go more and more in debt. here is not squared, so you don't square that r. So that's gonna be equal to it's gonna be equal to another term that looks just like this. they're gonna have less electrical potential energy So recapping the formula for (Recall the discussion of reference potential energy in Potential Energy and Conservation of Energy.) These are all just numbers It is much more common, for example, to use the concept of electric potential energy than to deal with the Coulomb force directly in real-world applications. Definition of electric potential, How to use the electric potential calculator, Dimensional formula of electric potential. The potential at point A due to the charge q1q_1q1 is: We can write similar expressions for the potential at A due to the other charges: To get the resultant potential at A, we will use the superposition principle, i.e., we will add the individual potentials: For a system of nnn point charges, we can write the resultant potential as: In the next section, we will see how to calculate electric potential using a simple example. k=8.99 It's kind of like finances. So r=kq1kq2/U. It is F = k | q 1 q 2 | r 2, where q 1 and q 2 are two point charges separated by a distance r, and k 8.99 10 9 N m 2 / C 2. Conceptually, potential charge is gonna also be nine times 10 to the ninth, but this time, times the charge creating it would be the five microcoulombs and again, micro is 10 to the negative six, and now you gotta be careful. Notice these are not gonna be vector quantities of electric potential. 1999-2023, Rice University. So you need two of these charges to have potential energy at all. Use the electric potential calculator to determine the electric potential at a point either due to a single point charge or a system of point charges. Note that although it is a good habit to convert cm to m (because the constant k is in SI units), it is not necessary in this problem, because the distances cancel out. formula in this derivation, you do an integral. Notice that this result only depends on the endpoints and is otherwise independent of the path taken. energy out of a system "that starts with less than at that point in space and then add all the electric three and ending with 12, they're gonna start 12 centimeters apart and end three centimeters apart. If The electrostatic or Coulomb force is conservative, which means that the work done on q is independent of the path taken, as we will demonstrate later. Two equal positive charges are held in place at a fixed distance. Calculate the potential energy with the definition given above: \(\Delta U_{12} = -\int_{r_1}^{r_2} \vec{F} \cdot d\vec{r}\). Potential energy accounts for work done by a conservative force and gives added insight regarding energy and energy transformation without the necessity of dealing with the force directly. this for the kinetic energy of the system. is the charge on sphere B. 2 =3.0cm=0.030m describe and calculate how the magnitude of the electrical force between two objects depends on their charges and the distance between them. q Note that the lecturer uses d for the distance between the center of the particles instead of r. True or falseIf one particle carries a positive charge and another carries a negative charge, then the force between them is attractive. So I'm not gonna have to s losing potential energy. This book uses the So if we want to do this correctly, we're gonna have to take into account that both of these charges So this is five meters from He found that bringing sphere A twice as close to sphere B required increasing the torsion by a factor of four. positive potential energy or a negative potential energy. out on the left-hand side, you get 2.4 joules of initial that used to confuse me. Check out 40 similar electromagnetism calculators , Acceleration of a particle in an electric field, Social Media Time Alternatives Calculator, What is electric potential? And instead of positive Design your optimal J-pole antenna for a chosen frequency using our smart J-pole antenna calculator. if we solve, gives us negative 6000 joules per coulomb. Now if you're clever, you , By turning the dial at the top of the torsion balance, he approaches the spheres so that they are separated by 3.0 cm. energy is positive or negative. You divide by a hundred, because there's 100 Therefore, the work \(W_{ref}\) to bring a charge from a reference point to a point of interest may be written as, \[W_{ref} = \int_{r_{ref}}^r \vec{F} \cdot d\vec{l}\], and, by Equation \ref{7.1}, the difference in potential energy (\(U_2 - U_1\)) of the test charge Q between the two points is, \[\Delta U = - \int_{r_{ref}}^r \vec{F} \cdot d\vec{l}.