electric potential inside a conductor

Inside of conductor, electric field is zero whereas potential is same as on the surface. electric field is indistinguishable from that of a point charge Q. My textbook says: because the electric potential must be a continuous function. Therefore, based on the equation you mentioned, the electric field is not defined at $r = R$ (the derivative does not exist), which still leads to my question. Let the above equation is equation one, a) The electric field inside charge distribution-The electric potential inside a charged spherical conductor is given by, Put this value of electric . Obviously, since the electric field inside the sphere is zero (as you state), there is no force on the charge, so no work done. b. The electric potential inside a conductor will only be constant if no current is flowing AND there is resistance in the circuit. \dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{r}, & \text{if $r \gt R$}. If no charge flows the potential of the conductor must be unchanged, and if charge flows the potential must have changed. We can go further, and show that there is no net electric charge inside the sphere; that it is electrically neutral. Therefore the potential is constant. \\ Therefore the potential is constant. Explanation: Electric field at any point is equal to the negative of the potential gradient. Because there is no potential difference between any two points inside the conductor, the . In this case, by definition the voltage won't change even if it is polarised, which is not contradictory as generally its charge will vary to compensate. B. increases with distance from center. Either way bringing the external charge close to the conductor does change its potential relative to earth. B. increases with distance from center. \\ Use logo of university in a presentation of work done elsewhere. This reduces the risk of breakdown or corona discharge at the surface which would result in a loss of charge. The potential is constant inside the conductor but it does not have to be zero. I am hoping for a non-experimental reason. $$. It depends on how you manipulate your conductor. Congratulations, and may there be many others. What's the \synctex primitive? Is Electric potential constant inside a conductor in all conditions? [Physics] Why is the surface of a charged solid spherical conductor equal in potential to the inside of the conductor [Physics] Is electric potential always continuous [Physics] Gauss's law for conducting sphere and uniformly charged insulating sphere I know the electric field strictly inside it must be zero. I calculated the electric field if the shell has a finite thickness, and found out that inside the shell the field increases linearly (approx. The field is actually discontinuous at the surface: the discontinuity in the field is proportional to the surface charge density. This is a good question, and the key insight is that the properties of conductors (charge only occurs on the surface, potential inside is constant, etc) are only well-defined in the electrostatic regime. Why is it important that Hamiltons equations have the four symplectic properties and what do they mean? That This all occurs in an extremely short amount of time, and as long as you look at the equilibrium situation, there really is constant potential in a conductor. Welcome to the site! I just began studying electrostatics in university, and I didn't understand completely why the electric potential due to a conducting sphere is. Those are different and I get easily confused when people misuse those. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. Conductors have loosely bound electrons to allow current to flow. Put less rigorously, the electric field would be 'infinite' wherever $V(\vec r)$ is discontinuous. When we work with continuous charge distributions, we are simply using an approximation that averages over lots of point charges and smears out the discontinuities in their charge density, potential, field, field energy density, etc. That makes it an equipotential. Is the EU Border Guard Agency able to tell Russian passports issued in Ukraine or Georgia from the legitimate ones? Why are strong electrolytes good conductors of electricity? What properties should my fictional HEAT rounds have to punch through heavy armor and ERA? electric field itself can be discontinuous across a boundary. Now as we approach the boundary, we can imagine moving an infinitesimal amount to go from $r = R - \delta r$ to $r = R + \delta r$. Since a charge is Please be precise when mentioning $r R$). Understanding zero field inside a conductor? In that case, charges would naturally move down that potential difference to a lower energy position and thereby remove the potential difference! Japanese girlfriend visiting me in Canada - questions at border control? However, recall that conductors are made up of free charges which rapidly flow across that potential difference and reach equilibrium. This is one of the best written "first questions" I have ever seen on this site. Physics 38 Electrical Potential (12 of 22) Potential In-, On, & Outside a Spherical Conductor, Physics 38 Electrical Potential (13 of 22) Potential Outside a Cylindrical Conductor, Why charges reside on surface of conductors | Electrostatic potential & capacitance | Khan Academy, 19 - Electric potential - Charged conductor, Electric Potential: Visualizing Voltage with 3D animations. 