## Electric potential

### Voltage

Charged pwrite-ups exert pressures on each other. The electric field E= F/q created by a charged pshort article at some place r in space is a measure of the force F the pwrite-up exerts on a test charge q, if we area the test charge at r. The electric field E is avector. The electrical field as a result of a charge circulation is the vector sum of the fields produced by the charges comprising the distribution. When we think around electrical power in daily life, we seldom think around the electrical fields. We are even more acquainted via the principles of voltage, existing, and also power.

### How is the voltage regarded the electric field?

When a pwrite-up through charge q is placed in an outside electric area E, (i.e. an electrical area created by other charges), then an electric pressure F= qE will certainly act on it. If this pressure is not well balanced by various other pressures, then the particle will certainly accelerate, and also its kinetic power will change. The electric force will certainly carry out positive job-related on the pshort article. If the electrical force is balanced by an additional outside force Fext = -qE, and if this external force moves the pwrite-up against the electric pressure, than the outside pressure Fext does positive job-related.

Recall!

The job-related W done on an object by a constant pressure is defined as W = Fd = F d cosθ. Here F is the magnitude of the force, d is the magnitude of the displacement vector, and also θ is the angle between the directions of the force and displacement vectors.

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The work done by a varying pressure in one dimension is identified as W = ΣxixfFx∆x, as ∆x becomes infinitesimallysmall. In three dimensions we compose W = Σ F∙∆r = ΣxixfFx∆x + Σyiyf Fy∆y + ΣzizfFz∆z.

Thus the occupational done done by an outside force balancing the electric force is

W = Σ Fext∙∆r = -q Σ E∙∆r.

In the sums we always assume that the displacements become infinitesimally small.

The work done by the external pressure Fext = -qE is equal to the adjust in theelectrostatic potential energy of the pshort article in the outside area. The readjust in the potential power of a charge q when being relocated from point A to suggest B, is the work-related done by Fext in relocating the charge.

∆U = UB - UA = -q ΣABE∙∆r.

The amount is taken along a details course. But the electrostatic pressure is a conservative force. The job-related is independent of the path. ∆U therefore counts only on the endpoints A and B of the route, not on the actual route itself.

Module 2: Question 1 (a) Which requires zero work, to move a positive suggest charge from point P to suggest 1, 2, 3, or 4? All those points are the very same distance from P.(b) Which requires the most positive work done by an outside pressure, to move a positive point charge from point P to allude 1, 2, 3, or 4?

Discuss this with your fellow students in the conversation forum!Review the principle of work in juniorg8.comics.Discuss the connection in between the work done by and versus the electrical field and also the electrostatic potential energy.

The readjust in potential energy is proportional to the charge q. Its authorize counts on whether the charge is positive or negative. We specify thepotential difference orvoltage ∆V as the potential energy distinction divided by the charge, or the potential energy difference per unit (positive) charge. The potential difference is the potential energy difference of a tiny, positive test charge, separated by the charge. As in the case of gravity, the zero of the potential energy and also therefore the zero of the potential are not uniquely defined. Only potential power differences and also potential distinctions are distinct. The SI unit of power is J, therefore the SI unit of potential is J/C. We defineVolt as V = J/C.

The electrostatic potential is a scalar, not a vector.

We say that a charge circulation, which produces an electrical field, likewise produces an electric potential. The electrostatic potential produced by a finite charge circulation is, by convention, collection to zero at infinity. Then the potential V(r) of the distribution is the work-related done per unit charge in bringing a little test charge from infinity to position r. For a charge distributions which extfinish to infinity, we cannot set the zero of the potential at infinity, because then the potential would certainly be limitless everywhere, and it would be a usemuch less concept. We then collection the zero at some convenient referral point, but we constantly have to specify the reference allude in addition to the potential, given that tbelow is no distinct convention. For many applications we select the ground to be the zero of the potential. The potential difference between the two terminals of an A, B, C, or D cell battery is 1.5 V. For eincredibly Coulomb of negative charge that is relocated from the positive to the negative terminal, 1.5 J of job-related must be done against electrical forces, and 1.5 J of some various other form of power is converted into electrostatic energy. When the terminals are associated by a wire, then the charge is free to relocate inside the wire, and also the electric area does occupational on the charge. This occupational deserve to now be reconverted right into some various other create of power.

The potential at a suggest r of a positive suggest charge situated at the origin is the work-related that should be done per unit charge in bringing a test charge from infinity to r.V(r) = -Σ∞r E∙∆r = ΣrE∙∆r .We deserve to lug the test charge along an arbitrary route, which we deserve to think of as being comprised of infinitesimal little measures, either in the radial direction or perpendicular to the radial direction. We perform not need to perform job-related as soon as we step perpendicular to the radial direction, because then E is perpendicular to ∆r. Along the radial direction E∙r = E∆r, bereason E and also r points both suggest outside. The amount becomes

V(r) = q/(4πε0r) = keq/r.

