Physics
Equipotential Lines
Equipotential lines are imaginary lines in a field where the potential at every point is the same. In physics, specifically in the context of electric fields, equipotential lines are perpendicular to the electric field lines. They are used to visualize and understand the distribution of electric potential in a given space.
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10 Key excerpts on "Equipotential Lines"
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
Given the electric field lines, the Equipotential Lines can be drawn simply by making them perpendicular to the electric field lines. Conversely, given the Equipotential Lines, as in Figure 19.10(a), the electric field lines can be drawn by making them perpendicular to the equipotentials, as in Figure 19.10(b). 744 Chapter 19 | Electric Potential and Electric Field This OpenStax book is available for free at http://cnx.org/content/col11406/1.9 Figure 19.9 The electric field lines and Equipotential Lines for two equal but opposite charges. The Equipotential Lines can be drawn by making them perpendicular to the electric field lines, if those are known. Note that the potential is greatest (most positive) near the positive charge and least (most negative) near the negative charge. Figure 19.10 (a) These Equipotential Lines might be measured with a voltmeter in a laboratory experiment. (b) The corresponding electric field lines are found by drawing them perpendicular to the equipotentials. Note that these fields are consistent with two equal negative charges. One of the most important cases is that of the familiar parallel conducting plates shown in Figure 19.11. Between the plates, the equipotentials are evenly spaced and parallel. The same field could be maintained by placing conducting plates at the Equipotential Lines at the potentials shown. Figure 19.11 The electric field and Equipotential Lines between two metal plates. An important application of electric fields and Equipotential Lines involves the heart. The heart relies on electrical signals to maintain its rhythm. The movement of electrical signals causes the chambers of the heart to contract and relax. When a person has a heart attack, the movement of these electrical signals may be disturbed. An artificial pacemaker and a defibrillator can be used to initiate the rhythm of electrical signals.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
A two-dimensional map of the cross-sectional plane that contains both charges is shown in Figure 7.34. The line that is equidistant from the two opposite charges corresponds to zero potential, since at the points on the line, the positive potential from the positive charge cancels the negative potential from the negative charge. Equipotential Lines in the cross-sectional plane are closed loops, which are not necessarily circles, since at each point, the net potential is the sum of the potentials from each charge. Figure 7.34 A cross-section of the electric potential map of two opposite charges of equal magnitude. The potential is negative near the negative charge and positive near the positive charge. View this simulation (https://openstaxcollege.org/l/21equipsurelec) to observe and modify the equipotential surfaces and electric fields for many standard charge configurations. There’s a lot to explore. One of the most important cases is that of the familiar parallel conducting plates shown in Figure 7.35. Between the plates, the equipotentials are evenly spaced and parallel. The same field could be maintained by placing conducting plates at the Equipotential Lines at the potentials shown. 322 Chapter 7 | Electric Potential This OpenStax book is available for free at http://cnx.org/content/col12074/1.3 Figure 7.35 The electric field and Equipotential Lines between two metal plates. Note that the electric field is perpendicular to the equipotentials and hence normal to the plates at their surface as well as in the center of the region between them. Consider the parallel plates in Figure 7.2. These have Equipotential Lines that are parallel to the plates in the space between and evenly spaced. An example of this (with sample values) is given in Figure 7.35. We could draw a similar set of equipotential isolines for gravity on the hill shown in Figure 7.2. If the hill has any extent at the same slope, the isolines along that extent would be parallel to each other.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler, Heath Jones, Matthew Collins, John Daicopoulos, Boris Blankleider(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
An equipotential surface is a surface on which the electric potential is the same everywhere. The easiest equipotential surfaces to visualise are those that surround an isolated point charge. According to equation 19.6, the potential at a distance r from a point charge q is V = kq/r. Thus, wherever r is the same, the potential is the same, and the equipotential surfaces are spherical surfaces centred on the charge. There are an infinite number of such surfaces, one for every value of r, and figure 19.11 illustrates two of them. The larger the distance r, the smaller is the potential of the equipotential surface. 520 Physics The net electric force does no work as a charge moves on an equipotential surface. This important characteristic arises because when an electric force does work W AB as a charge moves from A to B, the potential changes according to V B − V A = −W AB /q 0 (equation 19.4). Since the potential remains the same everywhere on an equipotential surface, V A = V B , and we see that W AB = 0 J. In figure 19.11, for instance, the electric force does no work as a test charge moves along the circular arc ABC, which lies on an equipotential surface. In contrast, the electric force does work when a charge moves between equipotential surfaces, as from A to D in the picture. The spherical equipotential surfaces that surround an isolated point charge illustrate another character- istic of all equipotential surfaces. Figure 19.12 shows two of the surfaces around a positive point charge, along with some electric field lines. The electric field lines give the direction of the electric field, and for a positive point charge the electric field is directed radially outwards. Therefore, at each location on an equipotential sphere the electric field is perpendicular to the surface and points outwards in the direction of decreasing potential, as the drawing emphasises.