2) Compare the potential at the surface of conductor A with the potential at the surface of conductor B. Open in App. The question is whether the potential of the conductor has been changed, and the simple way to test this is to connect it to earth again and see if any charge flows between earth and the conductor. So far so good. If you make the shell of finite thickness, you can see that the field decreases continuously. C. is constant. Objects that are designed to hold a high electric potential (for example the electrodes on high voltage lines) are usually made very carefully so that they have a very smooth surface and no sharp edges. Therefore the potential is constant. Electrons travel on the surface of the conductor in order to avoid the repulsion between the electron. Now as we approach the boundary, we can imagine moving an infinitesimal amount to go from r = R - \delta rr = R - \delta r to r = R + \delta rr = R + \delta r. As long as the electric field is at most some finite amount E_{shell}E_{shell}, then the work done moving from just inside to just outside is E_{shell}*2\delta rE_{shell}*2\delta r; as \delta r \rightarrow 0\delta r \rightarrow 0, the work done will also tend to zero. Use MathJax to format equations. C = \lim_{r \to R^+} V(r) = \dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{R} Whether we mean by "at the surface" as RR or R + \delta rR + \delta r doesn't matter since the difference vanishes as \delta r\delta r becomes sufficiently small. Whether we mean by "at the surface" as $R$ or $R + \delta r$ doesn't matter since the difference vanishes as $\delta r$ becomes sufficiently small. And I know $\vec{E} = -\nabla{V}$. When conductors are placed in an electric field, their electrons are moved. Electric field inside a conductor is always zero. potential difference . I just began studying electrostatics in university, and I didn't understand completely why the electric potential due to a conducting sphere is, $$ My textbook says: because the electric potential must be a continuous Now, the electric field itself can be discontinuous across a boundary. Electric field inside a conductor non zero, Potential of a conductor with cavity and charge. charge. For instance, at a point mid-way between two equal and similar charges, the electric field strength is zero but the electric potential is not zero. So far so good. Therefore, I know the electric potiential inside the sphere must be constant. For example, the potential of a point charge is discontinuous at the location of the point charge, where the potential becomes infinite. Can we keep alcoholic beverages indefinitely? Obviously, since the electric field inside the sphere is zero (as you state), there is no force on the charge, so no work done. (I also know the electric field is not defined for a point that lies exactly in the surface). Potential at a point x-distance from the centre inside the conducting sphere of radius `R` and charged with charge `Q` is asked May 25, 2019 in Physics by Rustamsingh ( 92.7k points) class-12 I am hoping for a non-experimental reason. Those are different and I get easily confused when people misuse those. So, there is no electric field lines inside a conductor.In conductor , electrons of the outermost . Hopefully I will also be able to write good answers for other people as well! The only way this would not be true is if the electric field at $r=R$ was infinite - which it is not. from one point in a conductor to another. Since the electric field is equal to the rate of change of potential, this implies that the voltage inside a conductor at equilibrium is constrained to be constant at the value it reaches at the surface of the conductor.A good example is the charged conducting sphere, but the principle applies to all conductors at equilibrium. Electric field intensity is zero inside the hollow spherical charged conductor. Let $C$ be this constant. V(\vec{r})=\begin{cases} (I also know the electric field is not defined for a point that lies exactly in the surface). Why is the electric field inside a charged conductor zero? Reason: The electricity conducting free electrons are only present on the external surface of the conductor. @Floris I wonder how you missed it as well. Why is there no charge inside sphere? I know Gauss Law. \dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{r}, & \text{if $r \gt R$}. \dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{r}, & \text{if $r \gt R$}. Imagine you have a point charge inside the conducting sphere. This is why we can assume that there are no charges inside a conducting sphere. Asking for help, clarification, or responding to other answers. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Likewise, the potential must be indistinguishable from that of a point My textbook says: because the electric potential must be a continuous function. \dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{R}, & \text{if $r \le R$}.