This expression also provides the potential because of any kind of spherically symmetric charge circulation external the circulation.

The potential outside a spherically symmetric charge distribution through full charge q is the exact same as that of a allude charge q, V(r) = keq/r. It is is proportional to the inverse of the distance from the suggest charge.

To find the potential due to a collection of charges, we use the principle of superposition and add the potentials because of the individual charges. Because the potential is a scalar, and not a vector, we just have to add numbers. To find the total electrostatic potential power of a repertoire of allude charges, sum over all pairs.

Problem: The charges in the teams A and also B below are all provided in units of Q. Which team of charges took less occupational by an external pressure to bring together from infinity?

Solution:

Reasoning:For instance A:We have just one pair of charges.W = ke 2Q*Q/d = 2 keQ2/d.For instance B:We have 3 pairs of charges.Bring the 2 lower charges together: W = ke Q2/d.Bring the optimal charge:Do work versus the pressure exerted by the two reduced chargesW = ke Q2/d + ke Q2/d = 2 ke Q2/d.Total Work: W = 3 keQ2/d = sum over all pairs.Group A took much less job-related by an external force to lug together from infinity than team B.

### Units:

1 electron volt (eV) = 1.6*10-19 J. 1 eV is the change in potential energy of a pwrite-up via charge qe = 1.6*10-9 C when the adjust in potential is 1 Volt (V). The unit was characterized so that as soon as you recognize the voltage in between two points in space, you recognize the change in potential energy of an elementary particle once it moves from one to the other suggest. (The authorize of the readjust in potential power relies on the authorize of the charge.) When a complimentary proton moves with a potential distinction of 1 V its kinetic energy decreases by -qV = (1.6*10-19 C)*(1 J/C) = -1.6*10-19 J = -1 eV. When a complimentary electron moves through the very same potential distinction of 1 V its kinetic power rises by -qV = -(-1.6*10-19 C)*(1 J/C) = 1.6*10-19 J = 1 eV.

Problem: Assume that the potential difference between the positive and also negative plate is 100 V. A proton travels from the positive to the negative plate. What happens to the proton"s kinetic energy?

Solution:

Reasoning:The electrostatic pressure is a conservative force. ∆KE + ∆U = 0.(change in kinetic power + change in potential power = 0.)Details of the calculation:The proton accelerates in the direction of the negative plate. Its potential energy decreases by 100 eV.Its kinetic power boosts by 1.6*10-17 J = 100 eV.

### Equipotential lines

Just as we explained the electric field roughly a charged object by area lines, we deserve to additionally define the electric potential pictorially withequipotential surfaces (contour plots). Each surconfront coincides to a various fixed worth of the potential. Equipotential lines are lines connecting points of the exact same potential. They regularly show up on area line diagrams.

Equipotential lines are always perpendicular to field lines.

Equipotential lines are therefore perpendicular to the pressure competent by a charge in the area. If a charge moves alengthy an equipotential line, no job-related is done. If a charge moves in between equipotential lines, work is done. Example:

Field lines and equipotential lines for a positive suggest charge are displayed in the figure. This is a 2-dimensional depiction, a cut via the 3-d surfaces.Please likewise explore this 3-dimensional representation listed below. Please click on the image! Problem:

In the Bohr model of the hydrogen atom, the electron orbits the proton at a distance of r = 5.29*10-11 m. The proton has charge +qe and also the electron has charge -qe, where qe = 1.6*10-19 C. How much work need to be done to entirely separate the electron and the proton

Solution:

Reasoning:The force on the electron is the Coulomb pressure in between the proton and also the electron. It pulls the electron in the direction of the proton. For the electron to relocate in a circular orlittle, the Coulomb pressure have to equal the centripetal pressure. We need keqe2/r2 = mv2/r.Details of the calculation:mv2 = keqe2/r, so the kinetic power of the electron is KE(r) = ½mv2 = ½keqe2/r. The potential energy of the electron in the area of the positive proton allude charge is U(r) = -qeV(r) = - keqe2/r. The full power is the amount of the electron"s kinetic power and its potential energy.KE(r) + PE(r) = -½keqe2/r = (-½) (9*109)(1.60*10-19) /(5.29*10-11) J = -2.18*10-18 J. This is normally stated in power units of electron volts (eV). 1 eV = 1.60*10-19 J.-2.18*10-18 J * 1eV/(1.60*10-19 J) = -13.6 eV.To rerelocate the electron from the atom, i.e. to move it extremely far away and offer it zero kinetic power, 13.6 eV of work should be done by an outside force. 13.6 eV is the ionization power of hydrogen. Problem:

An alpha pshort article containing two proloads is shot straight in the direction of a platinum nucleus containing 78 proloads from an extremely big distance with a kinetic power of 1.7*10-12 J. What will certainly be the distance of closest approach?

