As you recall, we identified the electric field, \\(\\vec E(\\vec r)\\), to be the electric pressure per unit charge. By defining an electric field all over in space, we were able to quickly determine the force on any kind of test charge, \\(q\\), even if it is the test charge is positive or an adverse (since the authorize of \\(q\\) will readjust the direction of the force vector, \\(q\\vec E\\)): \\<\\beginaligned \\vec E(\\vec r) &= \\frac\\vec F^E(\\vec r)q\\\\ \\therefore \\vec F^E(\\vec r)&=q\\vec E(\\vec r)\\endaligned\\> Similarly, we specify the electric potential, \\(V(\\vec r)\\), to be the electric potential energy per unit charge. This permits us to specify electric potential, \\(V(\\vec r)\\), almost everywhere in space, and then identify the potential energy of a details charge, \\(q\\), by simply multiplying \\(q\\) with the electric potential at that position in space. \\<\\beginaligned V(\\vec r) &= \\frac U(\\vec r)q\\\\ \\therefore U(\\vec r)&= q V(\\vec r)\\endaligned\\> The S.I. Unit for electrical potential is the “volt”, (V). Electrical potential, \\(V(\\vec r)\\), is a scalar ar whose worth is “the electric potential” at that place in space. A positive charge, \\(q=1\\textC\\), will thus have actually a potential power of \\(U=10\\textJ\\) if that is situated at a place in space where the electric potential is \\(V=10\\textV\\), due to the fact that \\(U=qV\\). Similarly, a negative charge, \\(q=-1\\textC\\), will certainly have an adverse potential energy, \\(U=-10\\textJ\\), in ~ the exact same location.

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Since only distinctions in potential energy are occupychristmas.orgically systematic (as readjust in potential energy is related to work), only changes in electric potential room occupychristmas.orgically meaningful (as electric potential is connected to electrical potential energy). A distinction in electric potential is generally called a “voltage”. One frequently makes a clear an option of wherein the electrical potential is zero (typically the ground, or infinitely much away), so the the ax voltage is supplied to explain potential, \\(V\\), rather of distinction in potential, \\(\\Delta V\\); this need to only it is in done once it is clear whereby the location of zero electric potential is defined.

We can define a free-falling mass by stating that the mass move from a an ar where it has actually high gravitational potential energy to a region of lower gravitational potential energy under the affect of the force of heaviness (the force associated with a potential energy always acts in the direction to decreases potential energy). The same is true for electric potential energy: charges will always experience a force in a direction come decrease their electric potential energy. However, positive charges will endure a pressure driving lock from areas of high electric potential to areas of low electrical potential, whereas an adverse charges will experience a force driving castle from regions of low electrical potential to areas of higher electric potential. This is because, for an adverse charges, the change in potential energy associated with moving through space, \\(\\Delta U\\), will be the an unfavorable of the corresponding readjust in electric potential, \\(\\Delta U=q\\Delta V\\), since the charge, \\(q\\), is negative.


Exercise \\(\\PageIndex1\\)

Electric potential rises along the \\(x\\) axis. A proton and also an electron are put at rest at the origin; in which direction perform the charges relocate when released?

the proton move towards an adverse \\(x\\), if the electron move towards confident \\(x\\). The proton move towards positive \\(x\\), if the electron moves towards negative \\(x\\). The proton and also electron move towards an unfavorable \\(x\\). The proton and also electron move towards positive \\(x\\). Answer

If the only force exerted on a bit is the electric force, and the particle moves in room such that the electric potential alters by \\(\\Delta V\\), we have the right to use preservation of power to identify the corresponding adjust in kinetic power of the particle:

\\<\\beginaligned \\Delta E&=\\Delta U+\\Delta K=0 \\\\ \\Delta U&=q\\Delta V \\endaligned\\>

\\<\\therefore \\Delta K=-q\\Delta V\\>

where \\(\\Delta E\\) is the adjust in complete mechanical power of the particle, which is zero when energy is conserved. The kinetic power of a hopeful particle increases if the fragment moves indigenous a region of high potential to a an ar of low potential (as \\(\\Delta V\\) would be negative and \\(q\\) is positive), and vice versa for a an adverse particle. This makes sense, because a hopeful and an unfavorable particle feel forces in the contrary directions.

