These are my complete notes for Electric Potential in Electromagnetism.
I color-coded my notes according to their meaning - for a complete reference for each type of note, see here (also available in the sidebar). All of the knowledge present in these notes has been filtered through my personal explanations for them, the result of my attempts to understand and study them from my classes and online courses. In the unlikely event there are any egregious errors, contact me at jdlacabe@gmail.com.
Summary of Electric Potential (Electromagnetism)
Table Of Contents
XV. Electric Potential.
XV.I Voltage.
Electric potential energy works much the same as gravitational potential energy. As an object has greater gravitational potential energy the farther away it is from the source of its atraction, two oppositely charged objects, attracted to one another, will have greater electric potential energy the farther they are from one another. Their attraction inherently transforms this potential energy into kinetic energy as they attract, a result of the Conservation of Mechanical Energy.
Obversely, two charged objects that repel will have their greatest potential energy when they are closest, increasing in kinetic energy as they farthern from one another.
This is the essence of electric potential energy: the potential for two charged objects to attract or repel from one another, to move as a result of the electric force produced by their interacting electric fields.
The electric field vector points from higher potential toward lower potential.
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P. Rule .
Change in Electric Potential Energy: SCALAR.
Units: Joules.
Equation:
$$\Delta U_e = -q \int_{i}^{f} E \cos\theta \, dr$$
∆Ue = The change in electric potential energy. It does not depend on the path taken between point A and point B - rather, all that is required is the displacement itself. Measured in Joules.
q = The magnitude of the charge/charged particle of the particular point being measured.
i = A generic starting point of a path taken in an electric field.
f = A generic ending point of a path taken in an electric field.
E = The magnitude of the electric field at a particular point.
θ = The angle between the directions of the electric field and the direction of the infinitesimally small displacement dr. When the charge is moving in the direction opposite that of the electric field, the angle will be 180°, and the negative of the resulting cosine will cancel out with the one outside the integral.
dr = The infinitesimally small displacement of the charge, essentially its position.
Definition: Finding an equation to express electric potential energy is no difficult task at all. As a conservative force, the electric force can have all of the existing equations for potential energy and work (see Rule 100) applied:
FE = -(dUe / dr)
E.g., the electric force equals the negative of the derivative of the "electric potential energy" with respect to position (the infinitesimally small change in position dr).
By simplifying this equation slightly, as in moving variables around, integrating both sides, and substituting in values for electric force, a new and improved Change in Electric Potential equation can be developed, seen above.
The potential energy of a positive charge increases as it moves opposite the direction of the electric field, while the potential energy of the negative charge will decrease as it moves opposite the direction of the field. E.g., change in potential energy will be positive for a positive charge, and negative for a negative charge.
Units: Joules.
Equation:
$$\Delta U_e = -q \int_{i}^{f} E \cos\theta \, dr$$
∆Ue = The change in electric potential energy. It does not depend on the path taken between point A and point B - rather, all that is required is the displacement itself. Measured in Joules.
q = The magnitude of the charge/charged particle of the particular point being measured.
i = A generic starting point of a path taken in an electric field.
f = A generic ending point of a path taken in an electric field.
E = The magnitude of the electric field at a particular point.
θ = The angle between the directions of the electric field and the direction of the infinitesimally small displacement dr. When the charge is moving in the direction opposite that of the electric field, the angle will be 180°, and the negative of the resulting cosine will cancel out with the one outside the integral.
dr = The infinitesimally small displacement of the charge, essentially its position.
Definition: Finding an equation to express electric potential energy is no difficult task at all. As a conservative force, the electric force can have all of the existing equations for potential energy and work (see Rule 100) applied:
FE = -(dUe / dr)
E.g., the electric force equals the negative of the derivative of the "electric potential energy" with respect to position (the infinitesimally small change in position dr).
By simplifying this equation slightly, as in moving variables around, integrating both sides, and substituting in values for electric force, a new and improved Change in Electric Potential equation can be developed, seen above.
The potential energy of a positive charge increases as it moves opposite the direction of the electric field, while the potential energy of the negative charge will decrease as it moves opposite the direction of the field. E.g., change in potential energy will be positive for a positive charge, and negative for a negative charge.
