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Fields in Physics
Connections between Gravitaitonal and Electric Fields
Gravitational fields are independent of the medium of the region, while electric fields are dependent on the pemittivity of the region.
Electric forces are orders of magnitude stronger than gravitational forces. This can be seen through their formulas, with the gravitational constant (G) being 6.67e-11, while Coulomb's constant (k) being 8.99e9.
Gravitaitonal fields only include attractive forces, while electric fields include both attractive and repulsive forces.
Both gravitational and electric fields involve some region in which a force is exerted on a test object (mass for gravitational, charge for electric).
Both gravitational and electric fields follow the inverse square law for measuring the intensity of their forces. The inverse square law states that the intensity of some physical quantity on an entity is inversely proportional to the squared distance between the entity and the source. For example, in both these forces, a doubling in the distance between the two masses/charges results in the gravitational/electrostatic force being quartered.
Visualization of the Inverse Square Law:
Both gravitational and electric fields are conservative (assuming the charges in the electric field are static. This means that work integral (variation of line integral of a vector field) only depends on the endpoints of the path, not the parameterization itself. This also means that parameterizations with the same initial and terminal point results in 0 work being done.
Connections between Electric and Magnetic Fields
Both electric and magnetic fields have attractive and repellant forces.
Both electric and magnetic fields are able to transform to the other in another reference frame.
The force in electric fields acts on charges regardless of the motion of the charges, while the force in magnetic fields only acts on moving charges given the cross product beween velocity and the magnetic field (B).
Both electric and magnetic fields are non-conservative (asumming the charges in the electric field are dynamic). This means that the work being doing over a path (defined by a parameterized line integral) will vary depending on the parameterization taken rather than just the initial and terminal points of the path.
The electric field (E) and magnet field (B) are orthogonal to one another. This is formally given by Maxwell's equations and intuitively given by the Right Hand Rule.
Maxwell's Equations:
Connections between Gravitational and Magnet Fields
Mangetic forces can be much stronger on a localized scale than gravitational forces, which are much weaker compared to the other fundamental forces (which include electromagnetic forces).
Gravitaitonal fields can be monopole fields and just point to one mass, while magnetic fields are always closed such that field lines move outwards of the North pole and inwards towards the South pole.
Gravitational fields have only attractive forces among masses, while magnetic fields have both attractive and repellant forces among magnets.
Gravitational fields are conservative vector fields, while magnet fields are not conservative vector fields. This means that the work done on a gravitational field is not reliant on the path/parameterization for gravitational fields (just the initial and terminal points of the path), while the work over magnetic fields is relliant of the path/parameterization taken.
Connections Between All 3 Fields
Superposition applies to all three fields, as any point thats affected by the vector fields created by multiple sources will have a resulting vector thats the sum of all the sources' effect on that point.
Conservative (Left) v.s. Non-conservative (Right) Vector Field:
Field lines never intersect. This is because fields are defined by vector fields and mathematically, each point can only have one associated vector with it. If multiple vectors existed on the same point (i.e. field lines intersecting) the definition of the field would be contradictory at that point.
Field lines can be straight or curved depending on the situation involving the field's entities. Field lines are generally straight when the attributed field is constant or uniform in some region. Field lines are generally curved when the attributed field in non-uniform; being influenced by some mass, charge, or current/polarity.
Field lines generally describe the stength and direction of the force field on any point of the region. This is more specfically denoted by the defined vector field of the region.
All three fields are defined to be vector fields of force; not scalar fields. This means that each point in the field (region/space) has a vector associated with it that has magnitude and direction.
Example of a Vector Field:
Electric Fields
Electric fields are force fields that span the region around charged particles (i.e. protons, electrons) that exert a force to other charged particles within the field. Like charges repel, while opposing charges attract one another. By convention of using a positive test charge, the test charge will be pushed away from regions with high potential/voltage (positive charges) and pulled into low potential/voltage (negative charges).
