# What is equipotential: Definition and 110 Discussions

In mathematics and physics, an equipotential or isopotential refers to a region in space where every point is at the same potential. This usually refers to a scalar potential (in that case it is a level set of the potential), although it can also be applied to vector potentials. An equipotential of a scalar potential function in n-dimensional space is typically an (n − 1)-dimensional space. The del operator illustrates the relationship between a vector field and its associated scalar potential field. An equipotential region might be referred as being 'of equipotential' or simply be called 'an equipotential'.
An equipotential region of a scalar potential in three-dimensional space is often an equipotential surface (or potential isosurface), but it can also be a three-dimensional mathematical solid in space. The gradient of the scalar potential (and hence also its opposite, as in the case of a vector field with an associated potential field) is everywhere perpendicular to the equipotential surface, and zero inside a three-dimensional equipotential region.
Electrical conductors offer an intuitive example. If a and b are any two points within or at the surface of a given conductor, and given there is no flow of charge being exchanged between the two points, then the potential difference is zero between the two points. Thus, an equipotential would contain both points a and b as they have the same potential. Extending this definition, an isopotential is the locus of all points that are of the same potential.
Gravity is perpendicular to the equipotential surfaces of the gravity potential, and in electrostatics and steady electric currents, the electric field (and hence the current, if any) is perpendicular to the equipotential surfaces of the electric potential (voltage).
In gravity, a hollow sphere has a three-dimensional equipotential region inside, with no gravity from the sphere (see shell theorem). In electrostatics, a conductor is a three-dimensional equipotential region. In the case of a hollow conductor (Faraday cage), the equipotential region includes the space inside.
A ball will not be accelerated left or right by the force of gravity if it is resting on a flat, horizontal surface, because it is an equipotential surface.
For the gravity of Earth, the corresponding geopotential isosurface (the equigeopotential) that best fits mean sea level is called the geoid.

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1. ### Equipotential surface (test questions)

The answer to the first question should be a sphere since for very large distances the multiple charges will act as a point charge. 1(a) is correct answer. For the second question, I find it slightly vague. How can equipotential surface be zero, may be it's asking for the potential of...
2. ### Gravitational equipotential multiple choice problem

Hello everyone, thank you for taking your time to read this. I was assigned a homework task of multiple choice questions to do with gravitational fields. This is one of the last questions and I have been pondering over it for some time now. I don't understand how any sort of answer is...
3. ### Equipotential surfaces for an electric dipole

The answer is given as (a), but I think it's not correct based on the equipotential surfaces diagram given in our book for an electric dipole as below. The red dashed lines, which are supposed to be the equipotential surfaces, are surely not representing a sphere centred at the dipole center...
4. ### I Do equipotential lines fall on the equiprobability contours?

For 2D charge distribution ρ(x,y)=Ne PDF(x,y), where PDF is the normalized probability density function with its peak on (0,0) and has standard deviations σ x. and σ y. Are the contours with the equal probability "PDF(x,y)=const" the same as the equipotiential contours?, I tend to think that...
5. ### Checking if a function is an equipotential surface

hi guys I came across that theorem that could be used to check if a surface represented by the function f(x,y,z) = λ could represent an equipotential surface or not, and it states that if this condition holds: $$\frac{\nabla^{2}\;f}{|\vec{\nabla\;f}|^{2}} = \phi(\lambda)$$ then f(x,y,z) could...
6. ### Question about equipotential lines and the work done moving along them

hi guys i have a conceptual question .As you know equipotential surfaces is one on which all point are the same potential there is no work required to move a charge from one point to the other . So my question is how can we change the locotion of a particle without using any force ?
7. ### What equations relate to the symmetry of equipotential lines?

This is the second quadrant of the equipotential line. I think this would be symmetry but I'm not sure what to do. Is this going to be the symmetry of both the x-axis and y-axis and the symmetry of the y-axis is ( - )? Sorry for the bad English! feel free to leave comment if you can't...
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### How do we show that the interior of a Faraday cage is an equipotential?

It is possible to show via Gauss' law that the the net flux through a surface within the Faraday cage must be zero, however this is not a sufficient condition for the electric field to be zero. For the electric field to be zero in the interior of the cage, all points within the cage must be at...
9. ### Removing connections between equipotential points in solving circuits?