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P. Rule .
Electric Potential: SCALAR (though an attribute of a vector Electric Field).
Units: Volts, e.g. Joules / Coulombs. The symbol for the volts unit, hilariously, is the same as the symbol for electric potential: V.
Equation:
V = (Ue / q)
V = Electric Potential, the measure of electric potential energy experienced per unit charge in an electric field. Measured in Volts.
Ue = The magnitude Electric Potential Energy, explained in detail in Rule 198 and in the introduction directly above. Note that this refers to the p.e. of a single point.
q = The magnitude of the charge/charged particle of the particular point being measured.
Definition: The Electric Potential is a measure of the electric potential energy experienced per unit charge in an electric field. V and ΔU are indeed different - they have the same relationship as electric field and electric force. The electric potential energy is what creates the electric potential.
Since Electric Potential is a scalar value, the net electric potential at a particular point is found by summing each electric potential values (from each electric field) at that point.
In the same manner in which an electric field is defined by the force experienced by a small, positive test charge (see Subsection XIII.I), the electric potential is defined by the energy experienced by a small, positive test charge.
Of considerable importance and inherent relation to electric potential is Electric Potential Difference, aka, voltage. For more information, see Rule 200.
Units: Volts, e.g. Joules / Coulombs. The symbol for the volts unit, hilariously, is the same as the symbol for electric potential: V.
Equation:
V = (Ue / q)
V = Electric Potential, the measure of electric potential energy experienced per unit charge in an electric field. Measured in Volts.
Ue = The magnitude Electric Potential Energy, explained in detail in Rule 198 and in the introduction directly above. Note that this refers to the p.e. of a single point.
q = The magnitude of the charge/charged particle of the particular point being measured.
Definition: The Electric Potential is a measure of the electric potential energy experienced per unit charge in an electric field. V and ΔU are indeed different - they have the same relationship as electric field and electric force. The electric potential energy is what creates the electric potential.
Since Electric Potential is a scalar value, the net electric potential at a particular point is found by summing each electric potential values (from each electric field) at that point.
In the same manner in which an electric field is defined by the force experienced by a small, positive test charge (see Subsection XIII.I), the electric potential is defined by the energy experienced by a small, positive test charge.
Of considerable importance and inherent relation to electric potential is Electric Potential Difference, aka, voltage. For more information, see Rule 200.
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P. Rule .
Voltage/Electric Potential Difference: SCALAR.
Units: Volts, e.g. Joules / Coulombs. The symbol for the volts unit, hilariously, is the same as the symbol for electric potential: V.
Equation:
$$\Delta V = \frac{\Delta U_e}{q}$$ $$\Delta V = - \int_{i}^{f} \vec{E} \cdot d\vec{r}$$
∆V = The Electric Potential Difference, the difference between the electric potentials of two points in an electric fields. Measured in Volts.
∆Ue = The change in electric potential energy. It does not depend on the path taken between point A and point B - rather, all that is required is the displacement itself. Measured in Joules.
q = The magnitude of the charge/charged particle of the particular point being measured.
i = A generic "starting" point in the electric field for measuring the electric potential difference.
f = A generic "ending" point in the electric field for measuring the electric potential difference.
E = The electric field at a particular point, the one that is generating the electric potential. When constant, it can be taken out of the integral, and the equation can resolve to ∆Vuniform = -Ed.
dr = The infinitesimally small displacement of the charge, essentially its position.
Definition: The change in electric potential, derived from the Electric Potential described in Rule 199, is arguably more important and used than electric potential by itself.
Represented by ∆V, the Electric Potential Difference (also known as Voltage) is the difference in the electric potential between two points: ∆V = Vf - Vi.
Equations representing ∆V, slightly reworked from those of Rule 199 & Rule 198, are given above. If Vi = 0, then these equations are merely calculating the electric potential at point f. When the electric field is uniform, then it can be taken out of the integral, and the equation can resolve to the following:
∆Vuniform = -Ed.
Inherently, a charge does not need to move from one point to another in order for the difference to exist; ∆V's existence is only dependent on the electric field itself.