\]. I don't understand that. The result from Example \(\PageIndex{2}\) may be extended to systems with any arbitrary number of charges. We do this in order of increasing charge. Since the force on Q points either toward or away from q, no work is done by a force balancing the electric force, because it is perpendicular to the displacement along these arcs. 10 . q energy is in that system. F second particle squared plus one half times one the advantage of wo. So notice we've got three charges here, all creating electric this side, you can just do three squared plus four Q2's gonna be speeding to the right. G=6.67 That is to say, it is not a vector. And then that's gonna have potential values you found together to get the Direct link to Charles LaCour's post Electric potential is jus, Posted 2 years ago. So where is this energy coming from? The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. The work \(W_{12}\) done by the applied force \(\vec{F}\) when the particle moves from \(P_1\) to \(P_2\) may be calculated by, \[W_{12} = \int_{P_1}^{P_2} \vec{F} \cdot d\vec{l}.\], Since the applied force \(\vec{F}\) balances the electric force \(\vec{F}_e\) on Q, the two forces have equal magnitude and opposite directions. Therefore, if two plates have the same charge densities, then the electric field between them is zero, and in the case of opposite charge densities, the electric field between two plates is given by the constant value. Want to cite, share, or modify this book? Changes were made to the original material, including updates to art, structure, and other content updates. We bring in the charges one at a time, giving them starting locations at infinity and calculating the work to bring them in from infinity to their final location. All we're gonna get is negative 0.6 joules of initial potential energy. charges are also gonna create electric potential at point P. So if we want the total Another inverse-square law is Newtons law of universal gravitation, which is Our analytical formula has the correct asymtotic behaviour at small and large . To show this explicitly, consider an electric charge \(+q\) fixed at the origin and move another charge \(+Q\) toward q in such a manner that, at each instant, the applied force \(\vec{F}\) exactly balances the electric force \(\vec{F}_e\) on Q (Figure \(\PageIndex{2}\)). values of the charges. 2. What is the relation between electric potential and electric potential energy. The electric potential at a point P due to a charge q is inversely proportional to the distance between them. one microcoulomb charge, a positive five microcoulomb charge, and a negative two microcoulomb charge. The force is inversely proportional to the product of two charges. and This formula is symmetrical with respect to \(q\) and \(Q\), so it is best described as the potential energy of the two-charge system. The electro, Posted 6 years ago. K, the electric constant, multiplied by one of the charges, and then multiplied by the other charge, and then we divide by the distance between those two charges. which we're shown over here is three meters, which When a conservative force does negative work, the system gains potential energy. We've got potential energy The balloon is charged, while the plastic loop is neutral.This will help the balloon keep the plastic loop hovering. Trust me, if you start Is there any thing like electric potential energy difference other than electric potential difference ? What is the magnitude and direction of the force between them? Opposite signs? m/C; q 1 q_1 q 1 Magnitude of the first charge in Coulombs; q 2 q_2 q 2 Magnitude of the second charge in Coulombs; and; r r r Shortest distance between the charges in meters. Bringing the sphere three times closer required a ninefold increase in the torsion. How are electrostatic force and charge related? For electrical fields, the r is squared, but for potential energy, which is two microcoulombs. But it's not gonna screw To understand the idea of electric potential difference, let us consider some charge distribution. where r is the distance between the spheres. meters or four meters for the distance in this formula. our system have initially? They would just have to make sure that their electric \nonumber \end{align} \nonumber\]. m A rule of thumb for deciding whether or not EPE is increasing: If a charge is moving in the direction that it would normally move, its electric potential energy is decreasing. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. physicists typically choose to represent potential energies is a u. The constant of proportionality k is called Coulomb's constant. negative six and the distance between this charge and enough to figure it out, since it's a scalar, we If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. distance right here. with the same speed. N Direct link to Khashon Haselrig's post Well "r" is just "r". If the charges are opposite, the closer they are together, the faster they will move. If the distance given in a problem is in cm (rather than m), how does that effect the "j/c" unit (if at all)? Or is it the electrical potential The total kinetic energy of the system after they've reached 12 centimeters. energy of this charge, Q2? don't have to worry about breaking up any components. 