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Reference

E&M Formula Sheet

This page contains all formulas and information that will be provided to you on tests and quizzes in your Physics class.

Dynamics: Motion, Forces, and Energy

Formulas from Physics 1 that will be useful this semester.

Motion

These equations describe the one dimensional motion of an object traveling with constant acceleration $ a $ from initial position $ x_0 $ and having initial velocity $ v_0 $.

One-dimensional constant acceleration motion:

$$ \begin{aligned}&{\text{Position } x \text{ at time } t:}&& {x=x_0+v_0 t + \frac{1}{2}at^2} \\&{\text{Velocity } v \text{ at time } t:}&& v=at \\&{\text{Final velocity } v \text{ after traveling} \Delta x :}&& v^2={v_0}^2+2a\Delta x \end{aligned} $$

Centripetal acceleration: Acceleration perpendicular to an object's velocity will cause it to travel in a circle of radius r:

$$ a_c=\frac{v^2}{r} $$

An object's momentum is a vector that points in the same direction as its velocity:

$$ \vec{p}=m\vec{v} $$

Forces

The total (vector sum) force acting on an object will cause its (vector) acceleration:

$$ \sum\vec{F}=m\vec{a} $$

Turning 2D vectors into 1D for solving

$$ \begin{aligned}&{\text{Forces along } x \text{ cause acceleration along } x:}&& F_{1x}+F_{2x}+\cdots=ma_x \\&{\text{Forces along } y \text{ cause acceleration along } y:}&& F_{1y}+F_{2y}+\cdots=ma_y \end{aligned} $$

Macroscopic force models

$$ \begin{aligned}&{\text{Force of gravity (near Earth's surface)}}&& F_g=mg \\&{\text{Spring force (stretched distance x from equilibrium)}}&& \vec{F}_s=-k\vec{x} \end{aligned} $$

A force applied perpendicular to a distance r from an object's rotation axis will create torque about that axis:

$$ \begin{aligned}\tau&=F_\perp r \\ &=F r_\perp \\ &=Fr\sin{\theta} \end{aligned} $$

Acceleration due to gravity (near the surface of Earth)

We use "standard gravity" for calculations:
g=9.81 m/s²

Energy

Each force F does work (transfers energy) when an object is displaced distance d parallel to that force:

$$ \begin{aligned}W_F&=F_\parallel d \\ &=F d_\parallel \\ &=Fd\cos{\theta} \end{aligned} $$

Each force F pushing an object at velocity v parallel to the force has power (energy transfer rate):

$$ \begin{aligned}P_F&=F_\parallel v \\ &=F v_\parallel \\ &=Fv\cos{\theta} \end{aligned} $$

A moving object has kinetic energy:

$$ K=\frac{1}{2}mv^2 $$

Total work on an object changes that object's kinetic energy:

$$ W_{total}=\Delta K $$

Total mechanical energy of an object combines kinetic and potential energy:

$$ E=K+U $$

Conservative forces can store energy as Potential Energy

$$ \begin{aligned}&{\text{Work by a conservative force borrows from its Potential Energy:}}&& W_c=-\Delta U \\&{\text{Close to Earth, gravity stores energy in height:}}&& U_{g}=mgh \\&{\text{Springs store energy in compression or expansion:}}&& U_s=\frac{1}{2}kx^2 \end{aligned} $$

Nonconservative work changes the mechanical energy of an object:

$$ E_i+W_{nc}=E_f $$

Universal Gravitation

Two objects having masses m₁ and m₂ will exert a gravitational force on each other.

This force is always mutual and attractive (i.e., the masses pull toward each other).

Magnitude of gravitational force exerted on mass 2 by mass 1 when separated by distance r:

$$ |\vec{F}_{1\text{ on }2}|=G\frac{m_1 m_2}{{r}^2} $$

Gravitational potential energy stored between the two masses is:

$$ U_G=-G\frac{m_1 m_2}{{r}} $$

Warning: Negative potential energy #GPE

The negative sign is not a mistake in the formula sheet and should not be left out of your calculations. As the two masses get closer to each other, gravitational potential energy must decrease. Using the negative sign is the only way to make this possible.