In stimulate to explain the energies of particles such together electrons, it is practically to usage a different unit of power than the Joule, so that the quantities connected are no orders the magnitude smaller sized than 1. A common an option is the “electron volt”, . One electron volt coincides to the power acquired by a fragment with a fee of \\(e\\) (the fee of the electron) as soon as it is sped up by a potential distinction of \\(1\\textV\\): \\<\\beginaligned \\Delta E &= q\\Delta V\\\\ 1\\texteV&=(e)(1\\textV)=1.6\\times 10^-19\\textJ\\endaligned\\> one electron that has increased from rest across a an ar with a \\(150\\textV\\) potential difference across it will have a kinetic the \\(150\\texteV=2.4\\times 10^-17\\textJ\\). Together you have the right to see, the is less complicated to describe the energy of an electron in electron volts than Joules.



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It is often helpful in occupychristmas.orgics to take previously learned concepts and compare lock to brand-new ones, in this case, gravitational potential energy and electric potential power can be contrasted to help understand the occupychristmas.orgical meaning of electrical potential.

Suppose that an object with a huge mass, \\(M\\), is sit in space. Currently place an item of a much smaller mass, \\(m\\), at any type of distance, \\(r\\), from the center of \\(M\\). The gravitational potential power of the little mass is offered by the following formula: \\<\\beginaligned U_g&=\\fracGMmr\\endaligned\\> i m sorry is very comparable to the formula for electric potential energy: \\<\\beginaligned U(\\vec r)&=\\frackQqr\\endaligned\\> Now, if we were to remove the massive \\(m\\) native its position, we would certainly no much longer have an object with gravitational potential energy. However, we could still describe the gravitational potential for the point, \\(r\\), i m sorry would result in gravitational potential energy when any mass \\(m\\) is inserted there. This is the gravitational indistinguishable to electric potential, and can be identified as: \\<\\beginaligned V_g&=\\fracU_gm\\endaligned\\> i m sorry is additionally very comparable to the formula for electrical potential: \\<\\beginaligned V_E&=\\fracU_Eq\\endaligned\\> This compare is shown in number \\(\\PageIndex1\\).

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Figure \\(\\PageIndex1\\): Gravitational potential energy and also gravitational potential (left) beside its electrical analogue (right).

Example \\(\\PageIndex1\\)

A proton and an electron relocate from a an ar of room where the electrical potential is \\(20\\textV\\) come a an ar of room where the electric potential is \\(10\\textV\\). If the electrical force is the only force exerted on the particles, what have the right to you say about their readjust in speed?

Solution:

The two particles move from a region of an are where the electric potential is \\(20\\textV\\) to a region of space where the electrical potential is \\(10\\textV\\). The adjust in electrical potential skilled by the corpuscle is thus: \\<\\beginaligned \\Delta V = V_final-V_initial=(10\\textV)-(20\\textV)=-10\\textV\\endaligned\\> and we take it the chance to emphasize that one have to be an extremely careful with signs when utilizing potential. The adjust in potential power of the proton, through charge \\(q=+e\\), is thus: \\<\\beginaligned \\Delta U_p=q\\Delta V = (+e)(-10\\textV)=-10\\texteV\\endaligned\\> The potential power of the proton hence decreases by \\(10\\texteV\\) (which you can easily transform to Joules). Since we room told the no other pressure is exerted top top the particle, the complete mechanical power of the bit (kinetic add to potential energies) must be constant. Thus, if the potential power decreased, climate the kinetic power of the proton has increased by the very same amount, and also the proton’s rate increases.

The change in potential power of the electron, through charge \\(q=-e\\), is thus: \\<\\beginaligned \\Delta U_e=q\\Delta V = (-e)(-10\\textV) = 10\\texteV\\endaligned\\> The potential energy of the electron hence increases by \\(10\\texteV\\). Again, the mechanical energy of the electron is conserved, for this reason that an increase in potential energy results in the same decrease in kinetic energy and also the electron’s speed decreases.