The electric field around a point charge has radial symmetry. This radial symmetry means that the electric field lines for a lone charge will be equally seperated circularly around the charge. The lines will point outward for positive charges and inward for negative charges.
Applicatons of Electric Fields
Particle accelerators use electric fields as the base to harness power by accelerating charged particles (i.e. protons and electrons) to extremely high speeds. The electric field is applied in multiple stages to increase the exerted force in the certain direction needed for the chrages to reach high velocities.
Electric Fields in Particle Accelerators:
The liquid crystal displays (LCDs) found in many monitors, TVs and other electronic devices use a liquid crystal layer to control the direction/orientation of light that passes through to provide accurate light information that should be outputted to the user. This is done with an electrode layer surrounding both sides of the electrode layer, providing electric potential (voltage) to any region of the liquid crystals and creating an electric field that enables precise control of light by orienting the liquid crystals in various ways.
Electric Fields in LCDs (for monitors and other electronic devices):
Electric fields are used in capacitors to store electrical energy through the charges being seperated to accumulate on conductive plates (with the plates themselves being seperated by some insulating material). The accumulated charges create the electric field within the capacitor.
Diagram of Electric Fields in Capacitors:
Electrostatic fields (Electric fields without dynamic charges) are conservative vector fields. This means that the work along some path of the field is not dependent on the paramterization taken, but rather just the initial and terminal points.
Electric fields follow the principle of superposition, in which any point on the field that is affected by multiple electric charges/sources will have its resulting vector be the sum of all the effects of each source's respective vector field.
There exists equipotential surfaces in which electric potential (voltage) is constant throughout the whole surface. These equipotential surfaces are othogonal to the electric field lines.
Electric Field Strength Formula:
Electric Field Stength for the special case of an electrostatic field:
Electric Field Lines:
Electric field lines will move away from positive charges and toward negative charges.
Electrostatic Force Formula:
The electrostatic force follows the inverse square law for its intensity measure. The inverse square law states that the intensity of some physical quantity (in this case electrostatic force) observed by an entity is inversely proportional to the square of the distance between the entity and the source. For example, a doubling in the distance between the charges results in the force being quartered.
Magnetic Fields
Magnetic Force Formula for Current Conducting Wire:
The magnetioc force formula for current conducting wire involves the cross product between the length of the wire and magnetic field (B). This means that the magnetude of the force is maxmimized when the length of the wire and the magnetic field are orthogonal to one another and minimized when they are parallel to one another.
Magnetic fields are force fields spanning the region around a magnetic object/current conductors, and moving charges, all experiencing a force due to magnets, electric currents, and changes in the electric field.
There are two different perspective when analyzing the direction of current in magnetic fields. The conventional perspective is when current flows from the postive terminal (high potential) to the negative terminal (low potential). The electron flow perspective is when current flows from the negative terminal (low potential) to positive (high potential). The right hand rule for various cases is based on the conventional flow of current. If the electron flow of current perspective is used, the respective left hand rules can be used.
The right hand rule for a moving charge. The thumb points in the direction of current, the index finger points in the direction of the magnetic field, and the middle finger points in the direction of magnetic force:
The right hand rule for a coiled conductor/solenoid. The thumb points in the direction of the magnetic field, while the fingers curl in the direction of current:
The magnetic field lines inside the coil are the strongest.
The right hand rule for a straight conductor. The thumb points in the direction of current while the other fingers curl in the direction of the magnetic field:
Applications of Magnetic Fields:
Magnetic fields are used in electromagnets, in which coiled conductive wire generates a strong electric current. This electric current can further be enhanced using a ferromagnetic core (such as iron).
Examples of how Magnet Fields are used in electromagnet systems:
Compass Navigation heavily uses the concept of magnetic fields. This is because in order to locate the directions of places relative to the North and South poles, compasses utilize the Earth's magnetic field and its own compass needle (which is a magnet itself) to get the south part of the needle to point to the North pole and the north part of the needle to point to the South pole.
