Potential Difference at One Location: Difference between revisions

Latest revision as of 16:41, 5 December 2015

Written by Alex George

The Main Idea

Potential difference is usually defined as the change in the potential between two distinct points. However, it is also useful to determine potential at a single location of interest.

In this case, we must define the potential at a location (ex. Location A) as the potential difference between a point infinitely far from all other charged particles and the location we defined.

As potential has a value at all locations at space, it is a scalar field, not a vector field.

A Mathematical Model

The Potential at One Location

[math]\displaystyle{ V_{a} = V_{a} - V_{\infty } }[/math] where [math]\displaystyle{ V_{a} }[/math] is the potential location of interest and [math]\displaystyle{ V_{\infty } }[/math] is the potential at a location infinitely far away.

This equation is only valid when [math]\displaystyle{ V_{\infty } }[/math] is equal to zero.

Potential near a Point Charge

Parameters: A point charge with charge [math]\displaystyle{ q }[/math]. We are a distance [math]\displaystyle{ x }[/math] from this point charge and we want to find the potential difference between our point charge and the potential at infinity.

[math]\displaystyle{ V_{x} = V_{x} - V_{\infty } = -\int_{\infty }^{x} \frac{1}{4\pi\varepsilon _{0} } \frac{q}{x^2} dx = \frac{1}{4\pi\varepsilon _{0} } (\frac{q}{x}-\frac{q}{\infty}) = \frac{1}{4\pi\varepsilon _{0}} \frac{q}{x} }[/math]

The sign of the potential depends on the sign of the charge on the particle:

if [math]\displaystyle{ q \lt 0, V_{x} \lt 0 }[/math]

if [math]\displaystyle{ q \gt 0, V_{x} \gt 0 }[/math]

Potential Energy at a Single Location

Once the value of the potential at one location is known, the potential energy of a system can be calculated.

If a charged object was placed at the location where the potential is known, then the potential energy of the system would be given the the equation:

[math]\displaystyle{ U_{A} = qV_{A} }[/math]

Substituting in the derived equation from the previous section for [math]\displaystyle{ V_{A} }[/math], we obtain the following:

[math]\displaystyle{ U = q_{1} (\frac{1}{4\pi\varepsilon _{0}} \frac{q_{2}}{x}) = (\frac{1}{4\pi\varepsilon _{0}} \frac{q_{1}q_{2}}{x}) }[/math] where [math]\displaystyle{ x }[/math] is the separation between the two particles, and [math]\displaystyle{ q_{1} }[/math] and [math]\displaystyle{ q_{2} }[/math] are the respective charges of the particles.

A Visual Model

A visualization of Distance vs. Potential from a point charge is given by Figure 1:

Figure 1. Potential vs. Distance for a Point charge with Charge [math]\displaystyle{ e }[/math]

The electric field of the point charge can be obtained by differentiation. The negative gradient of the potential with respect to distance would yield the electric field:

[math]\displaystyle{ E_{x} = -\frac{\partial V}{\partial x} }[/math]

Here is a video explaining the above concepts with more examples [1].

Examples

Here are a couple of problems that illustrate the concepts presented above.

Simple Examples

Example 1 - Potential at a Location Near a Sphere

What is the electric potential 5 cm away from a 1 cm diameter metal sphere that has a -3.00 nC static charge?

Solution:

As the distance from the center of the sphere is greater than the radius, we can treat the metal sphere as a point charge.

From the above derivation of the potential for a single point charge, we have:

[math]\displaystyle{ V_{x} = V_{x} - V_{\infty } = -\int_{\infty }^{x} \frac{1}{4\pi\varepsilon _{0} } \frac{q}{x^2} dx = \frac{1}{4\pi\varepsilon _{0} } \bigg(\frac{q}{x}-\frac{q}{\infty} \bigg) }[/math]

[math]\displaystyle{ V_{x} = 9\times 10^{9} \frac{N\cdot m^{2}}{C^{2} } \bigg(\frac{-3.00\times 10^{-9}\ C}{5.00\times 10^{-2}\ m}-\frac{-3.00\times 10^{-9}\ C}{\infty}\bigg) = 9\times 10^{9} \frac{N\cdot m^{2}}{C^{2} } \frac{-3.00\times 10^{-9}\ C}{5.00\times 10^{-2}\ m} = -539 V }[/math]

Example 2 - Charge Required to Produce a Specific Potential

What is the excess charge on a Van de Graff generator that has a 25.0 cm diameter metal sphere that produces a voltage of 50 kV near the surface?

