19.3: Electrical Potential Due to a Point Charge (2024)

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    Learning Objectives

    By the end of this section, you will be able to:

    • Explain point 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 distance.

    Point charges, such as electrons, are among the fundamental building blocks of matter. Furthermore, spherical charge distributions (like on a metal sphere) create external electric fields exactly like a point charge. The electric potential due to a point charge is, thus, a case we need to consider. Using calculus to find the work needed to move a test charge \(q\) from a large distance away to a distance of \(r\) from a point charge \(Q\), and noting the connection between work and potential \((W=-q\Delta V)\), we can define the electric potential \(V\) of a point charge:

    definition: ELECTRIC POTENTIAL \(V\) OF A POINT CHARGE

    The electric potential \(V\) of a point charge is given by

    \[V=\dfrac{kQ}{r}\: (\mathrm{Point\: Charge}). \label{eq1}\]

    where \(k\) is a constant equal to \(9.0 \times 10^{9}\, \mathrm{N}\cdot \mathrm{m^{2}/C^{2}}.\)

    The potential at infinity is chosen to be zero. Thus \(V\) for a point charge decreases with distance, whereas \(\mathbf{E}\) for a point charge decreases with distance squared:

    \[E=\dfrac{F}{q}=\dfrac{kQ}{r^{2}}.\]

    Recall that the electric potential \(V\) is a scalar and has no direction, whereas the electric field \(\mathbf{E}\) is a vector. To find the voltage due to a combination of point charges, you add the individual voltages as numbers. To find the total electric field, you must add the individual fields as vectors, taking magnitude and direction into account. This is consistent with the fact that \(V\) is closely associated with energy, a scalar, whereas \(\mathbf{E}\) is closely associated with force, a vector.

    Example \(\PageIndex{1}\): What Voltage Is Produced by a Small Charge on a Metal Sphere?

    Charges in static electricity are typically in the nanocoulomb \((\mathrm{nC})\) to microcoulomb \((\mu \mathrm{C})\) range. What is the voltage 5.00 cm away from the center of a 1-cm diameter metal sphere that has a \(-3.00 \mathrm{nC}\) 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 center. Thus we can find the voltage using Equation \ref{eq1}.

    Solution

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

    \[ \begin{align*} V&=k\dfrac{Q}{r} \\[5pt] &=(8.99 \times 10^{9} \, \mathrm{N}\cdot \mathrm{m^{2}/C^{2}}) \left(\dfrac{-3.00\times 10^{-9}\,\mathrm{C}}{5.00\times 10^{-2}\,\mathrm{m}}\right) \\[5pt] &= -539\, \mathrm{V}. \end{align*}\]

    Discussion

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

    Example \(\PageIndex{2}\): What Is the Excess Charge on a Van de Graaff Generator

    A demonstration Van de Graaff generator has a 25.0 cm diameter metal sphere that produces a voltage of 100 kV near its surface. (Figure \(\PageIndex{1}\)) What excess charge resides on the sphere? (Assume that each numerical value here is shown with three significant figures.)

    19.3: Electrical Potential Due to a Point Charge (2)

    Strategy

    The potential on the surface will be 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 can thus determine the excess charge using Equation \ref{eq1}.

    Solution

    Solving for \(Q\) and entering known values gives

    \[ \begin{align*} Q &=\dfrac{rV}{k} \\[5pt] &= \dfrac{(0.125 \,\mathrm{m})(100\times 10^{3}\, \mathrm{V})}{8.99\times 10^{9}\, \mathrm{N\cdot m^{2}/C^{2}}} \\[5pt] &= 1.39\times 10^{-6} \,\mathrm{C} \\[5pt] &= 1.39\, \mathrm{\mu C}.\end{align*}\]

    Discussion

    This is a relatively small charge, but it produces a rather large voltage. We have another indication here that it is difficult to store isolated charges.

    The voltages in both of these examples could be measured with a meter that compares the measured potential with ground potential. Ground potential is often taken to be zero (instead of taking the potential at infinity to be zero). It is the potential difference between two points that is of importance, and very often there is a tacit assumption that some reference point, such as Earth or a very distant point, is at zero potential. This is analogous to taking sea level as \(h=0\) when considering gravitational potential energy, \(\mathrm{PE_{g}}=mgh\).

    Summary

    • Electric potential of a point charge is \(V=kQ/r\).
    • Electric potential is a scalar, and electric field is a vector. Addition 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.
    19.3: Electrical Potential Due to a Point Charge (2024)

    FAQs

    What is the electric potential at a point due to a point charge? ›

    Electric potential of a point charge is V=kQr V = k Q r . Electric potential is a scalar, and electric field is a vector. Addition 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.