- Alex Kaufman, B. Anderson, Alex A. Kaufman(Authors)
- 2010(Publication Date)
- Elsevier Science(Publisher)
Then, the Equipotential Lines again tend to circles. It happens because at great distances, the potential of the two charges behaves as that of the single charge 2 e placed at the origin. At the point (x = 0, y = 0, z = 0) the electric field is equal to zero, that is, grad U = 0. This is actually a “saddle point” of the potential, where it increases in the two directions along the x -axis going toward the charges, but decreases in opposite directions pointing along the y -axis. By drawing lines perpendicular to the Equipotential Lines, or solving Eq. [1.99], we obtain the family of the vector lines of the field E (Fig. 1.8C). All vectors lines start from charges, while the other terminal points are located at infinity. It is useful to notice that in the vicinity of the plane x = 0 and not far from the origin the vector lines are almost parallel to this plane. Such “focusing” of the field due to the presence of two charges is often used in borehole geophysics. Figure 1.8D shows the behavior of the potential and electric field caused by charges of equal magnitude and opposite sign. In this case, each vector line has the same terminal points, x = ± a, and everywhere grad U ≠ 0 ; that is, the field is regular at all points except at the two point charges. 1.12.4 Behavior of Potential U and Field E far away from Charges Suppose that a point of observation p is far away from charges distributed in some volume V with density δ (q). This means that distances L qp from the point p to any point q of the volume is nearly constant. Correspondingly, Eq. [1.90] can be written as U (p) = 1 4 π ε 0 L qp ∫ V δ (q) d V = e 4 π ε 0 L qp if L qp → ∞. [1.107] Here q is any point inside of the volume and e is total charge
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
The electric field points in the direction of decreasing potential. 19.4 Equipotential Surfaces and Their Relation to the Electric Field 535 parallel to the surface. This field component would exert an electric force on a test charge placed on the surface. As the charge moved along the surface, work would be done by this component of the electric force. The work, according to Equation 19.4, would cause the potential to change, and, thus, the surface could not be an equipotential surface as assumed. The only way out of the dilemma is for the electric field to be perpendicular to the surface, so there is no component of the field parallel to the surface. We have already encountered one equipotential surface. In Section 18.8, we found that the direction of the electric field just outside an electrical conductor is perpendicular to the conduc- tor’s surface, when the conductor is at equilibrium under electrostatic conditions. Thus, the sur- face of any conductor is an equipotential surface under such conditions. In fact, since the electric field is zero everywhere inside a conductor whose charges are in equilibrium, the entire conduc- tor can be regarded as an equipotential volume. There is a quantitative relation between the electric field and the equipotential surfaces. One example that illustrates this relation is the parallel plate capacitor in Figure 19.15. As Section 18.6 discusses, the electric field E → between the metal plates is perpendicular to them and is the same everywhere, ignoring fringe fields at the edges. To be perpendicular to the electric field, the equi- potential surfaces must be planes that are parallel to the capacitor plates, which themselves are equipotential surfaces. The potential difference between the plates is given by Equation 19.4 as ΔV = V B − V A = −W AB /q 0 , where A is a point on the positive plate and B is a point on the negative plate.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Since the potential remains the same everywhere on an equipotential surface, V A = V B , and we see that W AB = 0 J. In Figure 19.11, for instance, the electric force does no work as a test charge moves along the circular arc ABC, which lies on an equipotential surface. In contrast, the electric force does work when a charge moves between equipotential surfaces, as from A to D in the picture. The spherical equipotential surfaces that surround an isolated point charge illus- trate another characteristic of all equipotential surfaces. Figure 19.12 shows two of the surfaces around a positive point charge, along with some electric field lines. The electric field lines give the direction of the electric field, and