Solution:

Reasoning:The electric charge of the alpha particle is q1 = 2qe and also that of the platinum nucleus is q2 = 78 qe. The alpha ppost and also the nucleus repel each various other. As the alpha pwrite-up moves in the direction of the nucleus, some of its kinetic energy will be converted into electrostatic potential power. At the distance of closest technique, the alpha particle"s velocity is zero, and also all its initial kinetic power has actually been converted into electrostatic potential power.Details of the calculation:The initial kinetic power of the alpha pwrite-up is KE = 1.7*10-12 J.The final potential power is U = kq1q2/d, wbelow d is the distance of closest strategy. We have keq1q2/d = 1.7*10-12J. This yieldsd = keq1q2/(1.7*10-12J) = (9*109*2*78*(1.6*10-19)2/(1.7*10-12)) m = 2.1*10-14 m for the distance of closest strategy. The alpha particle"s velocity is zero at d.The acceleration is in a direction ameans from the nucleus. The distance between the alpha ppost and also the nucleus will certainly rise aacquire, converting the potential energy ago into kinetic energy.Problem:

An evacuated tube supplies a voltage of 5 kV to acceleprice electrons from remainder to hit a phosphor screen.(a) How a lot kinetic energy does each electron gain?(b) What is the rate of an electron as soon as it hits the copper plate?

Solution:

Reasoning:1 eV is the change in potential energy of a ppost through charge qe = 1.6*10-9 C when the change in potential is 1 Volt (V). Details of the calculation:(a) Each electron loses 5 keV of potential energy and also gains 5 keV = (5000 eV)(1.6*10-19 J/eV) = 8*10-16 J of kinetic energy.(b) E = ½mev2. v2 = 2E/me = (2*8*10-16 J)/(9.1*10-31 kg) = 1.75*1015 (m/s)2.v = 4.2*107 m/s. This is more than 1/10 the rate of light.

The electrostatic potential V is concerned the electrostatic field E. If the electrical field E is well-known, the electrostatic potential V have the right to be derived utilizing V(r) = -Σ∞rE∙r.

### How have the right to we achieve the area from the potential? Consider the two points P1 and P2 presented in number over. Assume that they are separated by an infinitesimal distance ∆L. The readjust in the electrostatic potential between P1 and P2 is given by ∆V = -E∙∆L = -E ∆L cosθ.Here θ is the angle in between the direction of the electric area and the direction of the displacement vector ∆L. We deserve to recompose this equation as ∆V/∆L = -E cosθ = -EL.EL shows the component of the electrical area alengthy the direction of ∆L. If the direction of the displacement is liked to coincide through the x-axis, this becomes∆V/∆x = -Ex.For the displacements along the y-axis and z-axis we achieve ∆V/∆y = -Ey and also ∆V/∆z = -Ez.The total electric area E have the right to be derived from the electrostatic potential V by combining these three equations. We say that E is the negative gradientof the potential V.

In many electrostatic problems the electric field as a result of a specific charge distribution need to be evaluated. The calculation of the electrical field deserve to be lugged out making use of two various methods:

The electrical area deserve to be calculated by using Coulomb"s law and vector addition of the contributions from all charges in the charge circulation.The full electrostatic potential V have the right to be obtained from the algebraic amount of the potential as a result of all charges that comprise the charge distribution, and the field have the right to be uncovered by calculating the gradient of V.

In many cases the second method is easier, because the calculation of the electrostatic potential requires an algebraic amount, while the straight calculation of the field involves a vector sum. Problem:

The potential difference between the two plates of the capacitor shown listed below is 12 V. Equipotential surfaces are displayed. If the separation between the plates is 1 mm, what is the toughness of the electric field between the plates?Solution:

Reasoning:Let the y-axis point upward. V just varies with y. ∆V/∆x = ∆V/∆z = 0. E = Ey j = -∆V/∆y j.Details of the calculation:Let the y-axis allude upward. The area is unidevelop and points in the negative y-direction. ∆V/∆y = -Ey = (12 V)/(10-3 m). Ey = -12000 V/m.The strength of the electrical area between the plates is E = 12000 V/m = 12000 N/C.Problem: The graph on the right mirrors a contour map of the equipotential surdeals with because of 3 suggest charges. Estimate the magnitude and also direction of the electrical field at suggest P.

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Solution:

Reasoning:Point P lies in between two equipotential surdeals with. The surchallenge above P is at 40 V and also the surchallenge below is at 20 V. The electrical field points amethod from the positive charge, from greater to reduced potential. It points downward at P.Given the distance range, I estimate the perpendicular distance in between the 40 V and also 20 V equipontential surdeals with near point P to be ~2.5 cm.Details of the calculation:|E| = ∆V/∆y = 20 V/ 2.5*10-2 m = 800 V/m.The magnitude on of the electrical field at point P is ~800 V/m and it points downward.