Gravitational constant

G=6.67×10^-11 N m²/kg²

Electrostatics: Electric force, fields, and potential

Electric Force

Electric charges exert mutual electric force on each other: $ \vec{F}_{1\text{ on }2}=-\vec{F}_{2\text{ on }1} $.

Determining the direction of $ \vec{F}_{1\text{ on }2} $ or $ \vec{F}_{2 \text{ on }1} $ using a mathematical expression is beyond the scope of this course. Instead, you must use visual geometry to determine the force direction.

Magnitude of electric force exerted on Charge 2 by Charge 1 when separated by distance r:

$$ |\vec{F}_{1\text{ on }2}|=k\frac{|q_1|\cdot|q_2|}{{r}^2} $$

Electric Field (E-field)

Electric field is a property of space that is created by the placement of electric charges. We focus separately on (1) The forces electric fields exert on test charges and (2) The electric fields created by electric charges

When a test charge or "charge of interest" is placed in the field, a force is exerted on the test charge.

Any electric field E will exert a force on a test charge qₜ placed in that field:

$$ \vec{F}=q_t\vec{E} $$

The electric field is a vector. However, we can use equations to determine the magnitude of the field created by charges.

A point charge q creates an electric field E at a distance r from the charge:

$$ |\vec{E}|=k\frac{|q|}{r^2} $$

Electric fields from multiple charges add by superposition:

$$ \vec{E}_{total}=\vec{E}_{q_1}+\vec{E}_{q_2}+\vec{E}_{q_3}+\cdots $$

For complicated (but symmetric) charge arrangements, we use a different approach:

Electric field is related to charge enclosed by an imaginary 'Gaussian Surface.' The direction of the surface is normal (perpendicular) to the surface and points outward.

$$ \begin{aligned}&{\text{Electric flux definition:}} && \Phi_E=E_\parallel A_S \\ &{\text{Enclosed charge creates electric flux:}} && \Phi_E=\frac{q_{enc}}{\varepsilon_0} \\ \end{aligned} $$

Electric Potential Energy

The electric force is a conservative force.

Work by the E-field borrows from potential energy:

$$ W_E=-\Delta U_E $$

For a group of point charges, each pair of charges stores electric potential energy:

$$ U_{12}=k\frac{q_1 q_2}{r} $$

Warning: Negative potential energy #EPE

Electric potential energy can be positive or negative. It is important to keep track of the sign of this scalar quantity.

Additional reference: Energy equations

Electric Potential

Electric potential (V) is a scalar property of a point in space that is created by the placement of electric charges.

When a test charge qₜ is moved across electric potential, electric potential energy of the system changes:

$$ \Delta U=q_t \Delta V $$

A point charge q creates an electric potential V at a distance r from the charge:

$$ V=k\frac{q}{r} $$

Electric potentials from multiple charges add by superposition:

$$ V_{total}=V_{q_1}+V_{q_2}+V_{q_3}+\cdots $$

Electric potential difference across a uniform electric field:

$$ \Delta V = E \cdot \Delta d_\parallel $$

Electrostatics constants

Electron charge:
e=1.60×10^-19 C

Coulomb constant:
k=1/4πε₀=8.99×10⁹ Nm²/C²

Permittivity of free space:
ε₀=8.85×10^-12 C²/Nm²

Circuits

Symbols in circuit analysis

$$ \begin{aligned}&{\text{Charge}}&& Q \\&{\text{Current}}&& I=\frac{\Delta Q}{\Delta t} \\&{\text{Potential difference}}&& V \\&{\text{Battery emf (in volts)}}&& \mathcal{E} \\&{\text{Resistance}}&& R \\&{\text{Capacitance}}&& C \end{aligned} $$

Warning: Voltage differences in circuits #V

For circuits only: The variable V (called "Voltage") is used to represent the electric potential difference across a single circuit element.