Units: Volts, e.g. Joules / Coulombs. The symbol for the volts unit, hilariously, is the same as the symbol for electric potential: V.
Equation:
$$\Delta V = \frac{\Delta U_e}{q}$$ $$\Delta V = - \int_{i}^{f} \vec{E} \cdot d\vec{r}$$
∆V = The Electric Potential Difference, the difference between the electric potentials of two points in an electric fields. Measured in Volts.
∆Ue = The change in electric potential energy. It does not depend on the path taken between point A and point B - rather, all that is required is the displacement itself. Measured in Joules.
q = The magnitude of the charge/charged particle of the particular point being measured.
i = A generic "starting" point in the electric field for measuring the electric potential difference.
f = A generic "ending" point in the electric field for measuring the electric potential difference.
E = The electric field at a particular point, the one that is generating the electric potential. When constant, it can be taken out of the integral, and the equation can resolve to ∆Vuniform = -Ed.
dr = The infinitesimally small displacement of the charge, essentially its position.
Definition: The change in electric potential, derived from the Electric Potential described in Rule 199, is arguably more important and used than electric potential by itself.
Represented by ∆V, the Electric Potential Difference (also known as Voltage) is the difference in the electric potential between two points: ∆V = Vf - Vi.
Equations representing ∆V, slightly reworked from those of Rule 199 & Rule 198, are given above. If Vi = 0, then these equations are merely calculating the electric potential at point f. When the electric field is uniform, then it can be taken out of the integral, and the equation can resolve to the following:
∆Vuniform = -Ed.
Inherently, a charge does not need to move from one point to another in order for the difference to exist; ∆V's existence is only dependent on the electric field itself.
# Electric Field in terms of Electric Potential: VECTOR.
Units: Newtons / Coulombs. It is a type of electric field.
Equation:
Er = -(dV / dr)
Er = The magnitude of the electric field in the direction of r. It can be considered in the traditional units of N/C, but note that V/m are also legal units to apply.
dV = The derivative of Electric Potential (instantaneous electric potential at a point in space & time, though time is irrelevant since this is electrostatic).
dr = The infinitesimally small displacement of the charge, essentially its position.
Definition: Derived using the 2nd equation outlined in Rule 200, this new equation isolating Electric Field with respect to Electric Potential is very useful in a variety of circumstances, such as when a problem gives electric potential in the form of an equation.
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P. Rule .
Equipotential Surfaces/Lines:
An equipotential surface or line is one in which the electric potential is the same at every point along it. These lines form from (and have their shapes determined by) an electric field. Of course, this means that if the electric field is resulting from a negative charge, then each electric potential value along the equipotential line will be negative. If resulting from a positive charge, then each value will be positive. Simple as.
Equipotential Lines, also known as isolines, are always perpendicular to the electric field - thus, the electric field will have no components along the equipotential line.
In a uniform electric field, an equipotential line will be an infinite straight line. In an electric field around a charged particle, the equipotential lines will be circles concentric to the particle.
The electric potential difference between any point on one equipotential line to any point along another equipotential line will be the same.
When moving a charge along an equipotential line, the work required to do so is zero (since the electric potential difference will be zero - see Rule 207).
An equipotential surface or line is one in which the electric potential is the same at every point along it. These lines form from (and have their shapes determined by) an electric field. Of course, this means that if the electric field is resulting from a negative charge, then each electric potential value along the equipotential line will be negative. If resulting from a positive charge, then each value will be positive. Simple as.
Equipotential Lines, also known as isolines, are always perpendicular to the electric field - thus, the electric field will have no components along the equipotential line.
In a uniform electric field, an equipotential line will be an infinite straight line. In an electric field around a charged particle, the equipotential lines will be circles concentric to the particle.
The electric potential difference between any point on one equipotential line to any point along another equipotential line will be the same.
When moving a charge along an equipotential line, the work required to do so is zero (since the electric potential difference will be zero - see Rule 207).
XV.II Individual Electric Potentials.
