For those of you who are keen on learning geography, you may find that strange lines run through topographical maps. Sometimes these lines get very close together, and sometimes, they are far apart. These lines, unfortunately, don't appear in the real world when one is on terra firma, so they must mean something else. These lines represent regions of equal height; the closer they are together, the steeper the terrain. Valleys and cliffs will be shown as areas where many of these lines converge since the slope is significantly large. Open fields and farmlands are usually flat and are shown by lines that are spread far apart. These lines are known as isolines, and the terrain's gradient is constant along a single isoline. In this article, we will learn how isolines can also be used to represent electric and gravitational fields.
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Jetzt kostenlos anmeldenFor those of you who are keen on learning geography, you may find that strange lines run through topographical maps. Sometimes these lines get very close together, and sometimes, they are far apart. These lines, unfortunately, don't appear in the real world when one is on terra firma, so they must mean something else. These lines represent regions of equal height; the closer they are together, the steeper the terrain. Valleys and cliffs will be shown as areas where many of these lines converge since the slope is significantly large. Open fields and farmlands are usually flat and are shown by lines that are spread far apart. These lines are known as isolines, and the terrain's gradient is constant along a single isoline. In this article, we will learn how isolines can also be used to represent electric and gravitational fields.
The region around a charged particle is one in which another charged particle will interact with it. Like charges will repel each other while unlike charges attract. This indicates that a force exists between these particles, and we use this idea to define the electric field.
An electric field is a region of space in which a stationary, electrically charged particle experiences a force.
If a stationary electric charge feels a force around another charge, then they must both produce electric fields. The force experienced by a second charge will change depending on its magnitude and its position. If we draw lines of force that visually represent the magnitude and direction of the electric field at any point, we have drawn what are known as electric field lines.
Electric field lines are lines that represent the magnitude and direction of the electric field at different points in the region containing that field.
If the field lines around a charge are drawn, the interaction that another charge will experience in that field can be determined. Some rules apply to field lines, that is, electric field lines:
The definitions and rules only help our understanding to an extent but visualizing the field lines would be much more helpful. A positively charged particle can often be represented as a point charge in free space. We could use the rules above to aid in constructing the field lines, as in the example below.
Question: Draw the electric field lines around a positive point charge \(+q\) and a negative charge \(-q.\)
Answer: The electric force that another charged particle would experience would decrease at greater distances from the point charge, so the field lines would diverge outward. We can draw this as is done in Fig. 1 below.
The field lines point radially outward, beginning on the charge. As our rules suggest, the field lines become more spread out as the field gets weaker, and no two field lines will ever cross. If the positive charge were replaced with a negative one, the field lines would point radially inward, as in Fig. 2 below.
Note that spherical, charged objects can be treated as point charges with the charge concentrated at the center of the object.
The electric potential \(V\) due to a point charge at a distance \(r\) from it is given by \[V=\frac{1}{4\pi \varepsilon_0}\frac{q}{r},\] where the permittivity of free space \(\varepsilon_0=8.85\times 10^{-12}\,\mathrm{F\,m^{-1}}.\) The SI unit of measurement of potential is the \(\text{joule per coulomb,}\) \(\mathrm{J\,C^{-1}},\) which is equivalent to the \(\text{volt},\) \(\mathrm{V}.\) Conceptually, the potential is the work done per unit charge in the field. For a uniform electric field, the electric field lines will be parallel to each other and point in the same direction. This shows that the field strength is constant, and the direction is the same at any point in the region containing the field. That direction will be determined by the sign of the charge on the surface of the object generating the potential.
The equation for electric potential tells us that at different distances \(r\) from the surface containing the charge, there will be different potentials. However, along a line that is parallel to the surface, the potential will be constant, as all points on that line are equidistant from the surface. These lines of constant potential are called isolines, and for a uniform field, they appear as in Fig. 3 below.
Note that the isolines are always perpendicular to the field lines. This is always necessary since any component of the electric field along the direction of an isoline will cause an electric force on a charge along that line. Work would be done along that isoline and potential would not remain constant, which cannot occur.
The scenario is different for a point charge. The field lines would be radial, but we would require that the isolines always be perpendicular to them. The isolines would therefore form concentric circles centered on the point charge \(q.\) Fig. 4 below shows the field lines and isolines due to a positive point charge.