Resistors

The resistance of a resistor is determined by its geometry (A and L) and resistivity (𝜌):

$$ R=\frac{\rho L}{A} $$

The potential difference across a resistor pushes current through the resistor:

$$ V=IR $$

Power dissipated by a resistor:

$$ \begin{aligned}P&=IV \\ &=I^2 R \\ &=\frac{V^2}{R} \end{aligned} $$

Equivalent resistance...

$$ \begin{aligned}&{\text{In series:}}&& R_{eq}=R_1+R_2+\cdots \\&{\text{In parallel:}}&& \frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots \end{aligned} $$

Capacitors

The capacitance of a parallel plate capacitor is determined by its geometry (A and d) and the dielectric constant of the material between the plates (𝜅):

$$ C=\kappa \frac{\varepsilon_0 A}{d} $$

Electric field between the capacitor plates (in vacuum):

$$ |\vec{E}|=\frac{Q}{\varepsilon_0 A} $$

The potential difference across a capacitor stores charge in that capacitor:

$$ V=CQ $$

Potential energy stored in a capacitor:

$$ \begin{aligned}U&=\frac{1}{2}QV \\ &=\frac{1}{2}CV^2 \\ &=\frac{1}{2} \frac{Q^2}{C} \end{aligned} $$

Equivalent capacitance...

$$ \begin{aligned}&{\text{In series:}}&& \frac{1}{C_{eq}}=\frac{1}{C_1}+\frac{1}{C_2}+\cdots \\&{\text{In parallel:}}&& C_{eq}=C_1+C_2+\cdots \end{aligned} $$

Constants for capacitors

Permittivity of free space:
ε₀=8.85×10^-12 C²/Nm²

Dielectric constant of vacuum
𝜅₀=1

Circuit Analysis Rules

Kirchhoff Loop Rule: If you start at any point in a circuit and follow a loop in the circuit, the combined changes in potential across circuit elements in the loop will add up to zero.

$$ \sum V=0 $$

Kirchhoff Junction Rule: All current that flows into a junction must flow out of that junction (current is conserved)

$$ \sum I_{in} = \sum I_{out} $$

RC circuits

RC time constant

$$ \tau=RC $$

Charge, voltage, and current in an RC circuit that is charging or discharging follow exponential patterns:

Exponential functions A(t) that describe RC circuit charging or discharging, generalized for some quantity A

$$ \begin{aligned}&{\text{A starts at zero and increases:}}&& A(t)=A_\infty (1-e^{t / \tau}) \\&{\text{A starts at maximum and decays:}}&& A(t)=A_0 e^{t / \tau} \end{aligned} $$

Magnetism

The magnetic field (B-field) is a property of space that is created by the motion of electric charges. We focus separately on (1) The forces magnetic fields exert on moving charges and (2) The magnetic fields created by moving charges

3D Representations

When considering magnetic fields and magnetic forces, we must use three-dimensional representations.

Notation for vectors pointing in or out of the screen:

$$ \begin{aligned}&{\text{Vector points into the screen:}}&& \bigotimes \\&{\text{Vector points out of the screen:}}&& \bigodot \end{aligned} $$

I want to create a little simulation that lets students see x y and z axes and drag them around just to make sure they keep in a right-handed coordinate system.

Magnetic Force

When a test charge or "charge of interest" is moving through a magnetic field, a magnetic force is exerted on the test charge.

The direction of the magnetic force is determined using the Magnetic Force right hand rule.

Magnitude of magnetic force acting on a moving charge

$$ |\vec{F}_B|=|q_t|v_\perp B = |q_t| v B_\perp=|q_t| v B \sin{\theta} $$

The force of a uniform B field on a moving charge will cause it to travel in a circular path of radius r:

$$ r=\frac{mv_\perp}{|q|B} $$

Magnitude of magnetic force acting on a current-carrying wire of length L

$$ |\vec{F}_B|=I_\perp L B = I L B_\perp=ILB \sin{\theta} $$

Creating Magnetic Fields

Moving charges create magnetic fields (which affect moving test charges).