As occurred when the revelation of the Charge-Flux Law resulted in multitudes of new electric field equations being discovered (see Subsection XIV.III), reflecting all manners of electrical shapes and phenomena (dipoles, shells, rods, etc.), all derived from the same basic fundamental equations (namely the charge density equations and the Charge-Flux law itself), the same effectively occurs for Electric Potential in the subsection below.
In this section, there can be found the multitudes of equations that describe the electric potential of different shapes and assortments/distributions, from electric dipoles to continuous charge distributions and beyond.
Note that this section specifically refers to electric potential, and that things like electric potential energy are dealt with in Subsections XV.I and XV.III.
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P. Rule .
Electric Potential around a Point Charge: SCALAR.
Units: Volts, e.g. Joules / Coulombs. The symbol for the volts unit, hilariously, is the same as the symbol for electric potential: V.
Equation:
Vpoint charge = (kq / r)
Vpoint charge = The electric potential created and surrounded by a point charge.
k = The Coulomb constant, equal to 8.99 × 10⁹ (N × m²) / (C²).
q = The charge of the central particle, measured in Coulombs.
r = The distance between the particle and the test charge.
Definition: In circumstances in which you are calculating the Electric Potential generated by a single point charge (which, since Electric Potential is a scalar, just means ignoring the electric fields of any other variables in the equation (if any)), there is a single equation that you can use that works regardless of whether the charge is negative or positive (though you will have to factor this into the sign of q):
Of course, the equation above for the electric potential of the particle is effectively assigning zero electric potential to any points infinitely far away. Occasionally, a problem will state that a particle moves from "a great distance" toward its final position. This is the problem's way of telling you that the initial electric potential and electric potential energy (by Rule 199) of the particle is zero.
Note that the first equation of Rule 200 can be substituted into this equation to create an alternate definition for change in potential energy.
Units: Volts, e.g. Joules / Coulombs. The symbol for the volts unit, hilariously, is the same as the symbol for electric potential: V.
Equation:
Vpoint charge = (kq / r)
Vpoint charge = The electric potential created and surrounded by a point charge.
k = The Coulomb constant, equal to 8.99 × 10⁹ (N × m²) / (C²).
q = The charge of the central particle, measured in Coulombs.
r = The distance between the particle and the test charge.
Definition: In circumstances in which you are calculating the Electric Potential generated by a single point charge (which, since Electric Potential is a scalar, just means ignoring the electric fields of any other variables in the equation (if any)), there is a single equation that you can use that works regardless of whether the charge is negative or positive (though you will have to factor this into the sign of q):
Of course, the equation above for the electric potential of the particle is effectively assigning zero electric potential to any points infinitely far away. Occasionally, a problem will state that a particle moves from "a great distance" toward its final position. This is the problem's way of telling you that the initial electric potential and electric potential energy (by Rule 199) of the particle is zero.
Note that the first equation of Rule 200 can be substituted into this equation to create an alternate definition for change in potential energy.
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P. Rule .
Net Electric Potential: SCALAR.
Units: Volts, e.g. Joules / Coulombs. The symbol for the volts unit, hilariously, is the same as the symbol for electric potential: V.
Equation:
$$V_{net} = \sum_{i=1}^{n} V_i = k \sum_{i=1}^{n} \frac{q_i}{r_i}$$
Vnet = The net electric potential of the system of charged particles.
n = The total number of charged particles.
i = The index variable for each individual charged particle.
Vi = The electric potential of the charged particle at index i.
k = The Coulomb constant, equal to 8.99 × 10⁹ (N × m²) / (C²).
qi = The charge of the ith particle.
ri = The distance between the particle at index i to the test charge.
Definition: Akin to calculating the center of mass of a system, finding the net electric potential of a system of particles is lightwork. It basically just involves summing the q/r values of each individual particle, and feeding that back into the equation from Rule 202.
The potential resulting from each charge is effectively calculated individually, summed thereafter. This sum is an algebraic sum (or scalar sum), not a vector sum like the sum used to calculate the electric field resulting from a group of charged particles.
Remember that the distance from the point of the test charge (necessary for finding the strength of the electric potential at said point) is found for each particle individually.
