Physics Formulas and Concepts: Magnetism, Induction, and Optics

Chapter 29: The Magnetic Field

Magnetic Poles and Materials: Like poles repel (+ +), opposite poles attract (+ -). Paramagnetic materials are magnetized only in the presence of a strong external magnetic field. Magnetic flux flows from north to south.

Gauss's Law for Magnetism

The magnetic flux through any closed surface is always zero. There is no way to isolate a north or south magnetic pole (no magnetic monopoles).

The most elementary electric field is from a point charge (a static charge), while the most elementary magnetic flux density is from a magnetic dipole (a small magnet or magnetic bar).

The induced magnetic dipole always has an opposite pole facing the solenoid.

Solenoid:

An electric charge experiences a magnetic force in a magnetic field, but only while moving in a direction perpendicular to the field.

The Cross Product and Right-Hand Rule

Right-Hand Rule for Cross Product: Curl the fingers of your right hand from the first vector to the second, and your thumb will point in the direction of the new vector.

• x = current flowing into the figure
• o = current flowing out of the figure

The magnetic field may be defined in terms of the force it exerts on a charged particle moving perpendicular to it. An external magnetic field seeks to align the magnetic dipole moment along it.

Fundamental Laws of Electromagnetism

Gauss's Law:

(Electric charges create electric fields)

Gauss's Law for Magnetism:

(Magnetic charges create magnetic fields, only they do not exist)

Ampere's Law:

(Electric fields that change with time create magnetic fields)

Current Interactions and Loops

In the case of parallel currents (current flowing in the same direction), the wires attract. In the case of anti-parallel currents (currents flowing in opposite directions), they repel.

Magnetic Field of a Current Loop: Curl the fingers of your right hand around the loop in the direction of the current. Your thumb will point in the direction in which B leaves the loop.

A uniform magnetic field can be generated with a solenoid. The magnetic field is strongest and most uniform inside the solenoid. The magnetic field outside a solenoid looks like that of a bar magnet.

Chapter 30: Electromagnetic Induction

Flux: The amount of "something" flowing through a given area.

Induction Experiments

When one coil is placed directly above another, there is no current in the lower circuit while the switch is in the closed position. A momentary current appears whenever the switch is opened or closed. When a bar magnet is pushed into a coil of wire, it causes a momentary current. Holding the magnet inside the coil has no effect. A quick withdrawal of the magnet creates a momentary current. Holding it stationary creates no current.

An induced current flows through a conducting loop when there is a change in the number of magnetic field lines passing through it.

Faraday's Law and Lenz's Law

Faraday's Law: When there is a change in the number of magnetic field lines passing through a conducting loop, a current flows through it. If it is a closed surface, the EMF will be 0.

The changing magnetic flux generates an induced current which creates an induced magnetic field which, in turn, resists the change in magnetic flux.

Lenz's Law: The induced current from Faraday's Law is always in a direction such that the induced magnetic field from the induced current opposes the change in the magnetic flux through the loop.

Changing the magnetic field induces an EMF. An induced EMF is an induced voltage, which implies the existence of an induced electric field.

Chapter 33: Light and Wave Optics

Light from a point source is emitted in all directions. As light expands away from the source, the curvature decreases. Rays indicate the direction in which each portion of the wavefront and its energy travel. A parallel bundle of rays could represent a laser beam or light from a distant object.

Wave Interference and Diffraction

• Diffraction: The spreading of waves (e.g., water or light) through a hole or aperture.
• Constructive Interference: Waves are aligned in phase.
• Destructive Interference: Waves are out of phase.

(Constructive)

Bright fringes will occur at angles θ_m:

The positions of these bright fringes will occur at:

The maximum intensity of the bright fringes is:

Physics Formulas: Section 1

Magnetic Flux:
(Weber, Wb)
The magnetic flux is equal to zero when the surface is closed (counts the number of field lines piercing an area).
B = Magnetic field strength (Tesla, T)
A = Area (m²)

Magnetic Force:
(Newton, N)
q = Charge (Coulombs, C)
v = Velocity (m/s)
B = Magnetic field (Tesla, T)

Magnetic Force with Two Forces:
(Newton, N)

Electric Field:
E = F_E/q (N/C)
F_E = Electric force vector (Newtons, N)
q = Charge (Coulombs, C)

Magnetic Field:
(Tesla, T)
F_B = Magnetic force (Newton, N)
q = Charge (Coulomb, C)
v = Velocity (m/s)