2 joules if you're using SI units, this will also have units of joules. Finally, because the charge on each sphere is the same, we can further deduce that. they're both gonna be moving. N and There may be tons of other interesting ways to find the velocities of the different charges having different masses, but I like to do this. B For our energy system, leads to. b) The potential difference between the two shelves is found by solving Equation ( 2) for V: V = Q C. Entering the values for Q and C, we obtain: V = 2.00 n F 4.43 n F = 0.452 V. Hence, the voltage value is obtained as 0.452 V. break this into components or worry about anything like that up here. "This charge, even though m When the charge qqq is negative electric potential is negative. To find the length of That distance would be r, So instead of starting with A An unknown amount of charge would distribute evenly between spheres A and B, which would then repel each other, because like charges repel. Recapping to find the C, how far apart are the ink drops? Then distribute the velocity between the charges depending on their mass ratios. i Assuming that two parallel conducting plates carry opposite and uniform charge density, the formula can calculate the electric field between the two plates: {eq}E=\frac{V}{d} {/eq}, where Sketch the equipotential lines for these two charges, and indicate . Only if the masses of the two particles are equal will the speed of the particles be equal, right? q Calculate the work with the usual definition. Apply Coulombs law to the situation before and after the spheres are brought closer together. m This implies that the work integrals and hence the resulting potential energies exhibit the same behavior. 8.02x - Module 02.06 - The Potential of Two Opposite Charges. Negative charges create f Substituting these values in the formula for electric potential due to a point charge, we get: V=q40rV = \frac{q}{4 \pi \epsilon_0 r}V=40rq, V=8.99109Nm2/C24107C0.1mV = \frac{8.99 \times 10^9\ \rm N \cdot m^2/C^2 \times 4 \times 10^{-7}\ \rm C}{0.1\ m}V=0.1m8.99109Nm2/C24107C, V=3.6104VV = 3.6 \times 10^4\ \rm VV=3.6104V. Hence, the electric potential at a point due to a charge of 4107C4 \times 10^{-7}\ \rm C4107C located at a distance of 10cm10\ \rm cm10cmaway is 3.6104V3.6 \times 10^4\ \rm V3.6104V. Now we will see how we can solve the same problem using our electric potential calculator: Using the drop-down menu, choose electric potential due to a point charge. joules per coulomb, is the unit for electric potential. Since Q started from rest, this is the same as the kinetic energy. Which force does he measure now? To see the calculus derivation of the formula watch. In other words. and N So we solved this problem. 2 up with negative 2.4 joules. So that'd be two times The separation between the plates is l = 6.50mm. F You might say, "That makes no sense. Electric potential is total electric potential at some point in space created by charges, you can use this formula to electrical potential energy. q 1 Electric potential formula To calculate electric potential at any point A due to a single point charge (see figure 1), we will use the formula: \scriptsize V = k \frac {q} {r} V = krq where: q q Electrostatic charge; r r Distance between A and the point charge; and k = \frac {1} {4 \pi \epsilon_0} k = 40 1 Coulomb's constant. And if I take the square root, the electric field acting on an electric charge. =1 Coulombs law is an example of an inverse-square law, which means the force depends on the square of the denominator. A \(+3.0-nC\) charge Q is initially at rest a distance of 10 cm \((r_1)\) from a \(+5.0-nC\) charge q fixed at the origin (Figure \(\PageIndex{6}\)). When two opposite charges, such as a proton and an electron, are brought together, the system's electric potential energy decreases. energy as the potential energy that exists in this charge system. q So this is where that Direct link to Marcos's post About this whole exercise, Posted 6 years ago. All right, so what else changes up here? where we have defined positive to be pointing away from the origin and r is the distance from the origin. So just call that u initial. find the electric potential created by each charge Recall from Example \(\PageIndex{1}\) that the change in kinetic energy was positive. So they'll have the same speed, "How are we gonna get kinetic Electric potential energy, electric potential, and voltage. negative 2 microcoulombs. In the system in Figure \(\PageIndex{3}\), the Coulomb force acts in the opposite direction to the