Units: Volts, e.g. Joules / Coulombs. The symbol for the volts unit, hilariously, is the same as the symbol for electric potential: V.
Equation:
$$V_{net} = \sum_{i=1}^{n} V_i = k \sum_{i=1}^{n} \frac{q_i}{r_i}$$
Vnet = The net electric potential of the system of charged particles.
n = The total number of charged particles.
i = The index variable for each individual charged particle.
Vi = The electric potential of the charged particle at index i.
k = The Coulomb constant, equal to 8.99 × 10⁹ (N × m²) / (C²).
qi = The charge of the ith particle.
ri = The distance between the particle at index i to the test charge.
Definition: Akin to calculating the center of mass of a system, finding the net electric potential of a system of particles is lightwork. It basically just involves summing the q/r values of each individual particle, and feeding that back into the equation from Rule 202.
The potential resulting from each charge is effectively calculated individually, summed thereafter. This sum is an algebraic sum (or scalar sum), not a vector sum like the sum used to calculate the electric field resulting from a group of charged particles.
Remember that the distance from the point of the test charge (necessary for finding the strength of the electric potential at said point) is found for each particle individually.
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P. Rule .
Electric Potential of an Electric Dipole: SCALAR.
Units: Volts, e.g. Joules / Coulombs. The symbol for the volts unit, hilariously, is the same as the symbol for electric potential: V.
Equation:
$$V_{dipole} = k \frac{p \cos \theta}{r^2}$$
Vdipole = The magnitude of the electric potential of an electric dipole at distance r from the dipole midpoint.
k = The Coulomb constant, equal to 8.99 × 10⁹ (N × m²) / (C²).
p = The magnitude of the electric dipole moment (see Rule 182), e.g. the charge of either particle multiplied by the distance between them.
θ = The angle between the dipole moment axis/vector (which points toward the positive charge) and the line extending from the dipole midpoint to the test charge.
d = The distance between the particles. Not used in the equation, but necessary for understanding r.
r = The distance between the test charge (the point at which the electric field is being calculated, which can be ANYWHERE insofar that r >>> d) and the dipole midpoint (d/2). r MUST be >>> to d, for reasons detailed in the proof (see below).
Definition: Unlike the equation for the elcetric field created by an electric dipole, which is limited to points along the dipole axis (due to the intense complexity of calculating the field magnitude anywhere else), the electric potential of any point in the electric field produced by an electric dipole is rather simple.
The derivation of the given equation, which requires finding the net electric potential of the dipole system (as explained in Rule 203), is fairly rudimentary and is presented here in image form for those interested. Note that in the proof, point P is the test charge.
Units: Volts, e.g. Joules / Coulombs. The symbol for the volts unit, hilariously, is the same as the symbol for electric potential: V.
Equation:
$$V_{dipole} = k \frac{p \cos \theta}{r^2}$$
Vdipole = The magnitude of the electric potential of an electric dipole at distance r from the dipole midpoint.
k = The Coulomb constant, equal to 8.99 × 10⁹ (N × m²) / (C²).
p = The magnitude of the electric dipole moment (see Rule 182), e.g. the charge of either particle multiplied by the distance between them.
θ = The angle between the dipole moment axis/vector (which points toward the positive charge) and the line extending from the dipole midpoint to the test charge.
d = The distance between the particles. Not used in the equation, but necessary for understanding r.
r = The distance between the test charge (the point at which the electric field is being calculated, which can be ANYWHERE insofar that r >>> d) and the dipole midpoint (d/2). r MUST be >>> to d, for reasons detailed in the proof (see below).
Definition: Unlike the equation for the elcetric field created by an electric dipole, which is limited to points along the dipole axis (due to the intense complexity of calculating the field magnitude anywhere else), the electric potential of any point in the electric field produced by an electric dipole is rather simple.
The derivation of the given equation, which requires finding the net electric potential of the dipole system (as explained in Rule 203), is fairly rudimentary and is presented here in image form for those interested. Note that in the proof, point P is the test charge.
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P. Rule .
Electric Potential of a Continuous Charge Distribution: SCALAR.