Example of how compasses use the Earth's magnetic field to determine the locations of Earth's North and South poles:
Data Storage devices such as hard drive disks (HDD) and video tapes use ferromagnetic material (such as iron) disks that spin with read and write heads (which are magnets themselves) to retrieve or store data.
Example of how Magnetic Fields are used in hard drives storage devices:
Magnetic fields are not conservative vector fields. This means that the work done over a path will depend on the parameterization taken rather than just the initial and terminal points of the path. However, this is regardless of the fact that work is never done on charged particles on a magnetic field anyways because the force is always orthogonal to the velocity of the particle.
The superposition principle is applied to magnetic fields. This means that at any point within the magnetic field that is affected by multiple magnetic sources, the resultant vector at that point is the sum of the vector effects from all the sources applicable.
According to Faraday's Law, a changing magnetic field induces and electric field. In other words, as magnetic flux/magnetic field changes over time, an electric force is produced.
Faraday's Law Formula:
Magnetic Field Lines:
Magntic field lines form closed loops, always moving/directing away from the north pole of the magnet and into the south pole of the magnet.
Magnetic field lines represent the stength and direction of the magnetic field at any point. Magnetic field lines never intersect because of this notion since that would cause inconsistencies regarding direction in the vector field.
Magnetic Field Strength Formula:
By definition of the formula, the range of a magnetic field can technically be infinite; though the strength of the field will converge to zero.
The strength of a magnetic field is measured is teslas (T).
Magnetic Force Formula for Moving Charges:
The magnetic force formula for moving charges involves a cross product betwen the velocity of the particle and the magnetic field (B). This means that the magnitude of the magnetic force is maximized when the velocity of the particle and the magnetic field (B) are orthogonal to one other, and minimized when they are parallel to one another.
Gravitational Fields
Gravitational fields are force fields that exist in the space around all masses to extert an attraction force. Masses around each other will attract each other and experience each other's gravitational force.
Applications of Gravitatinal Fields
Ocean Tides:
Gravitational Fields can be used to describe the ocean tides experienced by the various large bodies of water found on Earth (i.e. Pacific, Atlantic, Indian Ocean). The ocean tides formed are a result of the gravitatioal force from the Sun and Moon.
Sun and Moon gravitational fields impacting the tidal waves of Earth's oceans:
Freefall due to Gravitational Acceleration:
Gravitational Fields can be used to describe the freefall acceleration coefficient on various planetary bodies. This is done by using the formula for the strength of a gravitational field (g=F/m).
Freefall due to the acceleration of gravity found by the formula for gravitational field strength (Earth being 9.81 m/s^2):
Planetary Motion/Orbits:
Gravitational Fields can be used to describe the movement of planets, stars, and other celestial objects under the influence of gravitational forces. Gravitational fields can be also used to explain and calculated the orbits of planets and satilites.
Orbit of a Satilite described by Gravitational field of the Earth:
Gravitational fields are conservative vector fields. Conservative vector fields are vector fields that are path independent, meaning the line integral (work) from one point to another is the same value regardless of the parameterization taken. This also means that parameterizations that have the same initial and terminal point will have a line integral value (work) of zero.
The property of superposition applies to gravitational force vector fields. The property of superposition denotes how any point in the field that is affected by multiple gravitational forces from the respective masses has the vector sum of all those fields created from each mass.
The force of gravity is relatively weak compared to the weak nuclear force, electromanetic force, and strong nuclear force. This means that unless the two objects are on orders of magnitude different masses, then the force of gravity is often negligible.
Gravitational Field Lines between the Earth and Moon:
Field Lines generally indicate the strength and direction of each point on the vector field.
Formula for Force of Gravity:
The force of gravity follows the inverse square law for its intensity measure. The inverse square law states that the intensity of some quantity (force of gravity in this case) observed by an entity is inversely proportional to the square of the distance between the entity and the source. For example, a doubling of distance results in force being quartered.
Gravitational Field Strength Formula:
By the definition of this formula, the force of gravity can technically have infinite range; though the magnitude of the force will approach zero.