Magnetic Field (Point Charge):
(Tesla, T)
μ₀ = Permeability of free space constant = 4π × 10^-7 T·m/A
q = Charge (Coulomb, C)
v = Velocity (m/s)
r = Unit vector (e.g., r = 3i + 4j → |r| = √(3² + 4²) = 5 → r̂ = (3i + 4j)/5)
r² = Distance between charge and measurement point

Magnetic Field (Straight Wire):
(Tesla, T)
μ₀ = Permeability of free space constant = 4π × 10^-7 T·m/A
I = Current in the wire (Amperes, A)
d = Perpendicular distance from wire to measurement point (meters, m)

Cyclotron Period:
(Seconds, s)
m = Mass of the charge (kg)
q = Charge (Coulomb, C)
B = Magnetic field (Tesla, T)

Magnetic Force on a Wire:
(Newtons, N)
i = Electric current (Amperes, A)
l = Length of the wire (meters, m)
B = Magnetic field (Tesla, T)
Note: The length vector of the wire must be in the same direction as the electric current.

Magnitude of the Magnetic Field:
(Newton, N)
i = Electric current (Amperes, A)
L = Length of the wire (meters, m)
B = Magnetic field (Tesla, T)

Magnetic Dipole Moment:
(Ampere-meter squared, A·m²)
I = Current (Ampere, A)
A = Area enclosed by loop (m²)
n = Unit vector perpendicular to the plane of the loop (Direction)
Unit vector (n): Determined by the Right-Hand Rule: curl fingers in the direction of the current I, and your thumb points in the direction of n.

Electric Dipole Moment:
(C·m)
q = Charge (Coulomb, C)
d = Displacement (pointing from negative to positive charge)

Potential Energy (Electric Dipole Moment):
(Joules, J)
p = Electric dipole moment (C·m)
E = Electric field (N/C)

Magnetic Potential Energy:
(Joules, J)
μ = Magnetic dipole (A·m²)
B = Magnetic field (Tesla, T)
Note: Speed of light = 3 × 10⁸ m/s

Magnetic Field (Solenoid):
(Tesla, T)
μ₀ = Permeability of free space constant = 4π × 10^-7 T·m/A
i = Current (Amperes, A)
N = Total number of turns (loops)
L = Length (meters, m)

Electric Flux (Scalar):
(N·m²/C)
E = Electric field (N/C)
A = Area (m²)
θ = Angle between electric field and the normal vector

Electric Flux (Vector):
(N·m²/C)
E = Electric field vector
A = Area vector

Magnetic Flux:
(Weber, Wb)
A_eff = Effective area
A = Surface area (m²)
B = Magnetic field (T)
θ = Angle between magnetic field and the normal vector

Induced Electromotive Force (EMF):
(Volts, V)

EMF (Faraday's Law):
ε = -N (Volts, V)

EMF (Uniform Magnetic Field):
(Volts, V)

EMF (General):
(Volts, V)

Magnetic Flux:
(Webers, Wb)
B = Magnetic field (Tesla, T)
A = Area (m²)

Induced EMF (Generator):
(Volts, V)
N = Total number of turns (loops)
A = Surface area (m²)
B = Magnetic field (T)
ω = Angular velocity (rad/s)
t = Time (s)

Lorentz Force Law:
(Newton, N)
q = Charge (Coulomb, C)
E = Electric field (N/C)
v = Velocity (m/s)
B = Magnetic field (Tesla, T)

Constructive Interference (Path Length Difference):
(meters, m)
m = Order number
λ = Wavelength (meters, m)

Destructive Interference (Path Length Difference):
(meters, m)
m = Order number
λ = Wavelength (meters, m)

Double-Slit (Path Length Difference):

d = Slit separation (meters, m)

Bright Fringe (Double Slit) (Constructive):

Dark Fringe (Double Slit) (Destructive):

Angle of a Bright Fringe:

Bright Fringe Position (Double Slit):
(meters, m)
y_m = Vertical distance (meters, m)
m = Order number
λ = Wavelength (m)
L = Distance from the slit to the screen (m)
d = Distance between the two slits
Δy = Fringe spacing

Intensity (Double Slit):
(W/m²)
I₁ = Intensity of one slit
d = Distance between slits (m)
y = Vertical distance from the center of the screen (m)
λ = Wavelength (m)
L = Distance to the screen (m)

Total Amplitude (Interference of Two Equal Amplitude Waves):

e = Amplitude of a single wave
ΔΘ = Phase constant difference (rad)

Intensity (General):
(W/m²)
C = Given constant
e = Amplitude

Phase Difference:
(rad)
Δr = Path difference (meters, m)
λ = Wavelength (m)
d = Slit separation (m)
y = Position on the screen (m)
L = Distance to the screen (m)