Units: Volts, e.g. Joules / Coulombs. The symbol for the volts unit, hilariously, is the same as the symbol for electric potential: V.
Equation:
$$V_{CCD} = k \int \frac{dq}{r}$$
VCCD = The electric potential that exists around a continuous charge distribution, calculated at point r in space.
dq = The infinitesimally small point charges, of which there is an infinite number of. This value can be substituted for any of the equivalent charge density derivatives (see Linear, Surface, and Volumetric), pursuant to the nature of a particular problem.
r = The distance between the infinitesimal charge dq and the point where the electric field is being calculated. Unlike the Law of Electric Force, this r is a function that varies depending on the charge, since there is technically an infinite number of charges (dq).
Definition: This equation is applicable for all objects with continuous charge distributions: spheres, rods, planes, what have you. Everything that isn't just a particle (or dipole), essentially.
The equation for the electrical potential of a CCD is remarkably similar to that of the electric field created by a CCD (see Rule 174) - minus the vector and the square on the distance.
Furthermore, the actual application of the equation is similar to that of electric field: That pesky 'dq' value within the integral can be substituted out by taking the derivative of any of the charge densities (see linear, surface, and volumetric), depending on what one is looking for. Additionally, in some circumstances 'r' itself has to be substituted in for another value as well, such as when a Pythagorean-theorem-style-event occurs relating to the variables given in the problem.
Units: Volts, e.g. Joules / Coulombs. The symbol for the volts unit, hilariously, is the same as the symbol for electric potential: V.
Equation:
$$V_{CCD} = k \int \frac{dq}{r}$$
VCCD = The electric potential that exists around a continuous charge distribution, calculated at point r in space.
dq = The infinitesimally small point charges, of which there is an infinite number of. This value can be substituted for any of the equivalent charge density derivatives (see Linear, Surface, and Volumetric), pursuant to the nature of a particular problem.
r = The distance between the infinitesimal charge dq and the point where the electric field is being calculated. Unlike the Law of Electric Force, this r is a function that varies depending on the charge, since there is technically an infinite number of charges (dq).
Definition: This equation is applicable for all objects with continuous charge distributions: spheres, rods, planes, what have you. Everything that isn't just a particle (or dipole), essentially.
The equation for the electrical potential of a CCD is remarkably similar to that of the electric field created by a CCD (see Rule 174) - minus the vector and the square on the distance.
Furthermore, the actual application of the equation is similar to that of electric field: That pesky 'dq' value within the integral can be substituted out by taking the derivative of any of the charge densities (see linear, surface, and volumetric), depending on what one is looking for. Additionally, in some circumstances 'r' itself has to be substituted in for another value as well, such as when a Pythagorean-theorem-style-event occurs relating to the variables given in the problem.
XV.III Kinetic Energy & Work.
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P. Rule .
Kinetic Energy through Charge Movement:
There are two equations that can be used to find the kinetic energy of a charged object with an electric potential difference, one for if only conservative forces are acting on it, and one for if nonconservative ones are as well.
If the charged object moves through an electric potential difference ∆V without an applied force acting on it (moving purely through electric attraction), then only conservative forces are acting upon it. Applying and simplifying the conservation of mechanical energy equation gives the following equation for change in kinetic energy:
∆K = -q × ∆V
∆K = Change in Electric Kinetic Energy.
q = The magnitude of the electric charge carried by the object, which is irrelevant to the positioning/movement of the object.
∆V = The electric potential difference between two points in an electric field.
If, instead, an external/applied force is acting on the charged object, creating nonconservative work, then the change in kinetic energy is as follows:
∆K = (-q × ∆V) + Wnonc
∆K = Change in Electric Kinetic Energy.
q = The magnitude of the electric charge carried by the object, which is irrelevant to the positioning/movement of the object.
∆V = The electric potential difference between two points in an electric field.
Wnonc = The work produced by the external, nonconservative force, breaking the conservation of mechanical energy.
A special case can emerge in which ∆K equals to zero. Then, and only then, can Wnonc be isolated in a very simple manner, just moving variables to the other side. For more details, see Rule 207 & Rule 208.