    How to calculate electric potential due to a point charge? ›

    Electric potential of a point charge is V=kQ/r V = k Q / r . Electric potential is a scalar, and electric field is a vector.

    What is the electric potential 15.0 cm from a point charge? ›

    Short Answer

    The electric potential 15.0 cm from a 3.00 μ C point charge is 1.80 × 10 5 V .

    How do you find the potential energy of a point charge? ›

    Potential energy of a point charge can be written as, W=V/Q ( since, work done is equal to energy , This is from ' WORK -><-ENERGY PRINCIPLE ). WKT, V= -Ed , this depends on the direction of E ( electric field ) . This is for electrons.

    What does electric potential at a point mean? ›

    The electric potential at a point in an electric field is the amount of work done moving a unit positive charge from infinity to that point along any path when the electrostatic forces are applied.

    What is the formula of potential? ›

    The electric potential formula is: V = W q . Also, electric potential or voltage can be calculated by using the equation: V = E ∗ r or the formula: V = R ∗ I depending on the information given in the excercise.

    What is the formula for electric potential due to two point charges? ›

    Step 1: Determine the distances r1 and r2 from each point charge to the location where the electric potential is to be found. Step 2: Apply the formula V = k Q r for both charges to calculate the potential due to each charge at the desired location. Step 3: Find the sum of the potentials of charges 1 and 2.

    What is the formula for potential energy due to charge? ›

    What is the electric potential energy formula ? The electrostatic potential energy formula, is written as U e = k q 1 q 2 r where U e stands for potential energy, r is the distance between the two charges, and k is the Coulomb constant which has a value of 8.99 ∗ 10 9 N m 2 / C 2 .

    What is the electric field formula due to a point charge? ›

    Example 1: Calculating the Electric Field of a Point Charge

    Calculate the strength and direction of the electric field E due to a point charge of 2.00 nC (nano-Coulombs) at a distance of 5.00 mm from the charge. We can find the electric field created by a point charge by using the equation E=kQ/r2 E = k Q / r 2 .

    What is the potential at a point due to a charge of 5 10 7 c located 10cm away? ›

    Thus, the electric potential at a point due to charge of 5 × 10⁻⁷ C located 10 cm away is​ 4.5 × 10⁴ volts.

    What is the electric potential a distance of 2.1 10 − 15 m away from a proton? ›

    The distance 'r' is given as 2.1 × 10^-15 m. Substituting these values in, we get: V = (8.99 × 10^9 N m^2/C^2)* (1.6 × 10^-19 C) * (1.6 × 10^-19 C) / (2.1 × 10^-15 m) which results in an electric potential energy of approximately 0.68 MeV (mega electron volts).

    What is the electric potential 16.0 cm from a 4.00 μC point charge? ›

    Final answer:

    The electric potential, V, of a point charge, is found using the formula V = kq/r. Given the charge of 4.00 μC and the distance of 16.0 cm, the electric potential comes out to approximately 2.24 x 10⁵ V.

    How to find electric potential due to a point charge? ›

    Summary
    1. Electric potential of a point charge is V=kQ/r.
    2. Electric potential is a scalar, and electric field is a vector. Addition 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.
    Feb 20, 2022

    How is the electric potential at a point given? ›

    The electric potential at a point (x,y,z) is given by V=-x^2y-xz^3 +4 the electric field E at that point is. Q.

    What is the electric field at point due to a point charge? ›

    The magnitude of the electric field E created by a point charge Q is E=k|Q|r2 E = k | Q | r 2 , where r is the distance from Q. The electric field E is a vector and fields due to multiple charges add like vectors.

    What is the electric field potential at a point? ›

    In an electric field the potential at a point is given by the following relation V=343r where r is distance from the origin.

    What is the electric potential due to a point charge and dipole? ›

    Ans: A dipole is a pair of opposite charges with equal magnitudes separated by a distance, d. The electric potential due to a point charge q at a distance of r from that charge is given by, V = (1/4πε0) q/r. Where ε0 is the permittivity of free space.

    What is the potential at a point due to a negative charge? ›

    It can be shown (see below for the derivation) that voltage is calculated by the formula [ k Q / R (or d) ] where k is Coulomb's Constant and Q is the amount of charge and R (or d) is the distance from the charge to where the potential is wished to be measured. With a negative charge, one gets a negative potential.

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