Discussion:

By making use of the electric potential, \\(V\\), we modelled the readjust in electric potential power of a proton and an electron as they both relocated from one region of an are to another.

We discovered that as soon as a proton moves from a region of high electric potential come a an ar of lower electric potential, that potential energy decreases. This is due to the fact that the proton has a hopeful charge and also a to decrease in electrical potential will also an outcome in a decrease in potential energy. Because no other pressures are exerted top top the proton, the proton’s kinetic power must increase. Due to the fact that the potential energy of the proton decreases, the proton is moving in the same direction together the electrical force, and the electric force does positive work-related on the proton to boost its kinetic energy.

Conversely, we found that as soon as an electron moves from a region of high electric potential come a an ar of lower electric potential, the potential power increases. This is due to the fact that it has actually a an adverse charge and also a decrease in electrical potential therefore results in rise in potential energy. Because no other pressures are exerted top top the electron, the electron’s kinetic power must decrease, and the electron slows down. This renders sense, since the force that is exerted on one electron will be in the contrary direction from the pressure exerted top top a proton.



Exercise \\(\\PageIndex3\\)

What causes a positive charged particle to gain speed when it is increased through a potential difference?:

The particle accelerates because it loser potential energy as it moves from high to low potential. The particle increases because it loses potential power as it move from short to high potential The particle increases because it gains potential energy. The particle speeds up because it moves towards an unfavorable charges. Answer

Example \\(\\PageIndex2\\)

What is the electric potential at the edge of a hydrogen atom (a street of \\(1\\) indigenous the proton), if one to adjust \\(0\\text V\\) in ~ infinity? If an electron is located at a distance of \\(1\\) from the proton, exactly how much energy is compelled to eliminate the electron; that is, exactly how much energy is compelled to ionize the hydrogen atom?

Solution:

We can easily calculate the electrical potential, a street of \\(1\\unicodexC5\\) from a proton, since this coincides to the potential from a point charge (with \\(C=0\\)): \\<\\beginaligned V(\\vec r)=\\frackQr=\\frac(9\\times 10^9\\textN\\cdot\\textm^2\\text/C^2)(1.6\\times 10^-19\\textC)(1\\times 10^-10\\textm)=14.4\\textV\\endaligned\\> We have the right to calculate the potential energy of the electron (relative come infinity, wherein the potential is \\(0\\text V\\), since we determined \\(C=0\\)): \\<\\beginaligned U=(-e)V=(-1.6\\times 10^-19\\textC)(14.4\\textV)=-14.4\\texteV=-2.3\\times 10^-18\\textJ\\endaligned\\> wherein we also expressed the potential energy in electron volts. In bespeak to remove the electron indigenous the hydrogen atom, we should exert a pressure (do work) until the electron is infinitely much from the proton. At infinity, the potential power of the electron will be zero (by our an option of \\(C=0\\)). When relocating the electron indigenous the hydrogen atom come an limitless distance away, we have to do positive work-related to counter the attractive force from the proton. The work that we must do is specifically equal to the adjust in potential power of the electron (and equal to the negative of the work done by the force exerted through the proton):

\\<\\beginaligned W=\\Delta U=(U_final-U_initial)=(0\\textJ--2.3\\times 10^-18\\textJ)=2.3\\times 10^-18\\textJ \\endaligned\\>

The positive job-related that we must do, exerting a pressure that is opposite come the electrical force, is positive and equal come \\(2.3\\times 10^-18\\textJ\\), or \\(14.4\\texteV\\). If friend look up the ionization power of hydrogen, girlfriend will uncover that that is \\(13.6\\texteV\\), so that this very simplistic model is quite specific (we might improve the version by adjusting the proton-electron distance so the the potential is \\(13.6\\textV\\)).

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Discussion:

In this example, we identified the electrical potential energy of one electron in a hydrogen atom, and also found that it is negative, when potential power is characterized to be zero at infinity. In stimulate to eliminate the electron indigenous the atom, we have to do positive work in stimulate to rise the potential power of the electron indigenous a an unfavorable value come zero (the potential energy at infinity). This is analogous come the work-related that need to be done on a satellite in a gravitationally bound orbit for it to reach escape velocity.