There are two equations that can be used to find the kinetic energy of a charged object with an electric potential difference, one for if only conservative forces are acting on it, and one for if nonconservative ones are as well.
If the charged object moves through an electric potential difference ∆V without an applied force acting on it (moving purely through electric attraction), then only conservative forces are acting upon it. Applying and simplifying the conservation of mechanical energy equation gives the following equation for change in kinetic energy:
∆K = -q × ∆V
∆K = Change in Electric Kinetic Energy.
q = The magnitude of the electric charge carried by the object, which is irrelevant to the positioning/movement of the object.
∆V = The electric potential difference between two points in an electric field.
If, instead, an external/applied force is acting on the charged object, creating nonconservative work, then the change in kinetic energy is as follows:
∆K = (-q × ∆V) + Wnonc
∆K = Change in Electric Kinetic Energy.
q = The magnitude of the electric charge carried by the object, which is irrelevant to the positioning/movement of the object.
∆V = The electric potential difference between two points in an electric field.
Wnonc = The work produced by the external, nonconservative force, breaking the conservation of mechanical energy.
A special case can emerge in which ∆K equals to zero. Then, and only then, can Wnonc be isolated in a very simple manner, just moving variables to the other side. For more details, see Rule 207 & Rule 208.
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P. Rule .
Work in moving a Charged Particle: SCALAR.
Units: Joules.
Equation:
Wparticle = q × ∆V (if ∆K = 0)
U = q × ∆V (if ∆K = 0)
Wparticle = The work performed upon the particle that causes to move, whether by an external, nonconservative force, or simply the force of an electric field. The work is equivalent to the electric potential energy.
U = The electric potential energy of the particle, equivalent to the work.
q = The magnitude of the electric charge carried by the particle, which is irrelevant to its positioning/movement.
∆V = The electric potential difference between two points in an electric field.
Definition: If a charged particle is moved from an initial point to a final point (differing points, obviously) via some force, then the force is performing work on the particle. Furthermore, this work will change the electric potential energy of the charge, making them equal (under the law established in Rule 100 relating work and potential energy).
The force that is causing the particle to move can either be an external, nonconservative force, or simply the force of an electric field. Either way, there is no net kinetic energy (prior to movement in the case of the electric field), and the same equation is used.
There is an equation that can be used to find this work value in relation to electric potential difference - however, it requires that there be no change in the kinetic energy of the charge. Thus, using this equation mandates either the breaking of the conservation of Mechanical Energy, or the determination of potential energy/work prior to any movement (prior to any conversion into kinetic energy).
Note that the given equation is the negative of the work produced by an electric field, which itself can be derived by applying Rule 100 equations for potential energy to the 1st equation found in Rule 200.
Units: Joules.
Equation:
Wparticle = q × ∆V (if ∆K = 0)
U = q × ∆V (if ∆K = 0)
Wparticle = The work performed upon the particle that causes to move, whether by an external, nonconservative force, or simply the force of an electric field. The work is equivalent to the electric potential energy.
U = The electric potential energy of the particle, equivalent to the work.
q = The magnitude of the electric charge carried by the particle, which is irrelevant to its positioning/movement.
∆V = The electric potential difference between two points in an electric field.
Definition: If a charged particle is moved from an initial point to a final point (differing points, obviously) via some force, then the force is performing work on the particle. Furthermore, this work will change the electric potential energy of the charge, making them equal (under the law established in Rule 100 relating work and potential energy).
The force that is causing the particle to move can either be an external, nonconservative force, or simply the force of an electric field. Either way, there is no net kinetic energy (prior to movement in the case of the electric field), and the same equation is used.
There is an equation that can be used to find this work value in relation to electric potential difference - however, it requires that there be no change in the kinetic energy of the charge. Thus, using this equation mandates either the breaking of the conservation of Mechanical Energy, or the determination of potential energy/work prior to any movement (prior to any conversion into kinetic energy).
Note that the given equation is the negative of the work produced by an electric field, which itself can be derived by applying Rule 100 equations for potential energy to the 1st equation found in Rule 200.
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P. Rule .
Work in moving a System of Particles (# > 1): SCALAR.
Units: Joules.