A sketch of the setup and the equivalent circuit are attached. I believe the correct way to solve this is to redraw the circuit as shown in Fig. 3 and then remove the connections between evidently equipotential points, which reduces the problem to a familiar setup of in parallel and in series...
10. ### What Is the Required Cross-Section for Equipotential Link Conductors Above 250A?

When the exposed conductive parts (door, protective screen, closing panel, etc.) do not have any devices or equipment fixed on them, the equipotential connection of these exposed conductive parts must be provided using a conductor, minimum cross-section 2.5 mm² as it's stated in IEC (see...
11. ### Finding the equipotential surfaces of a finite line of charge

Homework Statement Consider a line of charge stretching along the z-axis from -L to +L. Find the potential everywhere. What are the surfaces of constant potential. (The next question answers the previous question and says its a prolate ellipsoid. Homework Equations I will upload an image of...
12. ### Why electric field is always perpendicular to equipotential?

I have some understanding, but I'm not sure about how accurate it is: Electrostatic force is given by F = qE, where F and E are both vector quantities. If the dot product of either side and the displacement vector Δs along an equipotential line is taken, the equation becomes F⋅Δs = qE⋅Δs. F...
13. ### Equipotential surface and electric field

Homework Statement Homework EquationsThe Attempt at a Solution I know the relation between electric field and electric potential . I can also find Electric field if expression for potential is given and vica versa . But I do not know how to work with electric field and equipotential...
14. ### Conceptual Equipotential Surfaces Problems

Homework Statement True or False: 1.Equipotential surfaces intersect: 2.Electric field lines are found within equipotential surfaces: Homework Equations E=Vd The Attempt at a Solution 1. I think this is false since the same reasoning describes why electric field lines don't intersect 2. I'm...
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### Flow line vs equipotential line

Homework Statement It's stated in the notes that line fg is the flow line . Why is it so ? Homework EquationsThe Attempt at a Solution I think it's wrong . The flow line is defined as a line along which the water paticle travel from upstream to downstream in the permeable medium . However ...
16. ### Calculating work on an electron on an equipotential surface

This is not a homework, but a question formed in my mind after reading my textbook. Homework Statement Consider an electron (a charged particle) on a metallic equipotential surface. We know that all the points on the surface are equipotential, thus there will be no force on charged particles...
17. ### [Electromagnetics] E-Fields & Equipotential Surfaces

Homework Statement This is from the book Engineering Electromagnetics by Hayt & Buck.[/B] Homework Equations E = - (ΔV/ΔL)[/B]The Attempt at a Solution At part (a), I took the potential difference between point A and the point directly above at the higher surface (106 V) and plugged in the...
18. ### Resistors and equipotential lines?

Homework Statement We are suppose to make a lab report about and experemint that we did in physics lab. It consisted of drawing equipotential lines in a overbeck electric field mapping apparatus. Different desing plates were used, each having conductors, some were circular and other where...
19. ### Question about equipotential lines and electric fields

Homework Statement Hi, I have a question about my lab report that I am doing. The only question that I am having problems to understand is the first one, it says, "why are the equipotential lines near conductor surfaces parallel to the surface and why are they perpendicular to the insulator...
20. ### Equipotential surface magnetic field

Homework Statement True or not true:[/B] Equipotential surfaces of the magnetic field are no closed surfaces but extend to infinity Homework Equations - The Attempt at a Solution I think it is true, because they are perpendicular to the (closed) magnetic field lines, but I am not sure...
21. ### Show that the equipotential lines are circles

Homework Statement In a specific area of the space, an electrical potential is given as: $$V(x,y,z) = A(2x^2 - 3y^2 - 3z^2)$$ where A is a constant. a.) Determine the electrical field E for any given point in the area. A test charge q_0 is moved from the point...
22. ### Doubt on equipotential surfaces

i know that all conductors are equipotential,then how are charges flowing in a conductor?and at times in we say that charges won't flow since two points are equipotential(like in wheat stone bridge we say that charge won't flow across the capacitor/resistor since the ends of the 5th...
23. ### How to find equipotential surface for given case?

