what is the potential of point a relative to point c?

Learning Objectives

By the end of this section, y'all will be able to:

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

Point charges, such as electrons, are amidst the key building blocks of matter. Furthermore, spherical charge distributions (like on a metal sphere) create external electrical fields exactly like a point accuse. The electrical potential due to a point charge is, thus, a case we need to consider. Using calculus to find the piece of work needed to motility a test accuse q from a large distance away to a distance of r from a betoken charge Q , and noting the connection betwixt work and potential (W = −qΔV), it can be shown that the electric potential Five of a betoken charge is [latex]V=\frac{kQ}{r}\\[/latex] (Betoken Accuse), where k is a constant equal to 9.0 × 10^nine North · mⁱⁱ/C².

Electric Potential V of a Point Charge

The electric potential 5 of a point accuse is given by

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

The potential at infinity is chosen to be nada. Thus Five for a point charge decreases with altitude, whereas Due east for a betoken charge decreases with altitude squared:

[latex]\displaystyle{Due east}=\frac{F}{q}=\frac{kQ}{r^2}\\[/latex].

Recall that the electric potential V is a scalar and has no direction, whereas the electric field E is a vector. To find the voltage due to a combination of point charges, you add the private voltages as numbers. To observe the total electric field, you must add together the private fields every bit vectors , taking magnitude and direction into business relationship. This is consequent with the fact that 5 is closely associated with free energy, a scalar, whereas East is closely associated with force, a vector.

Example one. What Voltage Is Produced by a Minor 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 middle of a ane-cm diameter metal sphere that has a −three.00 nC static accuse?

Strategy

Equally we have discussed in Electric Accuse and Electric Field, charge on a metal sphere spreads out uniformly and produces a field like that of a point charge located at its centre. Thus nosotros tin observe the voltage using the equation [latex]V=one thousand\frac{Q}{r}\\[/latex].

Solution

Inbound known values into the expression for the potential of a signal charge, we obtain

[latex]\begin{assortment}{lll}V&=&chiliad\frac{Q}{r}\\\text{ }&=&\left(8.99\times10^nine\text{ Due north}\cdot\text{m}^two\text{/C}^two\right)\left(\frac{-3.00\times10^{-9}\text{ C}}{5.00\times10^{-2}\text{ thou}}\right)\\\text{ }&=&-539\text{ V}\end{assortment}\\[/latex]

Discussion

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

Case two. What Is the Excess Charge on a Van de Graaff Generator

Figure 1. The voltage of this demonstration Van de Graaff generator is measured between the charged sphere and footing. Earth's potential is taken to be nothing every bit a reference. The potential of the charged conducting sphere is the same as that of an equal point charge at its eye.

A demonstration Van de Graaff generator has a 25.0 cm diameter metal sphere that produces a voltage of 100 kV near its surface. (See Figure ane.) What excess charge resides on the sphere? (Assume that each numerical value here is shown with iii significant figures.)

Strategy

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

Solution

Solving for Q and inbound known values gives

[latex]\begin{array}{lll}Q&=&\frac{rV}{k}\\\text{ }&=&\frac{\left(0.125\text{ m}\right)\left(100\times10^{3}\text{ 5}\right)}{8.99\times10^9\text{ N}\cdot\text{one thousand}^2\text{/C}^2}\\\text{ }&=&ane.39\times10^{-half-dozen}\text{ C}=1.39\mu\text{C}\end{array}\\[/latex]

Discussion

This is a relatively small-scale accuse, but information technology produces a rather large voltage. We have another indication here that it is difficult to shop isolated charges.

The voltages in both of these examples could be measured with a meter that compares the measured potential with basis potential. Footing potential is oftentimes taken to be zero (instead of taking the potential at infinity to be zero). It is the potential deviation between two points that is of importance, and very often there is a tacit assumption that some reference betoken, such as Earth or a very distant point, is at aught potential. As noted in Electric Potential Free energy: Potential Difference, this is analogous to taking body of water level equally h=0 when considering gravitational potential energy, PE_g =mgh.

Section Summary

• Electric potential of a point charge is [latex]V=\frac{kQ}{r}\\[/latex] .
• Electric potential is a scalar, and electric field is a vector. Improver of voltages equally numbers gives the voltage due to a combination of point charges, whereas improver of individual fields as vectors gives the full electric field.

Conceptual Questions

1. In what region of space is the potential due to a uniformly charged sphere the same equally that of a bespeak accuse? In what region does it differ from that of a point charge?
2. Tin the potential of a non-uniformly charged sphere be the same as that of a point charge? Explain.

Issues & Exercises

1. A 0.500 cm diameter plastic sphere, used in a static electricity demonstration, has a uniformly distributed 40.0 pC charge on its surface. What is the potential near its surface?
2. What is the potential 0.530 × 10⁻¹⁰ m from a proton (the average altitude between the proton and electron in a hydrogen cantlet)?
3. (a) A sphere has a surface uniformly charged with 1.00 C. At what distance from its heart is the potential 5.00 MV? (b) What does your answer imply about the practical attribute of isolating such a large accuse?
4. How far from a 1.00 μC point charge will the potential exist 100 Five? At what distance will information technology be 2.00 × ten² V?
5. What are the sign and magnitude of a point accuse that produces a potential of −2.00 Five at a altitude of ane.00 mm?
6. If the potential due to a point charge is five.00 × x² 5 at a distance of 15.0 one thousand, what are the sign and magnitude of the accuse?
7. In nuclear fission, a nucleus splits roughly in half. (a) What is the potential 2.00 × 10^−xiv one thousand from a fragment that has 46 protons in it? (b) What is the potential energy in MeV of a similarly charged fragment at this distance?
8. A research Van de Graaff generator has a 2.00-thousand-diameter metal sphere with a charge of 5.00 mC on it. (a) What is the potential near its surface? (b) At what altitude from its center is the potential 1.00 MV? (c) An oxygen atom with three missing electrons is released well-nigh the Van de Graaff generator. What is its energy in MeV at this distance?
9. An electrostatic paint sprayer has a 0.200-k-diameter metallic sphere at a potential of 25.0 kV that repels pigment droplets onto a grounded object. (a) What charge is on the sphere? (b) What charge must a 0.100-mg drop of paint have to make it at the object with a speed of 10.0 m/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 blastoff nucleus was 5.00 MeV, how shut to the gold nucleus (79 protons) could information technology come before existence deflected?
11. (a) What is the potential betwixt two points situated 10 cm and 20 cm from a iii.0 µC point accuse? (b) To what location should the indicate at 20 cm be moved to increase this potential difference by a factor of two?
12. Unreasonable Results. (a) What is the final speed of an electron accelerated from rest through a voltage of 25.0 MV past a negatively charged Van de Graaff terminal? (b) What is unreasonable nigh this upshot? (c) Which assumptions are responsible?

Selected Solutions to Bug & Exercises

1. 144 V

3. (a) one.80 km; (b) A accuse of ane C is a very big amount of charge; a sphere of radius 1.80 km is not practical.

5. −2.22 × 10^−thirteen C

7. (a) iii.31 × 10⁶ V; (b) 152 MeV

9. (a) two.78 × 10^−vii C; (b) two.00 × ten^−ten C

12. (a) 2.96 × 10⁹ m/southward; (b) This velocity is far as well swell. It is faster than the speed of lite; (c) The assumption that the speed of the electron is far less than that of light and that the trouble does not require 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/