Equation:
Wsystem = (k × q1 × q2) / r (if ∆K = 0)
U = (k × q1 × q2) / r (if ∆K = 0)
Wsystem = The work performed upon the system of particles by a force (generally the electric force), equivalent to the electric potential energy.
U = The electric potential energy between a particle pair (see definition).
k = The Coulomb constant, equal to 8.99 × 10⁹ (N × m²) / (C²).
q1 = The magnitude of the electric charge of the first particle.
q2 = The magnitude of the electric charge of the second particle.
r = The distance between the two particles being considered (see definition).
Definition: The total potential energy of a system of particles is the sum of the potential energies for every pair of particles in the system. The given equation is for that of a two particle system - for systems involving more particles, simply sum the collective potential energies to get a "net" work or "net" electric potential energy, in the following fashion:
∆W = ∆U = (k / r) (q1 × q2) (q1 × q3) (q2 × q3)
The above example is for only three particles. Add more pairs as needed for bigger systems.
This equation works effectively like a combining of the equations from Rule 207 and Rule 202 - this is why the individual electric potential energies of each pair need to be summed to produce a net electric potential energy.
As known from Rule 100, Work is equal to Potential Energy when Kinetic Energy is equal to zero. Thus, using this equation mandates either the breaking of the conservation of Mechanical Energy, or the determination of potential energy/work prior to any movement (prior to any conversion into kinetic energy). As there are two particles that will naturally be drawn towards or away from eachother (depending on whether their signs match) by the electric force, there does not necessarily have to be an external force acting, and should never be assumed in a question if not explicitly stated. Either way, there is no net kinetic energy (prior to movement in the case of the electric field), and the same equation(s) can be used.
Units: Joules.
Equation:
Wsystem = (k × q1 × q2) / r (if ∆K = 0)
U = (k × q1 × q2) / r (if ∆K = 0)
Wsystem = The work performed upon the system of particles by a force (generally the electric force), equivalent to the electric potential energy.
U = The electric potential energy between a particle pair (see definition).
k = The Coulomb constant, equal to 8.99 × 10⁹ (N × m²) / (C²).
q1 = The magnitude of the electric charge of the first particle.
q2 = The magnitude of the electric charge of the second particle.
r = The distance between the two particles being considered (see definition).
Definition: The total potential energy of a system of particles is the sum of the potential energies for every pair of particles in the system. The given equation is for that of a two particle system - for systems involving more particles, simply sum the collective potential energies to get a "net" work or "net" electric potential energy, in the following fashion:
∆W = ∆U = (k / r) (q1 × q2) (q1 × q3) (q2 × q3)
The above example is for only three particles. Add more pairs as needed for bigger systems.
This equation works effectively like a combining of the equations from Rule 207 and Rule 202 - this is why the individual electric potential energies of each pair need to be summed to produce a net electric potential energy.
As known from Rule 100, Work is equal to Potential Energy when Kinetic Energy is equal to zero. Thus, using this equation mandates either the breaking of the conservation of Mechanical Energy, or the determination of potential energy/work prior to any movement (prior to any conversion into kinetic energy). As there are two particles that will naturally be drawn towards or away from eachother (depending on whether their signs match) by the electric force, there does not necessarily have to be an external force acting, and should never be assumed in a question if not explicitly stated. Either way, there is no net kinetic energy (prior to movement in the case of the electric field), and the same equation(s) can be used.
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P. Rule .
Electronvolts:
A new unit, more convenient for dealing with work & electric potential/kinetic energy, is the Electronvolt. An electronvolt is defined as the work required to move a single elementary charge e (such as that of an electron or proton) through an electric potential difference ΔV of exactly one volt, and can thus be used to represent very small amounts of energy.
1 eV = 1.602 × 10⁻¹⁹ J
A new unit, more convenient for dealing with work & electric potential/kinetic energy, is the Electronvolt. An electronvolt is defined as the work required to move a single elementary charge e (such as that of an electron or proton) through an electric potential difference ΔV of exactly one volt, and can thus be used to represent very small amounts of energy.
1 eV = 1.602 × 10⁻¹⁹ J