for a positive point charge the elec- tric field is directed radially outward. Therefore, at each location on an equipotential sphere the electric field is perpendicular to the surface and points outward in the direc- tion of decreasing potential, as the drawing emphasizes. This perpendicular relation is valid whether or not the equipotential surfaces result from a positive charge or have a spherical shape. Problem-Solving Insight The electric field created by any charge or group of charges is everywhere perpendicular to the associated equipotential surfaces and points in the direction of decreasing potential. For example, Animated Figure 19.13 shows the electric field lines (in red) around an electric dipole, along with some equipotential surfaces (in blue), shown in cross section. Since the field lines are not simply radial, the equipotential surfaces are no longer spherical but, instead, have the shape necessary to be everywhere perpendicular to the field lines. + - ANIMATED FIGURE 19.13 A cross-sectional view of the equipotential surfaces (in blue) of an electric dipole. The surfaces are drawn to show that at every point they are perpendicular to the electric field lines (in red) of the dipole.
eBook - ePub- A. L. Stanford, J. M. Tanner(Authors)
- 2014(Publication Date)
- Academic Press(Publisher)
x axis.We may determine the electric field componentExfrom the given potential V using Equation (13-25) :Then the required electric field isE x= −= −d Vd xdd x[k λ ln](x)x − LE x=k λ Lx(x − L)E =(13-26)k λ Lx(x − L)i ˆ(13-26)13.4 Equipotential Surfaces and Charged Conductors
Any surface along which the electric potential has a constant value is called an equipotential surface . Equation (13-6) , dV = −E · ds , requires that for a displacement ds along an equipotential surface, where dV = 0, the electric field E is perpendicular to the displacement, so that E · ds = 0. Therefore, the electric field at any point is perpendicular to the equipotential surface passing through that point.The relationship between an electric field and its corresponding equipotential surfaces is conventionally pictured in diagrams like those of Figure 13.10 , a point charge and an electric dipole. In this figure the electric field (solid lines) and equipotential surfaces (dashed lines) are displayed in a cross-sectional plane passing through all points on an axis of symmetry. For charge distributions with axes of symmetry, the equipotential surfaces are those surfaces of revolution generated by rotating the dashed lines about an axis of symmetry. Notice that the equipotential surfaces are perpendicular to the electric field lines at every point.Figure 13.10 Representations of electric fields (solid lines) and equipotential surfaces (dashed lines) for a positive point charge and an electric dipole. The equipotential surfaces are perpendicular to the lines of electric field at every point.The surface of a charged conductor is an equipotential surface, because the electric field immediately outside the surface is perpendicular to that surface. Inside a conducting material, where the electric field E is identically equal to zero, the change (dV = −E · ds
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
For example, Figure 19.13 shows the electric field lines (in red) around an electric dipole, along with some equipotential surfaces (in blue), shown in cross section. Since the field lines are not simply radial, the equipotential surfaces are no longer spherical but, instead, have the shape necessary to be everywhere perpendicular to the field lines. Problem-Solving Insight q 1 q 2 q 3 q 4 (a) 2 2 2 2 (b) 1 1 2 2 (c) 1 1 1 1 (d) 1 2 1 2 q 1 q 2 q 4 q 3 + q D B A C Equipotential surfaces Figure 19.11 The equipotential surfaces that surround the point charge 1q are spherical. The electric force does no work as a charge moves on a path that lies on an equipotential surface, such as the path ABC. However, work is done by the electric force when a charge moves between two equipotential surfaces, as along the path AD. + q Higher potential Lower potential 90° 90° 90° Electric field line Figure 19.12 The radially directed electric field of a point charge is perpendicular to the spherical equipotential surfaces that surround the charge. The electric field points in the direction of decreasing potential. 