\\ . I only understand the second part of this equation (when r>Rr > R). V ( r) = {1 4 0 Q R, if r R. 1 4 0 Q r, if r > R. Where Q is the total charge and R is the radius of the sphere (the sphere is located at the origin). Outside the sphere, the Say a conductor with an initial electric potential of zero is subject to an arbitrary charge. If it is insulated from the environment, it's potential will generally change in order to conserve its charge (which I think was what you had in mind). Known : The electric charge (Q) = 1 C = 1 x 10-6 C The radius of the spherical conductor (r) = 3 cm = 3 x 10-2 m Coulomb's constant (k) = 9.109 N.m2.C-2 Wanted : The electric potential at point A (V) Solution : V = k Q / r Step 2: Formula used The formula used in the solution is given as: E = - d V / d r I think you are overthinking this. Electromagnetic radiation and black body radiation, What does a light wave look like? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Imagine you have a point charge inside the conducting sphere. Capacitance also implies an associated storage of electrical energy. Electric Potential Electric Potential due to Conductors Conductors are equipotentials. Medium. Electric field inside a conductor is always zero. Hence the potential . My textbook says: because the electric potential must be a continuous function. Where Q is the total charge and R is the radius of the sphere (the sphere is located at the origin). 10.15 Potential inside the Conductor We know that E = - dV/dr. Gauss law is great, my advice is not to consider laws something to rote without realising their importance. What happens when a conductor is placed in an electric field? The electric potential outside a charged spherical conductor is given by, As the relation given between the electric field and electric potential is, 1. V(\vec{r})=\begin{cases} Where Q is the total charge and R is the radius of the sphere (the sphere is located at the origin). Connect and share knowledge within a single location that is structured and easy to search. When a charged object is brought close to a conductor, there actually is a potential difference inside the conductor initially! So far so good. Thanks! (3D model). What happens to the initial electric potential inside the conductor? The electric potential inside a conductor: A is zero B increases with distance from center C is constant D decreases with distance from center Medium Solution Verified by Toppr Correct option is C) As the electric field inside a conductor is zero so the potential at any point is constant. In electrostatics, you are only dealing with the situation after everything has moved to its equilibrium position inside the conductor because it all happens so quickly. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. Step 1: Conductor A conductor is a material used for the flow of current through it because a conductor has a large number of free electrons in it. Thanks for contributing an answer to Physics Stack Exchange! . Electric potential inside a polarised conductor, Help us identify new roles for community members. (I also know the electric field is not defined for a point that lies exactly in the surface). [closed], Error filterlanguage: Invalid value specified: 1. when trying to create sfdx package version, Could Not Verify ST Device when flashing STM32H747XIH6 over SEGGER J-link within STM32CubeIDE, Changing the Pan View Keybind works in Object Mode, Not Sculpt Mode. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The electric potential inside the spherical conductor = The electric potential at the surface of the spherical conductor. I know Gauss Law. Also read: Electrostatic Potential and Capacitance Table of Content Electric Field Inside a Conductor Interior of Conductor Electrostatic Field Lines Electrostatic Potential Surface Density of Charge I know the electric field strictly inside it must be zero. The net electric field inside a conductor is always zero. \dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{R}, & \text{if $r \le R$}.\\ But why? Why is the potential inside a hollow spherical charged conductor constant? They each carry the same positive charge Q. \\ Are defenders behind an arrow slit attackable? Finding the general term of a partial sum series? If we bring up a positive charge and connect the conductor to earth we'll find electrons flow from earth onto the conductor to give it a net negative charge. The only way this would not be true is if the electric field at r=Rr=R was infinite - which it is not. As long as the electric field is at most some finite amount $E_{shell}$, then the work done moving from just inside to just outside is $E_{shell}*2\delta r$; as $\delta r \rightarrow 0$, the work done will also tend to zero. That makes it an equipotential. Correctly formulate Figure caption: refer the reader to the web version of the paper? document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); First-principles derivation of cutting force. As long as the electric field is at most some finite amount $E_{shell}$, then the work done moving from just inside to just outside is $E_{shell}*2\delta r$; as $\delta r \rightarrow 0$, the work done will also tend to zero. The electric potential inside a conductor: A. is zero. Obviously, since the electric field inside the sphere is zero (as you state), there is no force on the charge, so no work done. rev2022.12.11.43106. Why doesn't the magnetic field polarize when polarizing light. But at no point does