Q:- Describe equipotential surface due to a uniform grid consisting of long equally spaced parallel charged wires in a plane. The answer given in my textbook is - Equipotential surface have shape which changes periodically. At far off distances it becomes parallel to the plane. Why the...
24. ### What is the Solution to the Equipotential Surface Problem?

Homework Statement [/B] Homework Equations ##\vec{E}##=##\frac{∂V}{∂r}## The Attempt at a Solution I have provided both problem and solution(almost)but the problem is I did not understand the solution.First of all I did not understand the question what we are told to determine? I locate the...
25. ### Equipotential Apparatus on the cheap?

Hi, I recently stumbled across the wonderful equipotential apparatus, which allows one to create equipotentials by hand for educational purposes. After browsing around and finding sales prices ranging between 150 (pasco) and 500 dollars, I'd like to just make one. It does not appear at all...
26. ### Distance between equipotential surfaces

Homework Statement A non-conducting sphere (radius 11.3 cm) has uniform charge density ρ = 0.596 μC/m3. Find the distance, in meters, between equipotential surfaces V1 = 16.2 Volts and V2 = 42.3 volts. (Distance is always positive.) Homework Equations V=kq/r ρ=Q/V The Attempt at a Solution...
27. ### Geophysics: Dynamic form factor and Equipotential surfaces

I've got the following problem which I need help with. I'm used to calculating coefficients when the problem is about ellipsoids and first order approximations. But when it comes to spheres and coefficients J_n I really don't know how to approach the problem. Can somebody help me out? Consider...
28. ### SOLVED: Equipotential surfaces for finite line of charge

Homework Statement Purcell 2.10 [/B][not the problem I'm asking about, but needed for Purcell 2.11 which I am asking about] A thin rod extends along the z axis from z = -d to z = d. The rod carries a charge uniformly distributed along its length with linear charge density \lambda. By...
29. ### Equipotential surface of a infinite plane?

Homework Statement So the equipotential surface of a point charge is sphere with the charge in the center, and the equipotential surface of a infinite line is a cylinder with the line of charge as the axis. I was wondering what is the shape of the equipotential surface of a infinite plane...
30. ### Electric field, Equipotential surfaces

Question : The top of the atmosphere is at about 400 kV with respect to the surface of the Earth, corresponding to an electric field that decreases with altitude. Near the surface of the Earth, the field is about 100 V/m. Why then we do not get electric shock as we step outside our house into...
31. ### Simple equation to show work done on an equipotential surface is zero

w = ΔKE + ΔPE For a particle moving at constant speed, there is no change in velocity so no change in KE. What about change in PE for a partcile moving at constant speed on a equipotential surface? Would I be right in stating that since the particle moves along the same surface (AKA same...
32. ### How Does Electric Field Direction Relate to Equipotential Lines?

Homework Statement The equipotential curves in a certain region of an equipotential diagram in an xy plane are parallel to the y-axis and are equally spaced. The potential is increasing in the +x direction. Which direction would an electric field vector point in this region? Homework...
33. ### How to graph equipotential lines of two charges?

The two charges are (-q) and q. I'm just curious as to how you go about graphing them. Thanks!
34. ### What is the Electric Field at Position D on the Equipotential Surfaces?

Homework Statement The drawing shows a graph of a set of equipotential surfaces in cross section. The grid lines are 2.0 cm apart. Determine the magnitude and direction of the electric field at position D. Specify whether the electric field points toward the top or the bottom of the drawing...
35. ### Equipotential surface / electric scalar potential problem (why )

equipotential surface / electric scalar potential problem (why!) Homework Statement A potential field is given by V = 3x^2*y - y*z. Is the following statement valid? "A unit normal to the equipotential surface V = -8 at P(2,-1,4) is <-0.83,0.55,0.07>"Homework Equations Gradient of a scalar...
36. ### Equipotential Surfaces physics problems

Homework Statement Homework Equations - Charge conservation - Equipotential surfacesThe Attempt at a Solution Let Q1 be the amount of charge on the inner sphere with radius c, and Q2 be the amount of charge on the outer sphere with radius b. Using Gauss's law, I figured out that Q1=+2Q and...
37. ### Find Net Resistance b/w A & B in Attach 1: Explaining Equipotential Points