19.4 | Equipotential Surfaces and Their Relation to the Electric Field 469 To see why an equipotential surface must be perpendicular to the electric field, consider Figure 19.14, which shows a hypothetical situation in which the perpendicular relation does not hold. If E B were not perpendicular to the equipotential surface, there would be a component of E B parallel to the surface. This field component would exert an electric force on a test charge placed on the surface. As the charge moved along the surface, work would be done by this component of the electric force. The work, accord- ing to Equation 19.4, would cause the potential to change, and, thus, the surface could not be an equipotential surface as assumed. The only way out of the dilemma is for the electric field to be perpendicular to the surface, so there is no component of the field parallel to the surface.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
In this module, we propose to go the other way — that is, to find the electric field when we know the potential. As Fig. 24-5 shows, solving this problem graphically is easy: If we know the potential V at all points near an assembly of charges, we can draw in a family of equipotential surfaces. The electric field lines, sketched perpendicu- lar to those surfaces, reveal the variation of E → . What we are seeking here is the mathematical equivalent of this graphical procedure. Figure 24-17 shows cross sections of a family of closely spaced equipo- tential surfaces, the potential difference between each pair of adjacent surfaces being dV. As the figure suggests, the field E → at any point P is perpendicular to the equipotential surface through P. Suppose that a positive test charge q 0 moves through a displacement d s → from one equipotential surface to the adjacent surface. From Eq. 24-6, we see that the work the electric field does on the test charge during the move is –q 0 dV. From Eq. 24-16 and Fig. 24-17, we see that the work done by the electric field may also be written as the scalar product (q 0 E → ) · d s → , or q 0 E(cos θ) ds. Equating these two expressions for the work yields –q 0 dV = q 0 E(cos θ) ds, (24-38) or E cos θ = − dV ds . (24-39) Since E cos θ is the component of E → in the direction of d s → , Eq. 24-39 becomes E s = − ∂V ∂s . (24-40) We have added a subscript to E and switched to the partial derivative symbols to emphasize that Eq. 24-40 involves only the variation of V along a specified axis (here called the s axis) and only the component of E → along that axis. In words, Eq. 24-40 (which is essentially the reverse operation of Eq. 24-18) states: s q 0 P θ Two equipotential surfaces + ds E Figure 24-17 A test charge q 0 moves a distance d s → from one equipotential surface to another.
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
2. The x, y, and z components of E → may be found from E x = − ∂V ___ ∂x ; E y = − ∂V ___ ∂y ; E z = − ∂V ___ ∂z . When E → is uniform, all this reduces to E = − Δ V ____ Δs , where s is perpendicular to the equipotential surfaces. 3. The electric field is zero parallel to an equipotential surface. LEARNING OBJECTIVES 704 CHAPTER 24 Electric Potential Calculating the Field from the Potential In Module 24.2, you saw how to find the potential at a point f if you know the electric field along a path from a reference point to point f. In this module, we propose to go the other way—that is, to find the electric field when we know the potential. As Fig. 24.2.2 shows, solving this problem graphically is easy: If we know the potential V at all points near an assembly of charges, we can draw in a family of equipotential surfaces. The electric field lines, sketched perpendicular to those surfaces, reveal the variation of E → . What we are seeking here is the math- ematical equivalent of this graphical procedure. Figure 24.6.1 shows cross sections of a family of closely spaced equipo- tential surfaces, the potential difference between each pair of adjacent surfaces being dV. As the figure suggests, the field E → at any point P is perpendicular to the equipotential surface through P. Suppose that a positive test charge q 0 moves through a displacement d s → from one equipotential surface to the adjacent surface. From Eq. 24.1.6, we see that the work the electric field does on the test charge during the move is –q 0 dV. From Eq. 24.2.2 and Fig. 24.6.1, we see that the work done by the electric field may also be written as the scalar product ( q 0 E → ) ⋅ d s → , or q 0 E(cos θ) ds. Equating these two expressions for the work yields –q 0 dV = q 0 E(cos θ) ds, (24.6.1) or E cos θ = − dV ___ ds . (24.6.2) Since E cos θ is the component of E → in the direction of d s → , Eq.
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