Solution

The potential of the surface will be the same as that of a point charge from the center of the sphere, 12.5 cm away. The excess charge can be derived from the equation for potential a distance away from a point charge:

[math]\displaystyle{ V_{x} = \frac{1}{4\pi\varepsilon _{0} } \bigg(\frac{q}{x}\bigg) }[/math]

[math]\displaystyle{ q = {4\pi\varepsilon _{0}} x V_{x} }[/math]

[math]\displaystyle{ q = \frac{1}{9\times 10^{9}} \frac {N\cdot m^{2}}{C^{2}} (12.5 \times 10^{-2} \ m) (50 \times 10^{3} \ V) = 6.94 \times 10^{-7} \ C }[/math]

Difficult Examples

Example 1 - Potential Along the Axis of a Ring

What is the potential a distance [math]\displaystyle{ z }[/math] from the center of the ring?

Solution:

The electric field of a ring is given by the following equation:

[math]\displaystyle{ E_{z,ring} = \frac{1}{4\pi\varepsilon _{0} } \frac {Qz}{(R^{2}+z^{2})^{\frac {3}{2}}} }[/math] where [math]\displaystyle{ z }[/math] is the distance from the center of the ring, [math]\displaystyle{ R }[/math] is radius of the ring, and [math]\displaystyle{ Q }[/math] is the charge on the ring.

This equation can be integrated with respect to [math]\displaystyle{ z }[/math] and the appropriate bounds to obtain the potential at a location [math]\displaystyle{ z }[/math] from the center of the ring:

[math]\displaystyle{ V_{z,ring} = -\int_{\infty}^{z} \frac{1}{4\pi\varepsilon _{0} } \frac {Qz}{(R^{2}+z^{2})^{\frac {3}{2}}} dz }[/math]

[math]\displaystyle{ V_{z,ring} = \frac{1}{4\pi\varepsilon _{0} } \frac {Q}{\sqrt{R^{2}+z^{2}}} }[/math]

Connectedness

How is this topic connected to something that you are interested in? This topic is related to the concept of Potential Energy, or the capacity to do work in a system. Electric Potential is a derivation of potential energy and is a convenient way to express potential in terms of electric potential energy per unit charge. I am interested in bio instrumentation, so this concept has direct relevance in terms of circuit analysis and design.

How is it connected to your major? My major is biochemistry and electric potential has several applications in biological systems. For instance, in mechanisms such as the firing of a neuron's action potential, the electric potential across the neuronal membrane must reach certain thresholds in order to propagate the neural signal.

Is there an interesting industrial application? Electric potential has various applications in industry, particularly in the aspects of circuit design and analysis. A particular implementation in medicine would be a defibrillator.

History

The SI unit for electric potential is the volt, which is named in honor of the Italian physicist Count Alessandro Volta (1745-1827), who invented the voltaic pile, the first chemical battery.

Volta had developed the voltaic pile using an effective pair of metals (zinc and silver) to create a steady electric current in the 1880s.

Today, the "conventional" volt is used and was defined in 1988 by the 18th general conference on Weights and Measures and adopted in 1990. The Josephson effect was employed to produce an exact frequency to voltage conversion and a cesium frequency standard is used to define the modern volt.

More information on Alessandro Volta can be found at Wikipedia [2] .