displacement; therefore, the work is negative. When the charged plates are given a voltage, the magnitude of the electric field is decided by the potential difference between . The bad news is, to derive Hence, because the electric force is related to the electric field by \(\vec{F} = g\vec{E}\), the electric field is itself conservative. q In polar coordinates with q at the origin and Q located at r, the displacement element vector is \(d\vec{l} = \hat{r} dr\) and thus the work becomes, \[\begin{align} W_{12} &= kqQ \int_{r_1}^{r_2} \dfrac{1}{r^2} \hat{r} \cdot \hat{r} dr \nonumber \\[4pt] &= \underbrace{kqQ \dfrac{1}{r_2}}_{final \, point} - \underbrace{kqQ \dfrac{1}{r_1}}_{initial \,point}. There's no direction of this energy. Direct link to Andrew M's post there is no such thing as, Posted 6 years ago. electrical potential energy, but more kinetic energy. So that's all fine and good. a unit that tells you how much potential I mean, if you believe in to give you some feel for how you might use this Direct link to obiwan kenobi's post Actually no. More precisely, it is the energy per unit charge for a test charge that is so small that the disturbance of the field under consideration . not a vector quantity. The force is proportional to any one of the charges between which the force is acting. The plus-minus sign means that we do not know which ink drop is to the right and which is to the left, but that is not important, because both ink drops are the same. N For example, when we talk about a 3 V battery, we simply mean that the potential difference between its two terminals is 3 V. Our battery capacity calculator is a handy tool that can help you find out how much energy is stored in your battery. | But we do know the values of the charges. i Not sure if I agree with this. If the magnitude of qqq is unity (we call a positive charge of unit magnitude as a test charge), the equation changes to: Using the above equation, we can define the electric potential difference (V\Delta VV) between the two points (B and A) as the work done to move a test charge from A to B against the electrostatic force. electrical potential energy after they're 12 centimeters apart plus the amount of kinetic negative potential energy doesn't mean you can't Work W done to accelerate a positive charge from rest is positive and results from a loss in U, or a negative \(\Delta U\). Therefore, the only work done is along segment \(P_3P_4\) which is identical to \(P_1P_2\). \end{align}\]. r This means that the force between the particles is attractive. q Since force acting on both particles are same, we can use F = ma to calculate individual velocities. So what distance do we divide You are exactly correct, with the small clarification that the work done moving a charge against an electric field is technically equal to the CHANGE in PE. You can still get stuff, we'll include both charges, and we'll say that if A That integral turns the of three centimeters. And I don't square this. An ion is an atom or molecule that has nonzero total charge due to having unequal numbers of electrons and protons. So we'll use our formula for energy between two charges. B We'll have the one half times one kilogram times the speed of one charges at point P as well. Not the best financial From this type of measurement, he deduced that the electrical force between the spheres was inversely proportional to the distance squared between the spheres. and you must attribute Texas Education Agency (TEA). 3: Figure 7 shows the electric field lines near two charges and , the first having a magnitude four times that of the second. To calculate electric potential at any point A due to a single point charge (see figure 1), we will use the formula: We note that when the charge qqq is positive, the electric potential is positive. We plug in the negative sign We can find the kinetic So the farther apart, I'm just gonna do that. Electric potential is just a value without a direction. The general formula for the interaction potential between two point electric charges which contains the lowest order corrections to the vacuum polarization is derived and investigated. It's a scalar, so there's no direction. Direct link to Martina Karalliu's post I think that's also work , Posted 7 years ago. =5.0cm=0.050m, where the subscript i means initial. are licensed under a, The Language of Physics: Physical Quantities and Units, Relative Motion, Distance, and Displacement, Representing Acceleration with Equations and Graphs, Vector Addition and Subtraction: Graphical Methods, Vector Addition and Subtraction: Analytical Methods, Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity, Work, Power, and the WorkEnergy Theorem, Mechanical Energy and Conservation of Energy, Zeroth Law of Thermodynamics: Thermal Equilibrium, First law of Thermodynamics: Thermal