what is the distance from the point charge to where you took the measurement?

Learning Objectives

By the cease of this section, you volition be able to:

• Explicate betoken charges and express the equation for electric potential of a point charge.
• Distinguish between electric potential and electric field.
• Determine the electric potential of a point charge given charge and altitude.

Bespeak charges, such as electrons, are among the fundamental building blocks of affair. Furthermore, spherical charge distributions (like on a metallic sphere) create external electrical fields exactly like a point accuse. The electric potential due to a point charge is, thus, a case we need to consider. Using calculus to find the piece of work needed to movement a test charge q from a large distance abroad to a altitude of r from a point accuse Q , and noting the connection between work and potential (Due west = −qΔV), it can be shown that the electric potential V of a point charge is [latex]Five=\frac{kQ}{r}\\[/latex] (Bespeak Accuse), where k is a constant equal to 9.0 × ten⁹ N · thousandⁱⁱ/C².

Electrical Potential V of a Signal Charge

The electric potential 5 of a betoken charge is given by

[latex]\displaystyle{V}=\frac{kQ}{r}\\[/latex] (Point Charge)

The potential at infinity is chosen to be nix. Thus 5 for a point charge decreases with distance, whereas E for a point charge decreases with distance squared:

[latex]\displaystyle{East}=\frac{F}{q}=\frac{kQ}{r^two}\\[/latex].

Retrieve that the electrical potential V is a scalar and has no direction, whereas the electrical field E is a vector. To find the voltage due to a combination of indicate charges, you add the individual voltages every bit numbers. To find the full electrical field, yous must add the private fields as vectors , taking magnitude and management into business relationship. This is consequent with the fact that V is closely associated with energy, a scalar, whereas E is closely associated with forcefulness, a vector.

Instance 1. What Voltage Is Produced by a Pocket-size Charge on a Metallic Sphere?

Charges in static electricity are typically in the nanocoulomb (nC) to microcoulomb (µC) range. What is the voltage 5.00 cm away from the center of a 1-cm diameter metal sphere that has a −three.00 nC static charge?

Strategy

As we have discussed in Electric Charge and Electric Field, charge on a metal sphere spreads out uniformly and produces a field like that of a point charge located at its eye. Thus we tin find the voltage using the equation [latex]Five=yard\frac{Q}{r}\\[/latex].

Solution

Entering known values into the expression for the potential of a point accuse, we obtain

[latex]\begin{array}{lll}V&=&g\frac{Q}{r}\\\text{ }&=&\left(8.99\times10^9\text{ N}\cdot\text{m}^2\text{/C}^2\correct)\left(\frac{-3.00\times10^{-9}\text{ C}}{v.00\times10^{-2}\text{ m}}\right)\\\text{ }&=&-539\text{ 5}\end{array}\\[/latex]

Discussion

The negative value for voltage ways a positive charge would exist attracted from a larger altitude, since the potential is lower (more negative) than at larger distances. Conversely, a negative accuse would be repelled, as expected.

Case two. What Is the Backlog Accuse on a Van de Graaff Generator

Effigy 1. The voltage of this demonstration Van de Graaff generator is measured between the charged sphere and ground. Earth'south potential is taken to be zip as a reference. The potential of the charged conducting sphere is the same as that of an equal point charge at its center.

A demonstration Van de Graaff generator has a 25.0 cm diameter metal sphere that produces a voltage of 100 kV most its surface. (Encounter Effigy 1.) What backlog accuse resides on the sphere? (Assume that each numerical value here is shown with three significant figures.)

Strategy

The potential on the surface will exist the same as that of a point charge at the center of the sphere, 12.5 cm away. (The radius of the sphere is 12.5 cm.) We tin thus determine the excess charge using the equation [latex]5=\frac{kQ}{r}\\[/latex].

Solution

Solving for Q and entering known values gives

[latex]\begin{array}{lll}Q&=&\frac{rV}{one thousand}\\\text{ }&=&\frac{\left(0.125\text{ m}\correct)\left(100\times10^{3}\text{ V}\correct)}{8.99\times10^nine\text{ N}\cdot\text{m}^2\text{/C}^2}\\\text{ }&=&1.39\times10^{-6}\text{ C}=i.39\mu\text{C}\end{array}\\[/latex]

Give-and-take

This is a relatively small charge, but information technology produces a rather large voltage. We have some other indication hither that it is difficult to store isolated charges.