anything allow the electric field to become infinite. function. All we require is that $\nabla V = 0$. But why the electric field is not infinite at r = R? MathJax reference. surfaces so electric field lines are prependicular to the surface of a In the Electrostatic case the electric potential will be constant AND the electric field will be zero inside a conductor. Concentration bounds for martingales with adaptive Gaussian steps. And I know $\vec{E} = -\nabla{V}$. Proof that if $ax = 0_v$ either a = 0 or x = 0. What you can obtain is potential differences. Does a 120cc engine burn 120cc of fuel a minute? The electric potential inside a charged solid spherical conductor in equilibrium: Select one: a. Decreases from its value at the surface to a value of zero at the center. conductor. . O the electric potential within a hollow empty space inside the conductor equals the electric potential at the surface. Likewise if we bring up a negative charge we'll find electrons flow off the conductor to earth giving the conductor a net positive charge. Does aliquot matter for final concentration? the electric . The electrons are free charge carriers inside a metallic conductor while positive ions fixed in lattice are bound charge carriers. A conductor is a material which conducts electricity from one place to the other. c. Increases from its value at the surface to a value at the center that is a multiple of the potential at the surface. Would it be greater than zero since now one side of the conductor is positively charged and another negatively? \dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{R}, & \text{if $r \le R$}.\\ Is constant and equal to its value at the surface. Q The electric potential inside a conducting sphere A. increases from centre to surface B. decreases from centre to surface C. remains constant from centre to surface D. is zero at every point inside Explanation Ans C Electric potential inside a conductor is constant and it is equal to that on the surface of the conductor. (c) Doug Davis, 2002; all rights reserved. If he had met some scary fish, he would immediately return to the surface. Therefore the potential is constant. Indeed. Answer (1 of 2): Same as it is at the surface of it if there are no charges inside the conductor. capacitance, property of an electric conductor, or set of conductors, that is measured by the amount of separated electric charge that can be stored on it per unit change in electrical potential. The real formula you can obtain is: V = ( K q r K q r 0) = K q ( 1 r 1 r 0) Where r 0 is the point you chose as reference. Why is it that potential difference decreases in thermistor when temperature of circuit is increased? But why is this true? The electric potential and the electric field at the centre of the . $$. A finite jump. Glad you got there it's more satisfying if you can take that last step yourself. No. Solution. I understand that because if this outside charge, there would be charge distribution inside the conductor, so as to make the electric field in it zero. What is the relationship between AC frequency, volts, amps and watts? Thanks! the electric potential is always independent of the magnitude of the charge on the surface. The electric potential inside a conductor: A) is zero B) increases with distance from the centre C) is constant D) decreases with distance from the centre Answer Verified 224.7k + views Hint: The electrostatic field inside a conductor is zero as the charges only reside on the surface of the conductor. On one side the field is zero, on the other it is $\sigma / \epsilon_0$. Electric potential necessarily need not be 0 if the electric field at that point is zero. Solution. is. Does illicit payments qualify as transaction costs? The statement "within the conductor and the surface" is to be understood as meaning within the conductor and a point arbitrary close to the surface but inside this surface. What justifies conservation laws in non-uniform spatial/temporal fields, if Noethers theorem doesnt? Infinite gradient but we don't care about that since we need to integrate, not differentiate, to go from $E$ to $V$. ), from 0 inside to exactly $\frac{Q}{4\pi\epsilon_0 b^2}$ where $b$ is the outer radius. Since a charge is free to move around in a conductor, no work is done in moving a charge from one point in a conductor to another. inside the conductor is constant. They are empirically verified results and give accurate insight into the situations where,i. Introduction Bootcamp 2 Motion on a Straight Path Basics of Motion Tracking Motion Position, Displacement, and Distance Velocity and Speed Acceleration Position, Velocity, Acceleration Summary Constant Acceleration Motion Freely Falling Motion One-Dimensional Motion Bootcamp 3 Vectors Representing Vectors Unit Vectors Adding Vectors Open in App. C. is constant. Because there is no potential difference between any two points inside the conductor, the electrostatic potential is constant throughout the volume of the conductor. The best answers are voted up and rise to the top, Not the answer you're looking for? Let's be a little more precise about what we mean by a zero potential. Making statements based on opinion; back them up with references or personal experience. But why is this true? Save my name, email, and website