Wikipedia.org

@@ Line 3: / Line 3: @@
 ==The Main Idea==
-Usually, we are interested in the potential difference between two different points. However, it is also useful to determine potential at a single location.
+Potential difference is usually defined as the change in the potential between two distinct points. However, it is also useful to determine potential at a single location of interest.
 In this case, we must define the potential at a location (ex. Location A) as the potential difference between a point infinitely far from all other charged particles and the location we defined.
-As potential has a value at all locations at space, it is a scalar field in it of itself.
+As potential has a value at all locations at space, it is a scalar field, not a vector field.
 ===A Mathematical Model===
-'''Potential at One Location'''
+'''The Potential at One Location'''
 <math>V_{a} = V_{a} - V_{\infty }</math> where '''<math>V_{a}</math>''' is the potential location of interest and '''<math>V_{\infty }</math>''' is the potential at a location infinitely far away.
 This equation is only valid when <math>V_{\infty }</math> is equal to zero.
 '''Potential near a Point Charge'''
@@ Line 29: / Line 28: @@
 if <math> q > 0, V_{x} > 0 </math>
 '''Potential Energy at a Single Location'''
@@ Line 41: / Line 39: @@
 Substituting in the derived equation from the previous section for <math>V_{A}</math>, we obtain the following:
-<math> U_{A} = q_{1} (\frac{1}{4\pi\varepsilon _{0}} \frac{q_{2}}{x}) = (\frac{1}{4\pi\varepsilon _{0}} \frac{q_{1}q_{2}}{x}) </math> where <math> x </math> is the separation between the two particles, and <math> q_{1} </math> and <math> q_{2} </math> are the respective charges of the particles.
+<math> U = q_{1} (\frac{1}{4\pi\varepsilon _{0}} \frac{q_{2}}{x}) = (\frac{1}{4\pi\varepsilon _{0}} \frac{q_{1}q_{2}}{x}) </math> where <math> x </math> is the separation between the two particles, and <math> q_{1} </math> and <math> q_{2} </math> are the respective charges of the particles.
 ===A Visual Model===
@@ Line 51: / Line 49: @@
 '''Figure 1.''' Potential vs. Distance for a Point charge with Charge <math> e </math>
-The electric field of the point charge can bet obtained by differentiation. The negative gradient of the potential with respect to distance would yield the electric field:
+The electric field of the point charge can be obtained by differentiation. The negative gradient of the potential with respect to distance would yield the electric field:
 <math> E_{x} = -\frac{\partial V}{\partial x} </math>
+Here is a video explaining the above concepts with more examples [https://youtu.be/70SsJNE3VFE].
 ==Examples==
@@ Line 59: / Line 59: @@
 Here are a couple of problems that illustrate the concepts presented above.
-===Simple===
+===Simple Examples===
-) What is the electric potential 5 cm away from a 1 cm diameter metal sphere that has a -3.00 nC static charge?
+'''Example 1 - Potential at a Location Near a Sphere'''
+What is the electric potential 5 cm away from a 1 cm diameter metal sphere that has a -3.00 nC static charge?
 '''Solution:'''
@@ Line 71: / Line 73: @@
 <math>V_{x} = V_{x} - V_{\infty } = -\int_{\infty }^{x} \frac{1}{4\pi\varepsilon _{0} } \frac{q}{x^2} dx = \frac{1}{4\pi\varepsilon _{0} } \bigg(\frac{q}{x}-\frac{q}{\infty} \bigg) </math>
-<math>V_{x} = 9\times 10^{9} \frac{N\dotm^{2}}{C^{2} } (\frac{-3.00\times 10^{-9} \quad C}{5.00\times 10^{-2} \quad m}-\frac{q}{\infty}) </math>
+<math>V_{x} = 9\times 10^{9} \frac{N\cdot m^{2}}{C^{2} } \bigg(\frac{-3.00\times 10^{-9}\ C}{5.00\times 10^{-2}\ m}-\frac{-3.00\times 10^{-9}\ C}{\infty}\bigg) = 9\times 10^{9} \frac{N\cdot m^{2}}{C^{2} } \frac{-3.00\times 10^{-9}\ C}{5.00\times 10^{-2}\ m} = -539 V </math>
+'''Example 2 - Charge Required to Produce a Specific Potential'''
-===Difficult===
+What is the excess charge on a Van de Graff generator that has a 25.0 cm diameter metal sphere that produces a voltage of 50 kV near the surface?
+'''Solution'''
+The potential of the surface will be the same as that of a point charge from the center of the sphere, 12.5 cm away. The excess charge can be derived from the equation for potential a distance away from a point charge:
+<math> V_{x} = \frac{1}{4\pi\varepsilon _{0} } \bigg(\frac{q}{x}\bigg) </math>
+<math> q = {4\pi\varepsilon _{0}} x V_{x} </math>
+<math> q = \frac{1}{9\times 10^{9}} \frac {N\cdot m^{2}}{C^{2}} (12.5 \times 10^{-2} \ m) (50 \times 10^{3} \ V) = 6.94 \times 10^{-7} \ C  </math>
+===Difficult Examples===
+'''Example 1 - Potential Along the Axis of a Ring'''
