Otis Singleton
Electric charge is the fundamental force behind nearly every macroscopic force studied in classical mechanics. It is conserved, meaning it can neither be created nor destroyed. Electric forces are often compared to gravity as they are the only two forces that act at a distance. However, unlike gravity, electric forces have two types of quantities responsible for the force, which are attractive and repulsive forces. When two of the same types of charges are brought together, the force is repulsive, while when different types of charges are brought together, the force is attractive. The force exerted by a moving electric charge is given by the Liénard-Wiechert potential, where the effect of the charge does not travel faster than light.
| Characteristics | Values |
|---|---|
| Nature of electrical forces | Attractive and repulsive forces |
| Comparison with gravitational forces | Similar to gravity, electrical forces act at a distance |
| Dependence on time | Formulas are valid only if time is constant |
| Propagation speed | The force propagates at the speed of light |
| Change in force | The force does not change instantaneously |
| Conservation of electric charge | Electric charge is conserved; it can be transferred but not created or destroyed |
| Role in classical mechanics | The electric force is the fundamental force behind most macroscopic forces in classical mechanics |
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What You'll Learn
Electric charge is conserved
Mathematically, the law of charge conservation can be expressed as a continuity equation:
${\displaystyle {\frac {\mathrm {d} Q}{\mathrm {d} t}}={\dot {Q}}_{\rm {IN}}(t)-{\dot {Q}}_{\rm {OUT}}(t).}$
In this equation, ${\displaystyle \mathrm {d} Q/\mathrm {d} t}$ represents the electric charge accumulation rate in a specific volume at time t, while ${\displaystyle {\dot {Q}}_{\rm {IN}}}$ and ${\displaystyle {\dot {Q}}_{\rm {OUT}}}$ represent the amount of charge flowing into and out of the volume, respectively, as functions of time.
The conservation of electric charge implies that the change in the amount of electric charge in a volume of space is equal to the amount of charge flowing into the volume minus the amount of charge flowing out. This can also be understood in terms of charge density and electric current density using vector calculus in electromagnetic field theory. The charge density continuity equation equates the rate of change of charge density at a point with the divergence of the current density at the same point, demonstrating that the charge density can only change if a current of charge flows into or out of that point.
Experimental tests, such as searches for particle decays that would occur if electric charge was not conserved, have provided strong evidence for the conservation of electric charge. For example, no observations have been made of an electron decaying into a neutrino and a single photon, as would be expected if charge was not conserved. Additionally, the fact that photons have zero mass, as demonstrated by experimental evidence, also supports charge conservation due to its link with gauge invariance in the electromagnetic field.
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Electric force vs gravity
Electric force and gravitational force are two fundamental forces of nature. While the former is a result of the interaction of charged particles, the latter is the force of attraction between two masses.
Electric forces act at a distance, but the force is not propagated instantly. It takes time for the information to get from one point to another. The force exerted by a moving electric charge is given by the Liénard-Wiechert potential, which shows that the effect of the charge does not travel faster than light.
The electric force is much stronger than the force of gravity. Consider two spheres, each with one kilogram of mass and one coulomb of electric charge. There will be electrical repulsion pushing them apart and gravitational attraction pulling them together. The electric force between these spheres is 1.35 x 10^20 times stronger than the gravitational force.
The gravitational force is so weak that it is surprising we have noticed it at all. It is always attractive and cumulative. All the atoms in the Earth pull us towards its centre, giving us weight. On the other hand, the electrical forces of the electrons and nuclei of atoms have opposite charges and cancel each other out, so we do not experience any "electrical weight" from the Earth.
In a thought experiment, we can compare the force of gravity to the electric force between two apples. A medium-sized apple weighs roughly 100 grams and exerts a downward force of about 1 newton when held in the hand. This is the force of attraction between the apple and the Earth. However, the electric force between two apples is zero because they have equal numbers of positive and negative charges, resulting in electrical neutrality.
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Attractive and repulsive forces
Electric charge is conserved, meaning it can be neither created nor destroyed. When particles change, the new particles' charges must add up to the same charge as the original particle. These charges reside on substances that can be divided into two types: conductors, where charges are free to move, and insulators, where charges are locked in place.
The electric force is the fundamental force behind nearly every macroscopic force studied in classical mechanics. When we observe two neutrally charged particles, we do not see a force, so it cannot be their masses causing it. In addition to an attractive force, there is also a repulsive force, which is not seen in gravity. This suggests that there must be two different types of quantities responsible for the force. Gravity has only one type—mass—but the electric force has two different types of "mass," referred to as electric charge.
When two charges of the same kind are brought together, the force is repulsive, while bringing together different types of charges results in an attractive force. This behaviour is similar to Newton's law of universal gravitation, where the force is proportional to the masses of the two bodies and inversely proportional to the square of the distance between them. However, unlike Newtonian physics, where forces change instantaneously, electromagnetic forces do not change instantaneously. It takes time for the information to travel from one point to another, and the effect of the charge does not travel faster than light.