Double Slit Intensity (Detailed):
(W/m²)
I₁ = Intensity of one slit
E = Combined amplitude of two waves
e = Amplitude
C = Given constant
d = Slit separation (m)
y = Position on the screen (m)
L = Distance to the screen (m)

Single Slit Diffraction (Minima, Dark Fringes):
(rad)
p = Order number
λ = Wavelength (m)
a = Width of the slit (m)

Physical Position (Minima, Dark Fringes, Single Slit):
(m)
p = Order number
λ = Wavelength (m)
a = Width of the slit (m)
L = Distance to the screen (m)

Width of Central Bright Fringe (Single Slit):
(m)
y₁ = Distance from center to the first dark fringe (m)
λ = Wavelength (m)
a = Width of the slit (m)
L = Distance to the screen (m)

Angular Resolution:
(rad)
λ = Wavelength (m)
D = Diameter (m)

Width of Central Bright Spot:
(m)
y₁ = Distance from center to the first dark fringe (m)
L = Distance to the screen (m)
λ = Wavelength (m)
D = Diameter (m)
θ₁ = Angle to the first dark ring

Path Length Difference (Michelson Interferometer):
(m)
L₂ = Length of the second arm
L₁ = Length of the first arm

Destructive Interference (Dark Fringe, Michelson Interferometer):

Diameter of Central Bright Spot:

D = Diameter (m)

Magnification Formula:

h_i = Height of the image
h_o = Height of the object
d_i = Image distance
d_o = Object distance

Thin Lens Equation:

s = Object distance
s' = Image distance
f = Focal length (+ for converging, - for diverging)

Lateral Magnification:

s = Object distance
s' = Image distance

Magnification:

h' = Height of image
h = Height of object

Snell's Law:

n = Index of refraction

Index of Refraction:

v = Velocity (m/s)

Critical Angle:

n₁ = Refractive index (denser material)
n₂ = Refractive index (less dense material)

Physics Formulas: Section 2

Magnetic Flux:
(Weber, Wb)
The magnetic flux is equal to zero when the surface is closed (counts the number of field lines piercing an area).
B = Magnetic field strength (Tesla, T)
A = Area (m²)

Magnetic Force:
(Newton, N)
q = Charge (Coulombs, C)
v = Velocity (m/s)
B = Magnetic field (Tesla, T)

Magnetic Force with Two Forces:
(Newton, N)

Electric Field:
E = F_E/q (N/C)
F_E = Electric force vector (Newtons, N)

Magnetic Field:
(Tesla, T)
F_B = Magnetic force (Newton, N)

Magnetic Field (Point Charge):
(Tesla, T)
μ₀ = Permeability of free space constant = 4π × 10^-7 T·m/A
r = Unit vector (e.g., r = 3i + 4j → |r| = √(3² + 4²) = 5 → r̂ = (3i + 4j)/5)
r² = Distance between charge and measurement point

Magnetic Field (Straight Wire):
(Tesla, T)
μ₀ = 4π × 10^-7 T·m/A
I = Current in the wire (Amperes, A)
d = Perpendicular distance from wire to measurement point (meters, m)

Cyclotron Period:
(Seconds, s)
m = Mass of the charge (kg)
B = Magnetic field (Tesla, T)

Magnetic Force on a Wire:
(Newtons, N)
i = Electric current (Amperes, A)
l = Length of the wire (meters, m)
B = Magnetic field (Tesla, T)
Note: The length vector of the wire must be in the same direction as the electric current.

Magnitude of the Magnetic Field:
(Newton, N)
i = Electric current (Amperes, A)
L = Length of the wire (meters, m)
B = Magnetic field (Tesla, T)

Magnetic Dipole Moment:
(Ampere-meter squared, A·m²)
I = Current (Ampere, A)
A = Area enclosed by loop (m²)
n = Unit vector perpendicular to the plane of the loop (Direction)

Unit Vector (n): Determined by the Right-Hand Rule: curl fingers in the direction of the current I, and your thumb points in the direction of n.