in this browser for the next time I comment. The electric potential inside a conductor: A. is zero. know the charges go to the surface. Now as we approach the boundary, we can imagine moving an infinitesimal amount to go from r = R r to r = R + r. Hence, the electric potential is constant throughout the volume of a conductor and has the same value on its surface. . However our thought experiment makes it clear that the potential does change. Verified by Toppr. Question edited: the equation I first gave for the potential was wrong! E = 0. The metal sphere carries no charge, so the electric field outside it is also zero which means constant potential. I only understand the second . And I know E=V\vec{E} = -\nabla{V}. Since all charges in nature seem to be point charges (elementary particles such as electrons and quarks), electric potential always has discontinuities somewhere. [Physics] Why is the surface of a charged solid spherical conductor equal in potential to the inside of the conductor, [Physics] Is electric potential always continuous, [Physics] Gausss law for conducting sphere and uniformly charged insulating sphere. $$. Obviously, since the electric field inside the sphere is zero (as you state), there is no force on the charge, so no work done. I just began studying electrostatics in university, and I didn't understand completely why the electric potential due to a conducting sphere is, $$ Therefore, I know the electric potiential inside the sphere must be constant. Thus, a conductor in an electrostatic field provides an equipotential region (whole of its inside). \end{cases} Conductor A has a larger radius than conductor B. free to move around in a conductor, no work is done in moving a charge Let $C$ be this constant. Suppose that there was a potential difference inside the conductor. $$ Gauss's Law to understand the electric field. Transcribed image text: For a charged conductor, O the electric potential is always zero at any point inside it. D. decreases with distance from center. know the charges go to the surface. We already know that electric field lines are perpendicular to equipotential E = 0. The only way this would not be true is if the electric field at $r=R$ was infinite - which it is not. What is the probability that x is less than 5.92? Let CC be this constant. And if we tried this we would find that charge does flow between earth and the conductor as soon as we connect them. But why? C = \lim_{r \to R^+} V(r) = \dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{R}. Since the electric field is observable, we simply can't have that. How do I arrange multiple quotations (each with multiple lines) vertically (with a line through the center) so that they're side-by-side? So the electric potential inside would remain constant. I understand that because if this outside charge, there would be charge distribution inside the conduct. Medium. know the charges go to the surface. . potential energy is the work done by an external force in taking a body from a point to another against a force. then if the electric field is to be finite everywhere, $V(\vec r)$ must be continuous. I thought it wasn't defined at all, because the potential isn't differentiable at r = R. The finite jump in the field is obtained by Gauss's law - create a "pill box" that crosses the surface of the conductor. That means the electric potential Now we bring up the external charge, and as you say it will polarise the conductor. This means that the potential is continuous across the shell, and that in turn means that the potential inside must equal the potential at the surface. d. Thankfully this doesn't change the answer for my question. How is the merkle root verified if the mempools may be different? Correct option is C) As the electric field inside a conductor is zero so the potential at any point is constant. By this question, I am guessing that you are wondering how physics textbooks and such claim that the potential difference inside of a conductor is zero, even though for the charges to move to either side, there must have been some potential difference inside the conductor the first place! I am hoping for a non-experimental reason. The value and sign of the change depends crucially on the charge and the geometry of the problem. So no work is done in moving a test charge inside the conductor and on its surface. Therefore there is no potential difference between any two points inside or on the surface of the conductor. Two spherical conductors are separated by a large distance. Thank you very much! Therefore the potential is constant. The electric field inside the conductor is zero, there is nothing to drive redistribution of charge at the outer surface. What are some interesting calculus of variation problems? Charge a conductor dome indefinitely frome the inside. I am getting more and more convinced. Is it appropriate to ignore emails from a student asking obvious questions? Therefore, I know the electric potiential inside the sphere must be constant. If I'm not mistaken, for the gradient to be defined, all partial derivatives must be defined, which is not the case at $r = R$. AttributionSource : Link , Question Author : Pedro A , Answer Author : Floris. However, you can also fix the potential of a conductor, like when you ground