The voltages in both of these examples could exist measured with a meter that compares the measured potential with ground potential. Ground potential is oftentimes taken to be zilch (instead of taking the potential at infinity to be goose egg). It is the potential difference between 2 points that is of importance, and very often there is a tacit assumption that some reference indicate, such as Earth or a very distant point, is at null potential. As noted in Electrical Potential Energy: Potential Divergence, this is analogous to taking sea level equally h=0 when considering gravitational potential free energy, PE_g =mgh.

Section Summary

• Electric potential of a point charge is [latex]5=\frac{kQ}{r}\\[/latex] .
• Electric potential is a scalar, and electric field is a vector. Add-on of voltages as numbers gives the voltage due to a combination of point charges, whereas addition of individual fields as vectors gives the total electric field.

Conceptual Questions

1. In what region of space is the potential due to a uniformly charged sphere the aforementioned as that of a betoken accuse? In what region does it differ from that of a betoken accuse?
2. Tin the potential of a not-uniformly charged sphere exist the same equally that of a point charge? Explain.

Problems & Exercises

1. A 0.500 cm diameter plastic sphere, used in a static electricity demonstration, has a uniformly distributed 40.0 pC accuse on its surface. What is the potential near its surface?
2. What is the potential 0.530 × 10^−ten yard from a proton (the boilerplate distance between the proton and electron in a hydrogen atom)?
3. (a) A sphere has a surface uniformly charged with 1.00 C. At what distance from its center is the potential five.00 MV? (b) What does your answer imply almost the practical aspect of isolating such a large charge?
4. How far from a 1.00 μC point accuse will the potential exist 100 V? At what distance will information technology be two.00 × 10² V?
5. What are the sign and magnitude of a indicate accuse that produces a potential of −2.00 V at a distance of 1.00 mm?
6. If the potential due to a point charge is 5.00 × x² V at a distance of 15.0 m, what are the sign and magnitude of the charge?
7. In nuclear fission, a nucleus splits roughly in half. (a) What is the potential 2.00 × 10⁻¹⁴ grand from a fragment that has 46 protons in it? (b) What is the potential free energy in MeV of a similarly charged fragment at this altitude?
8. A research Van de Graaff generator has a 2.00-m-diameter metal sphere with a accuse of five.00 mC on it. (a) What is the potential nearly its surface? (b) At what distance from its center is the potential ane.00 MV? (c) An oxygen atom with three missing electrons is released near the Van de Graaff generator. What is its energy in MeV at this altitude?
9. An electrostatic pigment sprayer has a 0.200-k-diameter metal sphere at a potential of 25.0 kV that repels paint aerosol onto a grounded object. (a) What accuse is on the sphere? (b) What accuse must a 0.100-mg drop of pigment have to make it at the object with a speed of 10.0 thou/southward?
10. In one of the archetype nuclear physics experiments at the beginning of the 20th century, an alpha particle was accelerated toward a gold nucleus, and its path was substantially deflected by the Coulomb interaction. If the energy of the doubly charged alpha nucleus was v.00 MeV, how close to the gold nucleus (79 protons) could it come earlier being deflected?
11. (a) What is the potential between two points situated ten cm and xx cm from a three.0 µC bespeak charge? (b) To what location should the point at 20 cm exist moved to increment this potential difference by a gene of 2?
12. Unreasonable Results. (a) What is the final speed of an electron accelerated from rest through a voltage of 25.0 MV by a negatively charged Van de Graaff terminal? (b) What is unreasonable about this upshot? (c) Which assumptions are responsible?

Selected Solutions to Problems & Exercises

1. 144 V

iii. (a) 1.80 km; (b) A accuse of 1 C is a very large corporeality of accuse; a sphere of radius 1.80 km is not practical.

five. −2.22 × x^−thirteen C

7. (a) 3.31 × 10^vi V; (b) 152 MeV

9. (a) ii.78 × x⁻⁷ C; (b) two.00 × ten⁻¹⁰ C

12. (a) ii.96 × 10^nine chiliad/south; (b) This velocity is far too swell. It is faster than the speed of lite; (c) The supposition that the speed of the electron is far less than that of lite and that the problem does not crave a relativistic treatment produces an answer greater than the speed of light.

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Source: https://courses.lumenlearning.com/physics/chapter/19-3-electrical-potential-due-to-a-point-charge/

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