Find the Net Resistance between A and B in attachment 1. The circuit in attachment 1 simplifies to attachment 2 from the fact that the points C and E are equipotential. Same goes for D and F. So we remove CE and F as no current passes through equipotential points. But why? How are these 2...
38. ### Equipotential Field line equation derivation

Hello, This is not a homework question! I am simply trying to understand for my own amusement. A while back I synched Solidworks and Excel, and am using this to create 3d models of different things such as dipoles for visual interpretation. So in doing this I needed to find potential...
39. ### Clarification about a conductor as an equipotential

Clarification about a conductor as an equipotential... Homework Statement This is not a specific problem, but a general question I have. In Griffiths 2.5 (pg. 97 for Third Edition), it states that "A conductor is an equipotential. For if a and b are any two points within (or at the...
40. ### Equipotential of transmission or telegraph line

Take a telegraph line or transmission line which is long. Now slowly the wave propagates through the wire in the forward direction. In the beginning one end of the wire will have a non-zero voltage while the other end will have zero voltage and it takes some more time to get settled after some...
41. ### Uniqueness Theorem, Concentric Equipotential Cylinders & Moving Charge

Homework Statement Consider two electrodes 2 mm apart in vacuum connected by a short wire. An alpha particle of charge 2e is emitted by the left plate and travels directly towards the right plate with constant speed 106 m/s and stops in this plate. Make a quantitative graph of the current in...
42. ### Electric Field and Potential in a Conductor: Zero at the Surface?

I understand that if two points are inside a conductor they're at the same potential. I also understand that if two points lie on the surface of the conductor they're at the same potential. If point a lies on the surface of a conductor and point b lies inside is there still zero potential...
43. ### Equipotential line and electric field

Homework Statement The equation of an equipotential line in an electric field is y=2x, then the electric field strength vector at (1,2) may be a)4i+3j b)4i+8j c)8i+4j d)-8i+4jHomework Equations The Attempt at a Solution Since its an equipotential line, the electric field vector should...
44. ### Equipotential lines and electric field lines

Homework Statement I am given the equation for the potential of an arbitrary dipole. I need to draw the electric field lines for this dipole in a plane, and also show that these lines are perpendicular to the equipotential lines. I have already derived the equation for the electric field using...
45. ### Equipotential lines and electric field lines

Homework Statement I am given the equation for the potential of an arbitrary dipole. I need to draw the electric field lines for this dipole in a plane, and also show that these lines are perpendicular to the equipotential lines. I have already derived the equation for the electric field using...
46. ### Electric Potential and Equipotential Contour

Homework Statement The electric potential V in a region of space is given by the following expression, where A is a constant. V(x, y, z) = A (6x^2 - 7y^4 + 6z^2) (a) Derive an expression for the electric field vector E at any point in this region. (Use the following as necessary: A, x, y, and...
47. ### Equipotential with Spherical Conductors

For electrostatics, I know that conductors have 0 electric field inside. And I know that the surface of a spherical conductor has equipotential, (Maybe this is true for all shape of conductor in equilibrium right? ). So my question is, is the potential 0 inside a conductor as well? Is it...
48. ### Cassini ovals being the equipotential lines?

edit: If you cannot understand a passage it is probably because of my bad English. Just ask. Dear board, it is said oftentimes that the cassini ovals are the equipotential lines when considering two (equal) point charges in the oval's foci. (see e.g...
49. ### What are the correct answers to this Equipotential Contours Homework statement?

Homework Statement 2. Relevant thoughts Here, I numbered them down and put why I though those answers were what I have above 1- since 0V, its PE is zero 2- Closest contour lines, largest charge 3- Q1 is farther back from 0v, making it the largest negative 4- above 0V so it has a positive...
50. ### Equipotential, can someone check my work?

A solid sphere of +20C with radius a=1m. A conducting spherical shell of -30C with radius b=2m and c=3m is concentric with the solid sphere. 1. Find the inner and outer surface charge of the conducting spherical shell. My attempt: I just eyeballed it and said -20C is on the inner shell...