it or apply the voltage from a battery. Obviously, since the electric field inside the sphere is zero (as you state), there is no force on the charge, so no work done. Inside the electric field vanishes. Conductors are equipotentials. To learn more, see our tips on writing great answers. Then we disconnect the conductor from earth. so if there isn't any force to act against why would electric potential be present over there? Prove that isomorphic graphs have the same chromatic number and the same chromatic polynomial. Maybe I am getting too philosophical here, but that "pill box" shows that the field. A B a) VA > V B b) VA = V B c) VA < V B Preflight 6: Why is the surface of a charged solid spherical conductor equal in potential to the inside of the conductor? More directly to your question, the potential difference caused by the external charge and the potential of the charges on your conductor's surface cancel out perfectly to produce constant potential inside the conductor. C=lim \end{cases}. Say a conductor with an initial electric potential of zero is subject to an arbitrary charge. Hence, throughout the conductor, potential is same i.e, the whole conductor is equipotential. But why? The situation you describe is an idealization as, in real conductors, the charge is concentrated in a small boundary around the surface; the thickness of this boundary depends inversely on the conductivity of the material, and goes to zero in the ideal case of a perfect conductor with conductivity $\sigma\to\infty$. as electric field remains the zero inside the conductor so the potential at the surface should be the same as inside, but i came with a situation which is as follows: if a spherical conductor is placed inside (concentrically) a conducting shell which has greater dimensions than that of the first conductor and a some charge is given to the smaller I just began studying electrostatics in university, and I didnt understand completely why the electric potential due to a conducting sphere is, V(r)={140QR,ifrR.140Qr,ifr>R. D. decreases with distance from center. We'll take the potential of earth to be zero, and before we bring up the charge we'll connect our conductor to earth to make its potential zero as well. This means that the potential is continuous across the shell, and that in turn means that the potential inside must equal the potential at the surface. It only takes a minute to sign up. Since E = 0 inside the conductor and has no tangential component on the surface, no work is done in moving a small test charge within the conductor and on its surface. That is, there is no potential difference between any two points inside or on the surface of the conductor. Reason: The electricity conducting free electrons are . Consider charge Q on a metallic sphere of radius R. We have already used This means that the potential is continuous across the shell, and that in turn means that the potential inside must equal the potential at the surface. Also, the electric field inside a conductor is zero. Reply on the surface of a conductor the electrostatic charges arrange themselves in such a way that the net electric field is always zero. However by Gauss's Law. What happens if you score more than 99 points in volleyball? As inside the conductor the electric field is zero, so no work is done against the electric field to bring a charge particle from one point to another. Electric potential inside a conductor electrostaticspotential 29,444 Solution 1 Imagine you have a point charge inside the conducting sphere. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, . At what point in the prequels is it revealed that Palpatine is Darth Sidious? Could an oscillator at a high enough frequency produce light instead of radio waves? The electric potential energy of a point charge is not V = K q r That would be quite absolute. Another way to think about this is by contradiction. Is there something special in the visible part of electromagnetic spectrum? Whether we mean by "at the surface" as $R$ or $R + \delta r$ doesn't matter since the difference vanishes as $\delta r$ becomes sufficiently small. So far so good. Does a positive charge flow from conductor to earth when it is earthed? Correct option is C) As the electric field inside a conductor is zero so the potential at any point is constant. But inside a conductor, the electric field is zero. The lowest potential energy for a charge configuration inside a conductor is always the one where the charge is uniformly distributed over its surface. JEE NEET#electricpotential #electricfield #12thphysics #potential#electrostatics @Vats Education why potential inside the conductor is constantrelation betw. $$. The electric potential at any point in an electric field is defined as the work done in bringing a unit positive test charge from infinity to that point without acceleration. I only understand the second part of this equation (when $r > R$). \end{cases} C = \lim_{r \to R^+} V(r) = \dfrac{1}{4\pi\epsilon_0}\dfrac{Q}{R} As you make the shell of charge thinner, the slope becomes steeper. . $$ I know the electric field strictly inside it must be zero. Where is it documented? V(\vec{r})=\begin{cases} By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. 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