Electric Dipole Moment:
(C·m)
q = Charge (Coulomb, C)
d = Displacement (pointing from negative to positive charge)

Potential Energy (Electric Dipole Moment):
(Joules, J)
p = Electric dipole moment (C·m)
E = Electric field (N/C)

Magnetic Potential Energy:
(Joules, J)
μ = Magnetic dipole (A·m²)
B = Magnetic field (Tesla, T)
Note: Speed of light = 3 × 10⁸ m/s

Magnetic Field (Solenoid):
(Tesla, T)
μ₀ = 4π × 10^-7 T·m/A
i = Current (Amperes, A)
N = Total number of turns (loops)
L = Length (meters, m)

Electric Flux (Scalar):
(N·m²/C)
E = Electric field (N/C)
A = Area (m²)
θ = Angle between electric field and the normal vector

Electric Flux (Vector):
(N·m²/C)
E = Electric field vector
A = Area vector

Magnetic Flux:
(Weber, Wb)
A_eff = Effective area
A = Surface area (m²)
B = Magnetic field (T)
θ = Angle between electric field and the normal vector

Induced Electromotive Force (EMF):
(Volts, V)

EMF (Faraday's Law):
ε = -N (Volts, V)

EMF (Uniform Magnetic Field):
(Volts, V)

EMF (General):
(Volts, V)

Magnetic Flux:
(Webers, Wb)
B = Magnetic field (Tesla, T)
A = Area (m²)

Induced EMF (Generator):
(Volts, V)
N = Total number of turns (loops)
A = Surface area (m²)
B = Magnetic field (T)
ω = Angular velocity (rad/s)
t = Time (s)

Lorentz Force Law:
(Newton, N)
E = Electric field (N/C)
v = Velocity (m/s)
B = Magnetic field (Tesla, T)

Constructive Interference (Path Length Difference):
(meters, m)
m = Order number
λ = Wavelength (meters, m)

Destructive Interference (Path Length Difference):
(meters, m)
m = Order number
λ = Wavelength (meters, m)

Double-Slit (Path Length Difference):

d = Slit separation (meters, m)

Bright Fringe (Double Slit) (Constructive):

Dark Fringe (Double Slit) (Destructive):

Angle of a Bright Fringe:

Bright Fringe Position (Double Slit):
(meters, m)
y_m = Vertical distance (meters, m)
m = Order number
λ = Wavelength (m)
L = Distance from the slit to the screen (m)
d = Distance between the two slits
Δy = Fringe spacing

Intensity (Double Slit):
(W/m²)
I₁ = Intensity of one slit
d = Distance between slits (m)
y = Vertical distance from the center of the screen (m)
λ = Wavelength (m)
L = Distance to the screen (m)

Total Amplitude (Interference of Two Equal Amplitude Waves):

e = Amplitude of a single wave
ΔΘ = Phase constant difference (rad)

Intensity (General):
(W/m²)
C = Given constant
e = Amplitude

Phase Difference:
(rad)
Δr = Path difference (meters, m)
λ = Wavelength (m)
d = Slit separation (m)
y = Position on the screen (m)
L = Distance to the screen (m)

Double Slit Intensity (Detailed):
(W/m²)
I₁ = Intensity of one slit
E = Combined amplitude of two waves
e = Amplitude
C = Given constant
d = Slit separation (m)
y = Position on the screen (m)
L = Distance to the screen (m)

Single Slit Diffraction (Minima, Dark Fringes):
(rad)
p = Order number
λ = Wavelength (m)
a = Width of the slit (m)

Physical Position (Minima, Dark Fringes, Single Slit):
(m)
p = Order number
λ = Wavelength (m)
a = Width of the slit (m)
L = Distance to the screen (m)

Width of Central Bright Fringe (Single Slit):
(m)
y₁ = Distance from center to the first dark fringe (m)
λ = Wavelength (m)
a = Width of the slit (m)
L = Distance to the screen (m)

Angular Resolution:
(rad)
λ = Wavelength (m)
D = Diameter (m)

Width of Central Bright Spot:
(m)
y₁ = Distance from center to the first dark fringe (m)
L = Distance to the screen (m)
λ = Wavelength (m)
D = Diameter (m)
θ₁ = Angle to the first dark ring

Path Length Difference (Michelson Interferometer):
(m)
L₂ = Length of the second arm
L₁ = Length of the first arm

Destructive Interference (Dark Fringe, Michelson Interferometer):

Diameter of Central Bright Spot:
D = Diameter (m)

Magnification Formula:

h_i = Height of the image
h_o = Height of the object
d_i = Image distance
d_o = Object distance

Thin Lens Equation:

s = Object distance
s' = Image distance
f = Focal length (+ for converging, - for diverging)

Lateral Magnification:

s = Object distance
s' = Image distance

Magnification:

h' = Height of image
h = Height of object

Snell's Law:

n = Index of refraction

Index of Refraction:

v = Velocity (m/s)

Critical Angle:

n₁ = Refractive index (denser material)
n₂ = Refractive index (less dense material)