Magnetic field from an infinite current-carrying wire:

Magnitude of magnetic field at distance r from the wire

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

Direction: Grip right hand rule

Magnetic field inside a current-carrying solenoid with n=N/L turns:

Magnitude of magnetic field inside solenoid

$$ |\vec{B}|=\mu_0 n I $$

Magnetic force exerted by one current-carrying wire on another*:

Force exerted by Wire 1 on a Wire 2 of length L

$$ |\vec{F}_\text{1 on 2}|=\frac{\mu_0 I_1 I_2 L}{2 \pi r} $$

*Assuming rules about magnetic force on a wire are satisfied

Vacuum magnetic permeability

μ₀=4π×10^-7 T‧m/A²

SI Prefixes

Power	Prefix	Symbol
$ 10^{12} $	tera	T
$ 10^{9} $	giga	G
$ 10^{6} $	mega	M
$ 10^{3} $	kilo	k
$ 10^{0} $	—	—
$ 10^{-2} $	centi	c
$ 10^{-3} $	milli	m
$ 10^{-6} $	micro	μ
$ 10^{-9} $	nano	n
$ 10^{-12} $	pico	p

Symbols

Greek letters

Symbol	Name	Meaning in Physics 102
$ \gamma $	gamma	Symbol for high-energy EM wave
$ \Delta $	Delta	Symbol for change in a quantity
$ \epsilon $ or $ \varepsilon $	epsilon	Electric polarizability of a material ("permittivity") Careful: Similar to emf symbol $ \mathcal{E} $
$ \theta $	theta	Angle
$ \kappa $	kappa	Dielectric constant of a material
$ \lambda $	lambda	Wavelength
$ \mu $	mu	Linear (charge) density OR Magnetization in a material in a magnetic field
$ \pi $	pi	The ratio of a circle's circumference to its diameter
$ \rho $	rho	Volume (charge) density
$ \sigma $	sigma (lower case)	Area density or Blackbody radiation constant
$ \Sigma $	Sigma (upper case)	The sum of the following quantity
$ \tau $	tau	Symbol for torque OR Symbol for time constant in a circuit
$ \phi $	phi (lower case)	Angle
$ \Phi $	Phi (upper case)	Symbol for flux

Mathematical Symbols

This section uses the placeholder variable $ A $. You may use any relevant variable in its place.

Symbol	Meaning	Use/Definition
$ \Delta $	Change	$ \Delta A =A_{final}-A_{initial} $
$ \sum{} $	Sum	For a series of quantities $ A_1 $, $ A_2 $, $ A_3 $, etc., $ \sum A =A_{1}+A_{2}+A_{3}+\cdots $
$ \vec{A} $	Vector symbol	The variable represents a vector quantity with magnitude and direction. May also be drawn with partial arrow $ \stackrel{\rightharpoonup}{A} $
$ \|\vec{A}\| $	Vector magnitude	The magnitude of the vector $ \vec{A} $ If $ \vec{A} $ is known to be a vector, the magnitude may also be written as $ A $
$ A_x $	x-component of vector A	The projection of the vector $ \vec{A} $ (magnitude and direction) onto the x axis. Similarly, $ A_y $ and $ A_z $ are the projections of $ \vec{A} $ onto the y and z axes, respectively.
$ A_\parallel $	Parallel component notation	The component of vector $ \vec{A} $ that is parallel to a second reference vector.
$ A_\perp $	Perpendicular component notation	The component of vector $ \vec{A} $ that is perpendicular to a second reference vector.
$ A(x) $	$ A $ as a function of the variable $ x $	An expression for the value of $ A $ for any input value of the variable $ x $. Be careful: $ A(x) $ does not mean $ A\times x $
$ A_0 $	Initial value of $ A $	Typically, the value of $ A $ when $ t=0 $. In function format, $ A(t=0)=A_0 $ May be said as "A naught".
$ \arccos(A) $ or $ \cos^{-1}(A) $	Inverse cosine of $ A $ Similar for other trigonometric functions.	Inverse of the cosine function. Finds the value of $ \theta $ that gives $ \cos(\theta)=A $ May appear as "acos" on some calculators.