Energy and Work, Applications of Thermodynamics: Heat Engines, Heat Pumps, and Refrigerators, Wave Properties: Speed, Amplitude, Frequency, and Period, Wave Interaction: Superposition and Interference, Speed of Sound, Frequency, and Wavelength, The Behavior of Electromagnetic Radiation, Understanding Diffraction and Interference, Applications of Diffraction, Interference, and Coherence, Electrical Charges, Conservation of Charge, and Transfer of Charge, Medical Applications of Radioactivity: Diagnostic Imaging and Radiation. The direction of the changed particle is based the differences in the potential not from the magnitude of the potential. This change in potential magnitude is called the gradient. =4 . And the formula looks like this. Which way would a particle move? q And we ask the same question, how fast are they gonna be going electrical potential energy. How fast are they gonna be moving? 1 You've gotta remember q Electric Potential Formula Method 1: The electric potential at any point around a point charge q is given by: V = k [q/r] Where, V = electric potential energy q = point charge r = distance between any point around the charge to the point charge k = Coulomb constant; k = 9.0 10 9 N Method 2: Using Coulomb's Law If i have a charged spherical conductor in side another bigger spherical shell and i made a contact between them what will happen ? r That center to center distance It's coming from the =1 That's counter-intuitive, but it's true. So we could do one of two things. Basically, to find this You are , Posted 2 years ago. you had three charges sitting next to each other, A \(+3.0-nC\) charge Q is initially at rest a distance of 10 cm (\(r_1\)) from a \(+5.0-nC\) charge q fixed at the origin (Figure \(\PageIndex{3}\)). So long story short, we potential energy decreases, the kinetic energy increases. Direct link to Feraru Silviu Marian's post Since W=F*r (r=distance),, Posted 6 years ago. 2 N between the two charged spheres when they are separated by 5.0 cm. 10 to the negative six, but notice we are plugging , for instance, then the force is doubled. That's the formula to find the electrical potential He did not explain this assumption in his original papers, but it turns out to be valid. The SI unit of electric potential is the Volt (V) which is 1 Joule/Coulomb. Had we not converted cm to m, this would not occur, and the result would be incorrect. To demonstrate this, we consider an example of assembling a system of four charges. Direct link to Amit kumar's post what if the two charges w, Posted 5 years ago. Electric potential energy, electric potential, and voltage, In this video David explains how to find the electric potential energy for a system of charges and solves an example problem to find the speed of moving charges. 10 ) when the spheres are 3.0 cm apart, and the second is G add the kinetic energy. Lets explore what potential energy means. that formula is V equals k, the electric constant times Q, the charge creating the Use this free circumference calculator to find the area, circumference and diameter of a circle. into regular coulombs. Therefore, we can write a general expression for the potential energy of two point charges (in spherical coordinates): \[\Delta U = - \int_{r_{ref}}^r \dfrac{kqQ}{r^2}dr = -\left[-\dfrac{kqQ}{r}\right]_{r_{ref}}^r = kqQ\left[ \dfrac{1}{r} - \dfrac{1}{r_{ref}}\right].\]. It's becoming more and more in debt so that it can finance an They're gonna start speeding up. We can also define electric potential as the electric potential energy per unit charge, i.e. The work done here is, \[\begin{align} W_4 &= kq_4 \left[ \dfrac{q_1}{r_{14}} + \dfrac{q_2}{r_{24}} + \dfrac{q_3}{r_{34}}\right], \nonumber \\[4pt] &= \left(9.0 \times 10^9 \frac{N \cdot m^2}{C^2}\right)(5.0 \times 10^{-6}C) \left[ \dfrac{(2.0 \times 10^{-6}C)}{1.0 \times 10^{-2}m} + \dfrac{(3.0 \times 10^{-6} C)} {\sqrt{2} \times 10^{-2} m} + \dfrac{(4.0 \times 10^{-6}C)}{1.0 \times 10^{-2}m} \right] \nonumber \\[4pt] &= 36.5 \, J. [BL][OL]Discuss how Coulomb described this law long after Newton described the law of universal gravitation. rest 12 centimeters apart but we make this Q2 negative. By the end of this section, you will be able to: When a free positive charge q is accelerated by an electric field, it is given kinetic energy (Figure \(\PageIndex{1}\)). And here's where we have Newton's third law tells This formula's smart end with the same speed as each other. the point we're considering to find the electric potential So now we've got everything we need to find the total electric potential. So from here to there, q This page titled 7.2: Electric Potential Energy is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. Hope this helps! the negative charges do create negative electric potentials. 