The circular isolines mean that the potential is constant along a circular path of radius \(r\) surrounding the point charge. If we think classically and assume that electrons orbit the nucleus of an atom in a circular path, this would be why the nucleus does not work on electrons. The magnitude of the average electric field is given by \[\left|\vec{E}\right|=\left|\frac{\Delta V}{\Delta r}\right|,\] in a region between two points separated by a distance \(\Delta r\) and having a potential difference \(\Delta V\) between them. The SI unit of measurement of the electric field strength is \(\text{volts per meter},\) \(\mathrm{V\,m^{-1}}.\)
There are a few things to consider when drawing an isoline of equipotential. Firstly, the isolines are circular rather than polygonal because there are many field lines not drawn in the diagram. The only way for the isolines to be perpendicular to them all is if they are circular. The following steps can be taken when asked to draw isolines:
Note that there are many field lines, but the number is not definite; it is only used to compare the strengths of two or more fields. There are infinitely many isolines since there should be one for every value of the energy.
Electric fields are not the only type of field in physics, so it would be difficult to believe that electric field lines would be the only type of field lines. In fact, gravitational fields are quite similar to electric fields. The field lines are radial for point masses, and the equipotential lines are always perpendicular to the field lines. The equipotential lines are lines of constant potential energy per unit mass rather than per unit charge as in the case of electric fields.
On a large scale, we can consider the Earth to be a point mass with its mass being concentrated at its center (called the center of mass). The field lines, as viewed from afar, would be radially inward. Unlike charge, mass can only be positive, and so field lines can only ever point inward to represent the attractive force of gravity. The field lines represent the direction in which another mass would move when entering the field. Fig. 8 below shows the gravitational field lines and gravitational equipotential lines for an isolated mass \(m.\)
The field lines and equipotential lines may look the same, and even Coulomb's law shares similarities with Newton's law of gravitation, \(F\propto \frac{1}{r^2},\) but there are many significant differences between the two fields. The table below describes some of the differences between the electric field and the gravitational field.
Table 1 - Differences between Electric Fields and Gravitational Fields
Electric Fields | Gravitational Fields |
Electric fields exist in a region around charges. | Gravitational fields exist in a region around masses. |
Field lines show the force on a positive test charge in the region. | Field lines show the force on a test mass in the region. |
Charges can be positive or negative meaning that field lines can point inward or outward. | Masses can only be positive meaning that field lines can only point inward. |
Lines of electric equipotential are lines of constant potential energy per unit charge. | Lines of gravitational equipotential are lines of constant potential energy per unit mass. |
Field lines are used to represent lines of force for all types of fields and are not restricted to only electric and gravitational fields.
Now that we have seen illustrations of the field lines and equipotential isolines for the electric field, we can test our knowledge on the following example.
Question: The potential difference between two oppositely-charged, parallel plates is \(120\,\mathrm{V}.\) If the plates are separated by a distance of \(0.50\,\mathrm{m},\) calculate the magnitude of the average electric field strength in the region between the plates.
Answer: To be able to draw the electric field lines between the plates, we must note that the field will be uniform in the region between the plates. Fig. 9 below shows the field lines for this arrangement.
The parallel field lines indicate the uniform nature of the field. We can find the magnitude of the average electric field strength as follows, \[\begin{align} \left|\vec{E}\right|&=\left|\frac{\Delta V}{\Delta r}\right|\\[4 pt]&=\left|\frac{120\,\mathrm{V}}{0.50\,\mathrm{m}}\right|\\[4 pt]&=240\,\mathrm{V\,m^{-1}}. \end{align}\] The average strength of the electric field between the plates is \(240\,\mathrm{V\,m^{-1}}.\)
Electric field lines are lines that represent the magnitude and direction of the electric field at different points in the region containing that field.
The field lines represent the direction in which a positive test charge will move when entering the field.
Electric field lines begin on positive charge and end on negative charge.
Electric field lines point from positive to negative.
Electric field lines can be straight or curved.
In what type of electrical system are uniform electric fields produced?
Parallel oppositely charged plates.
Define a uniform electric field.
A uniform electric field is an electric field whose field strength does not vary with position.
An isoline within an electric field is a region of ...
Increasing potential.
Which of these statements about isolines in a uniform electric field is NOT true?
The distance between isolines increases as you move away from the middle of the field.
What is the equation for the strength of the uniform field \(E\) between two plates a distance \(d\) apart with a potential difference \(V\) between them.
\(E=\frac{V}{d}\).
How will the electric field strength between two parallel plates change if you move them closer together, keeping the potential difference constant?
It will increase.
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