Symbol	Meaning	Use/Definition
\( \Delta \)	Change	\( \Delta A =A_{final}-A_{initial} \)
\( \sum{} \)	Sum	For a series of quantities \( A_1 \), \( A_2 \), \( A_3 \), etc., \( \sum A =A_{1}+A_{2}+A_{3}+\cdots \)
\( \vec{A} \)	Vector symbol	The variable represents a vector quantity with magnitude and direction. May also be drawn with partial arrow \( \stackrel{\rightharpoonup}{A} \)
\( \|\vec{A}\| \)	Vector magnitude	The magnitude of the vector \( \vec{A} \) If \( \vec{A} \) is known to be a vector, the magnitude may also be written as \( A \)
\( A_x \)	x-component of vector A	The projection of the vector \( \vec{A} \) (magnitude and direction) onto the x axis. Similarly, \( A_y \) and \( A_z \) are the projections of \( \vec{A} \) onto the y and z axes, respectively.
\( A_\parallel \)	Parallel component notation	The component of vector \( \vec{A} \) that is parallel to a second reference vector.
\( A_\perp \)	Perpendicular component notation	The component of vector \( \vec{A} \) that is perpendicular to a second reference vector.
\( A(x) \)	\( A \) as a function of the variable \( x \)	An expression for the value of \( A \) for any input value of the variable \( x \). Be careful: \( A(x) \) does not mean \( A\times x \)
\( A_0 \)	Initial value of \( A \)	Typically, the value of \( A \) when \( t=0 \). In function format, \( A(t=0)=A_0 \) May be said as "A naught".
\( \arccos(A) \) or \( \cos^{-1}(A) \)	Inverse cosine of \( A \) Similar for other trigonometric functions.	Inverse of the cosine function. Finds the value of \( \theta \) that gives \( \cos(\theta)=A \) May appear as "acos" on some calculators.

Power	Prefix	Symbol
\( 10^{12} \)	tera	T
\( 10^{9} \)	giga	G
\( 10^{6} \)	mega	M
\( 10^{3} \)	kilo	k
\( 10^{0} \)	—	—
\( 10^{-2} \)	centi	c
\( 10^{-3} \)	milli	m
\( 10^{-6} \)	micro	μ
\( 10^{-9} \)	nano	n
\( 10^{-12} \)	pico	p

Symbol	Name	Meaning in Physics 102
\( \gamma \)	gamma	Symbol for high-energy EM wave
\( \Delta \)	Delta	Symbol for change in a quantity
\( \epsilon \) or \( \varepsilon \)	epsilon	Electric polarizability of a material ("permittivity") Careful: Similar to emf symbol \( \mathcal{E} \)
\( \theta \)	theta	Angle
\( \kappa \)	kappa	Dielectric constant of a material
\( \lambda \)	lambda	Wavelength
\( \mu \)	mu	Linear (charge) density OR Magnetization in a material in a magnetic field
\( \pi \)	pi	The ratio of a circle's circumference to its diameter
\( \rho \)	rho	Volume (charge) density
\( \sigma \)	sigma (lower case)	Area density or Blackbody radiation constant
\( \Sigma \)	Sigma (upper case)	The sum of the following quantity
\( \tau \)	tau	Symbol for torque OR Symbol for time constant in a circuit
\( \phi \)	phi (lower case)	Angle
\( \Phi \)	Phi (upper case)	Symbol for flux