+What is the potential a distance <math> z </math> from the center of the ring?
+'''Solution:'''
+The electric field of a ring is given by the following equation:
+<math> E_{z,ring} = \frac{1}{4\pi\varepsilon _{0} } \frac {Qz}{(R^{2}+z^{2})^{\frac {3}{2}}} </math> where <math> z </math> is the distance from the center of the ring, <math> R </math> is radius of the ring, and <math> Q </math> is the charge on the ring.
+This equation can be integrated with respect to <math> z </math> and the appropriate bounds to obtain the potential at a location <math> z </math> from the center of the ring:
+<math> V_{z,ring} = -\int_{\infty}^{z} \frac{1}{4\pi\varepsilon _{0} } \frac {Qz}{(R^{2}+z^{2})^{\frac {3}{2}}} dz</math>
+<math> V_{z,ring} = \frac{1}{4\pi\varepsilon _{0} } \frac {Q}{\sqrt{R^{2}+z^{2}}} </math>
 ==Connectedness==
-#How is this topic connected to something that you are interested in?
+#'''How is this topic connected to something that you are interested in?''' This topic is related to the concept of [[Potential Energy]], or the capacity to do work in a system. [[Electric Potential]] is a derivation of potential energy and is a convenient way to express potential in terms of electric potential energy per unit charge. I am interested in bio instrumentation, so this concept has direct relevance in terms of circuit analysis and design.
-#How is it connected to your major?
-#Is there an interesting industrial application?
+#'''How is it connected to your major?''' My major is biochemistry and electric potential has several applications in biological systems. For instance, in mechanisms such as the firing of a neuron's action potential, the electric potential across the neuronal membrane must reach certain thresholds in order to propagate the neural signal.
+#'''Is there an interesting industrial application?''' Electric potential has various applications in industry, particularly in the aspects of circuit design and analysis. A particular implementation in medicine would be a defibrillator.
 ==History==
-Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.
+The SI unit for electric potential is the volt, which is named in honor of the Italian physicist [[Count Alessandro Volta]] (1745-1827), who invented the voltaic pile, the first chemical battery.
+Volta had developed the voltaic pile using an effective pair of metals (zinc and silver) to create a steady electric current in the 1880s.
+Today, the "conventional" volt is used and was defined in 1988 by the 18th general conference on Weights and Measures and adopted in 1990. The Josephson effect was employed to produce an exact frequency to voltage conversion and a cesium frequency standard is used to define the modern volt.
+More information on Alessandro Volta can be found at Wikipedia [https://en.wikipedia.org/wiki/Alessandro_Volta] .
 == See also ==
-Are there related topics or categories in this wiki resource for the curious reader to explore?  How does this topic fit into that context?
+[[Electric Potential]]
+[[Potential Difference in a Uniform Field]]
+[[Potential Difference Path Independence]]
+[[Potential Difference of point charge in a non-Uniform Field]]
+[[Sign of Potential Difference]]
+[[Potential Difference in an Insulator]]
 ===Further reading===
-Books, Articles or other print media on this topic
+Chabay, Ruth W. ''Matter and Interactions, Ch. 16.7.'' 4th ed. New York: Wiley, 2015. Print.
+Urone, Paul Peter., et al. ''College Physics, Ch. 19.3''. Houston, Texas: OpenStax College, Rice University, 2013. Print.
 ===External links===
-[http://www.scientificamerican.com/article/bring-science-home-reaction-time/]
+[https://youtu.be/70SsJNE3VFE Youtube Video on Electric Potential of a Point Charge]
+[https://en.wikipedia.org/wiki/Alessandro_Volta Alessandro Volta]
 ==References==
-This section contains the the references you used while writing this page
+Chabay, Ruth W. ''Matter and Interactions.'' 4th ed. New York: Wiley, 2015. Print.
+Urone, Paul Peter., et al. ''College Physics''. Houston, Texas: OpenStax College, Rice University, 2013. Print.
+Wikipedia.org
-[[Category:Which Category did you place this in?]]
+[[Category:Electric Potential]]

Potential Difference at One Location: Difference between revisions

Latest revision as of 16:41, 5 December 2015

Contents

The Main Idea

A Mathematical Model

A Visual Model

Examples

Simple Examples

Difficult Examples

Connectedness

History

See also

Further reading

External links

References

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