnetic force acting on a moving charged particle. It is named after the Dutch physicist Hendrik Lorentz, who unified the concepts of electric and magnetic forces acting on charged particles into a single framework.

Full Lorentz Force Law:

$$
\vec{F}_{\text{total}} = q \vec{E} + q \vec{v} \times \vec{B}
$$

Where:

$$
\vec{F}_{\text{total}} = \text{Total force on the charge (Newtons, N)}
$$

$$
q = \text{Charge of the particle (C)}
$$

$$
\vec{E} = \text{Electric field vector (N/C)}
$$

$$
\vec{v} = \text{Velocity of the particle (m/s)}
$$

$$
\vec{B} = \text{Magnetic field vector (Tesla, T)}
$$

For this section, we focus solely on the magnetic component of the Lorentz force:

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

What This Really Means Physically

The magnetic force only acts on moving charges. No motion? No force.
The direction of the force is always perpendicular to both the velocity vector and the magnetic field vector.
Because the force is perpendicular to motion, it does not do work – it cannot speed up or slow down a particle. It only changes direction.
Imagine a skateboarder entering a circular bowl. The walls of the bowl don’t push the skater forward or backward – they just steer them. The magnetic field behaves similarly.

Magnetic Force Magnitude

$$
F = |q| v B \sin\theta
$$

Where:

$$
F = \text{Magnitude of the magnetic force (Newtons, N)}
$$

$$
|q| = \text{Absolute value of the charge (C)}
$$

$$
v = \text{Speed of the particle (m/s)}
$$

$$
B = \text{Magnetic field strength (Tesla, T)}
$$

$$
\theta = \text{Angle between } \vec{v} \text{ and } \vec{B}
$$

Interpretations:

θ = 0° or 180°: F = 0 meaning motion is parallel to the field
θ = 90°: F is maximized, meaning motion is perpendicular to the field
This is why particles curve in magnetic fields. The force redirects them without changing their speed.

Right-Hand Rule: Direction of Magnetic Force

For positive charges:

Point your fingers in the direction of the particle’s velocity v
Curl your fingers toward the magnetic field B
Your palm points in the direction of the magnetic force Fb

For negative charges (e.g., electrons), the force is in the opposite direction.

Magnetic Force Causes Circular Motion

When the magnetic field is perpendicular to the particle’s velocity, the resulting force is always directed toward the center of a circular path, acting as a centripetal force.

Derivation:

$$
q v B = \frac{m v^2}{r} \;\Rightarrow\; r = \frac{m v}{q B}
$$

Where:

$$
r = \text{Radius of the circular path (meters, m)}
$$

$$
m = \text{Mass of the particle (kg)}
$$

$$
v = \text{Speed of the particle (m/s)}
$$

$$
q = \text{Charge of the particle (C)}
$$

$$
B = \text{Magnetic field strength (Tesla, T)}
$$

Remember: Heavier particles or faster particles travel in larger circles. Stronger fields pull more sharply, reducing radius.

When a charged particle enters the field at an angle (not 0° or 90°), its motion splits into:

A circular component (perpendicular to B)
A linear component (parallel to B)

The combination results in a helical or spiral path.

MCAT-Relevant Applications

Mass Spectrometer

Charged particles are deflected in a magnetic field.
Used to determine mass-to-charge ratio (m/q).

$$
r = \frac{m v}{q B} \;\Rightarrow\; \text{Larger } \frac{m}{q} \Rightarrow \text{Wider deflection}
$$

Velocity Selector

Combines electric and magnetic fields.
Selects particles with a specific velocity:

$$
qE = qvB \;\Rightarrow\; v = \frac{E}{B}
$$

Particles that satisfy this condition pass through undeflected.

MCAT Strategy Highlights

Direction-first questions: Use right-hand rule carefully – account for charge sign.
Work-energy comparisons: Magnetic fields do no work. Speed remains constant.
Centripetal force questions: Use qvB = mv²/r to solve for any variable.
Be ready to interpret setups with:
- Proton beams in magnetic fields
- Electrons in cyclotrons
- Fields in perpendicular planes