The Timpson Lyceum · University Physics II (2026) – Electricity & Magnetism

Introduction to University Physics II – Lecture 0 (University Physics II)

Introduction to University Physics II – Lecture 0

Overview

This opening lecture sets up the whole course (Electricity & Magnetism, i.e. University Physics II) by building the mathematical and physical toolkit you will need before any circuits or fields appear. It compresses an entire first calculus course — limits, continuity, derivatives, and integrals — into a single intuition-first pass. It then grounds those ideas in physics through Galileo's falling-body experiment, showing how position, velocity, and acceleration are linked by derivatives and integrals. Finally it introduces the principle of relativity and the distinction between scalars and vectors. The goal is double exposure: meet everything once now, so it feels familiar when it returns at a slower pace.

Key Concepts

Electricity & magnetism as a fundamental force. Of the four fundamental forces (gravity, electromagnetism, weak nuclear, strong nuclear), nearly every everyday phenomenon that is not gravity is electromagnetic: sound (electrons in vibrating matter repelling air molecules), sight (photons emitted and absorbed by electrons changing energy level), and all of chemistry.

Limit. The value a function's output approaches as the input gets arbitrarily close to a point — not the value at the point itself. For most points the limit equals the function's value, but they can differ where the function has a hole. $\lim_{x \to a} f(x) = L$

Continuity. A function is continuous if you can draw it without lifting your pencil. Precisely, $f$ is continuous at $a$ when the value equals the limit: $f(a) = \lim_{x \to a} f(x)$

Slope of a line. Rise over run, the same everywhere on a line: $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$ This yields the point–slope form $y - y_1 = m(x - x_1)$ and the slope–intercept form $y = mx + b$ , where $b$ is the $y$ -intercept.

Derivative. The slope of a function at a point. Since slope needs two points, take a second point a distance $\Delta x$ away, form the secant slope, and let $\Delta x \to 0$ : $f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$ Notations $f'(x)$ and $\dfrac{dy}{dx}$ mean the same thing; the $dy/dx$ form ("change in $y$ with respect to $x$ ") is preferred here because deltas turn into derivatives when you take the limit.

Power rule. A fast shortcut for derivatives — bring the exponent down and subtract one: $\frac{d}{dx}\,x^n = n\,x^{n-1}$ The derivative of a constant is $0$ (a constant has zero slope).

Integral. The area under a curve from $a$ to $b$ . Approximate it with $n$ rectangles of base $\Delta x$ and height $f(x)$ , then let $n \to \infty$ : $\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{k=1}^{n} f\!\left(a + k\tfrac{b-a}{n}\right)\frac{b-a}{n}$ Shortcut (reverse of the power rule) — add one to the exponent and divide: $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$

Fundamental Theorem of Calculus. Differentiation and integration undo each other; the integral of a derivative recovers the original function.

Kinematics. Position $x(t)$ , velocity, and acceleration form a derivative chain:

Velocity = rate of change of position: $\displaystyle v = \frac{dx}{dt}$ (the slope of the position graph).
Acceleration = rate of change of velocity: $\displaystyle a = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ (a second derivative of position).

Galileo's falling body. Ignoring air resistance, objects accelerate downward at a constant $g \approx 9.8\ \text{m/s}^2$ (often rounded to $10\ \text{m/s}^2$ for easy arithmetic), independent of mass. With $a = 10$ , integrating once gives $v(t) = 10t$ and again gives $x(t) = 5t^2$ .

Relativity (Galilean). Position and velocity are relative — they can only be stated with respect to some other object or event; there is no absolute position or absolute velocity. Acceleration, however, is not relative: you can always feel and agree on who is accelerating.

Scalars vs. vectors. A scalar is a single magnitude (distance, speed). A vector needs magnitude and direction, requiring more than one number (position, displacement, velocity), and is drawn as an arrow. Displacement (end minus start position) can be negative and tracks direction; distance (total path length) cannot.

Worked Example

Find the slope (derivative) of $f(x) = x^2 - 1$ at a general point $x$ using the limit definition, then confirm with the power rule.

\begin{aligned} f'(x) &= \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \\ &= \lim_{\Delta x \to 0} \frac{\big[(x+\Delta x)^2 - 1\big] - \big[x^2 - 1\big]}{\Delta x} \\ &= \lim_{\Delta x \to 0} \frac{x^2 + 2x\,\Delta x + \Delta x^2 - 1 - x^2 + 1}{\Delta x} \\ &= \lim_{\Delta x \to 0} \frac{2x\,\Delta x + \Delta x^2}{\Delta x} \\ &= \lim_{\Delta x \to 0} \,(2x + \Delta x) \\ &= 2x \end{aligned}

The power rule gives the same result instantly: $\dfrac{d}{dx}(x^2 - 1) = 2x$ . So the slope at $x = 1$ is $2$ and at $x = 2$ is $4$ , matching the graph.

Summary

Everyday non-gravitational phenomena are electromagnetic; E&M is University Physics II, picking up the textbook at the chapter where mechanics leaves off.
A limit is what a function approaches near a point; a function is continuous where its value equals its limit.
A derivative is the slope at a point — the limit of rise over run as the spacing shrinks; the power rule $\frac{d}{dx}x^n = nx^{n-1}$ computes it instantly.
An integral is the area under a curve — an infinite sum of thin rectangles; the Fundamental Theorem of Calculus says integration and differentiation are inverses.
In kinematics, velocity is the derivative of position and acceleration the derivative of velocity; free fall has constant $g \approx 9.8\ \text{m/s}^2$ , giving $v = 10t$ and $x = 5t^2$ .
Position and velocity are relative while acceleration is absolute; scalars carry only magnitude, vectors carry magnitude and direction.