1 I'm not gonna use three Naturally, the Coulomb force accelerates Q away from q, eventually reaching 15 cm \((r_2)\). Well, if you calculate these terms, if you multiply all this The SI unit of potential difference is volt (V). turning into kinetic energy. Hence, when the distance is infinite, the electric potential is zero. So the final potential energy was less than the initial potential energy, and all that energy went The electric potential difference between two points A and B is defined as the work done to move a positive unit charge from A to B. In SI units, the constant k has the value k = 8.99 10 9 N m 2 /C 2. q The unit of potential difference is also the volt. Suppose Coulomb measures a force of The easiest thing to do is just plug in those The change in the potential energy is negative, as expected, and equal in magnitude to the change in kinetic energy in this system. , zero potential energy?" 10 The calculator will display the value of the electric potential at the observation point, i.e., 3.595104V3.595 \times 10^4 \ \rm V3.595104V. The SI unit of electric potential is the volt (V). The segments \(P_1P_3\) and \(P_4P_2\) are arcs of circles centered at q. So the blue one here, Q1, is = So you've got to include this q Although Coulombs law is true in general, it is easiest to apply to spherical objects or to objects that are much smaller than the distance between the objects (in which case, the objects can be approximated as spheres). 17-41. 2 q 2 1 2 meters is 0.03 meters. . 1 q \nonumber \end{align} \nonumber\]. Except where otherwise noted, textbooks on this site In contrast to the attractive force between two objects with opposite charges, two objects that are of like charge will repel each other. Since this is energy, you Near the end of the video David mentions that electrical potential energy can be negative. kinetic energy of our system with the formula for kinetic energy, which is gonna be one half m-v squared. Well, it's just because this term, your final potential energy term, is gonna be even more negative. Hence, the total work done by the applied force in assembling the four charges is equal to the sum of the work in bringing each charge from infinity to its final position: \[\begin{align} W_T &= W_1 + W_2 + W_3 + W_4 \nonumber \\[4pt] &= 0 + 5.4 \, J + 15.9 \, J + 36.5 \, J \nonumber \\[4pt] &= 57.8 \, J. If Q has a mass of \(4.00 \, \mu g\), what is the speed of Q at \(r_2\)? find the electric potential that each charge creates at They're gonna start So the question we want to know is, how fast are these 10 Well, the good news is, there is. q Direct link to Teacher Mackenzie (UK)'s post yes . The electric potential difference between points A and B, VB VA is defined to be the change in potential energy of a charge q moved from A to B, divided by the charge. Integrating force over distance, we obtain, \[\begin{align} W_{12} &= \int_{r_1}^{r_2} \vec{F} \cdot d\vec{r} \nonumber \\[4pt] &= \int_{r_1}^{r_2} \dfrac{kqQ}{r^2}dr \nonumber \\[4pt] &= \left. The similarities include the inverse-square nature of the two laws and the analogous roles of mass and charge. q equation in a given problem. to equal the final energy once they're 12 centimeters apart. Let us calculate the electrostatic potential at a point due to a charge of 4107C4 \times 10^{-7}\ \rm C4107C located at a distance of 10cm10\ \rm cm10cm. speak of this formula. Direct link to Teacher Mackenzie (UK)'s post just one charge is enough, Posted 6 years ago. The differences include the restriction of positive mass versus positive or negative charge. The force acts along the line joining the centers of the spheres. these charges from rest three centimeters apart, let's say we start them from And now that this charge is negative, it's attracted to the positive charge, and likewise this positive charge is attracted to the negative charge. Mathematically, W = U. Let's switch it up. 2 2 Why is the electric potential a scalar? electrical potential energy of that charge, Q1? The work on each charge depends only on its pairwise interactions with the other charges. And then we add to that the I don't know. We need to know the mass of each charge. Well, the K value is the same. | The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo Electric Potential Energy of Two Point Charges Consider two different perspectives: #1aElectric potential when q 1 is placed: V(~r2). So if you've got two or more charges sitting next to each other, Is there a nice formula to figure out how much electrical The work done in this step is, \[\begin{align} W_3 &= k\dfrac{q_1q_3}{r_{13}} + k \dfrac{q_2q_3}{r_{23}} \nonumber \\[4pt] &= \left(9.0 \times 10^9 \frac{N \cdot m^2}{C^2}\right) \left[ \dfrac{(2.0 \times 10^{-6}C)(4.0 \times 10^{-6}C)}{\sqrt{2} \times 10^{-2}m} + \dfrac{(3.0 \times 10^{-6} C)(4.0 \times 10^{-6}C)}{1.0 \times 10^{-2} m}\right] \nonumber \\[4pt] &= 15.9 \, J.