The Liénard-Wiechert potential describes the electromagnetic field of a moving electric charge and shows that the effect of the charge does not propagate instantly. This is further supported by the concept of retarded potentials, where the field propagates at the speed of light. These potentials are important in understanding the behaviour of electromagnetic fields and forces at a distance.
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Conductors and insulators
Conductors are materials that allow electric charges to move through them with ease. Examples of conductors include metals and salty water. In these materials, some electrons are not bound to individual atoms or specific sites within the material. These free electrons can move through the material in a similar way to air moving through loose sand.
Insulators, on the other hand, do not allow electric charges to escape or move through them easily. Examples of insulators include glass, pure water, and dry table salt. In insulators, electrons and ions are bound to the structure and cannot move freely. In fact, the movement of charges in insulators can be up to 10^23 times slower than in conductors.
The key difference between conductors and insulators is the mobility of their electrons. In conductors, free electrons can move through the material, while in insulators, electrons and ions are bound and cannot move easily. This difference has important implications for how these materials interact with electric forces.
When a charged object is brought near a neutral substance, whether it is a conductor or an insulator, the distribution of charges in the atoms and molecules of the neutral substance is slightly shifted. Opposite charges are attracted to the external charged object, while like charges are repelled. However, the way this plays out differs between conductors and insulators.
In the case of an insulator, the charged object must physically touch the insulator to transfer charge. For example, when a positively charged glass rod touches an electroscope, electrons are attracted to the top of the electroscope and are transferred to the rod, leaving the electroscope with a net positive charge.
Conductors, on the other hand, can be influenced by charged objects from a distance. This is because the electrostatic force decreases with distance, and the attraction of unlike charges is stronger than the repulsion of like charges. So, even without direct contact, a charged object can still influence the distribution of charges in a conductor.
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Liénard-Wiechert potential
Electric fields are important in many areas of physics and are fundamental to electrical technology. The electric field is defined in terms of force, and force is a vector with magnitude and direction. Electric fields act between two charges, similar to how gravitational fields act between two masses. Both electrostatic and gravitational forces obey an inverse-square law with distance.
In the context of "action at a distance in physics", if two charges are at a very large distance, and if either charge moves, the force associated with the charges would change instantaneously. However, according to Einstein, no information can travel faster than the speed of light, and photons (the carriers of information in electromagnetic force) cannot deliver information instantaneously. This presents a contradiction, as the force associated with the charges should not be able to change instantaneously if the information about the change in motion of the charge cannot be transmitted faster than the speed of light.
The Liénard-Wiechert potentials provide a solution to this contradiction. They describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. These potentials, stemming directly from Maxwell's equations, describe the complete, relativistically correct, time-varying electromagnetic field for a point charge in arbitrary motion. The Liénard-Wiechert potentials show that the effect of the charge does not travel faster than light, and thus, the force does not change instantaneously.
The Liénard-Wiechert potentials are expressed as:
${\displaystyle \mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\left({\frac {q\mathbf {v} }{|\mathbf {r} -\mathbf {r} _{s}|(1-{\boldsymbol {\beta }}_{s}\cdot (\mathbf {r} -\mathbf {r} _{s})/|\mathbf {r} -\mathbf {r} _{s}|)}}\right)_{t_{r}}={\frac {\mu _{0}c}{4\pi }}\left({\frac {q{\boldsymbol {\beta }}_{s}}{(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})|\mathbf {r} -\mathbf {r} _{s}|}}\right)_{t_{r}}}$
Where:
- ${\displaystyle \mathbf {A} }$ is the vector potential
- ${\displaystyle \mu _{0}}$ is the permeability of free space
- ${\displaystyle q}$ is the charge
- ${\displaystyle \mathbf {v} }$ is the velocity of the charge
- ${\displaystyle \mathbf {r} }$ is the position vector of the observation point
- ${\displaystyle \mathbf {r} _{s}}$ is the position vector of the charge
- ${\displaystyle t_{r}}$ is the retarded time
- ${\displaystyle c}$ is the speed of light
- ${\displaystyle {\boldsymbol {\beta }}_{s}={\frac {\mathbf {v} }{c}}}$ is the velocity of the charge divided by the speed of light
The Liénard-Wiechert potentials are non-trivial to calculate and require several steps. They are essential in describing the electromagnetic field of a moving electric point charge and ensuring that the effects of the charge propagate at the speed of light, in accordance with Einstein's theory of relativity.
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Frequently asked questions
Electric force is the fundamental force behind nearly every macroscopic force studied in classical mechanics.
There are two types of electric forces: attractive and repulsive.
Electric forces are conserved, meaning they cannot be created nor destroyed. When particles change, the new particles' charges must add up to the original particle's charge.
Yes, electric forces act at a distance, similar to gravitational forces. However, unlike gravitational forces, electric forces can be either attractive or repulsive.
The strength of electric forces is inversely proportional to the square of the distance between the charges. As the distance between charges increases, the force decreases, and vice versa.
