UA PHYS 241 - Physics 241 Lab: Electric Field Mapping

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Physics 241 Lab: Electric Field Mapping http://bohr.physics.arizona.edu/~leone/ua_spring_2009/phys241lab.html Name:____________________________ from “Sphinx” It is your fate, she has often said, to endure my riddling. Your fate to live at the mercy of my conundrum, which, in truth, is only a kind of psychic joke. No, you shall not leave this place. (Consider anyway the view from here.) In time, you will come to regard my questioning with a certain pained amusement; in time, get so you would hardly find it possible to live without my joke and me. -Robert Hayden Important: • In this course, every student has an equal opportunity to learn and chance of success. • How smart you are at physics is a simple function of how hard you work. Your work ethic is a matter of practice. Teach yourself to work hard in E&M by setting aside voluminous amounts of study time, making a study schedule and sticking to it. • Form study groups and meet as often as possible. Trade emails with several others and start working problems together immediately. Be sure to be inclusive in creating your study groups. • Join professional organizations. • Physicists help people: science => technology => jobs.Section 1: 1.1. One view of particle dynamics is to think about how a particle responds to the forces exerted on it (the vector force field). Another equally valid point of view is to may think of how a particle responds to the potential energy landscape that surrounds it (the scalar energy field). An example of how particles move in one dimension is shown below. Imagine that some external force creates the “hilly” potential energy graph shown, and that four particles are placed at locations A through D. The particle at A is at a minimum of the potential energy curve so that if it is moved, the particle will rise in potential energy. Therefore, the particle at A remains at rest. The particle at B sits at a part of the potential energy curve that has a positive slope. If it moves to the left, it can lower its potential energy. Therefore, the particle at B feels a force pushing it to the left. The particle at C sits at a precarious position. The slope of the potential energy curve at C is zero so that the particle feels no force. However, any small perturbation in the particles position will cause it to “tumble” from its unstable equilibrium. The particle at D has been placed where the slope of the potential energy curve is negative. If the particle moves to the right, it will lower its potential energy. Therefore, the particle at D feels a force pushing it to the right. Notice that the magnitude of the slope at B is greater than at D. Thus the particle at B does not have to move as far to lower its energy as the particle at D. This corresponds to the particle at B feeling a stronger force than the particle at D. The big idea here is that given a graph of potential energy, you can find the corresponding force at each location by realizing that Force equals the negative slope of the potential energy, ! r F (x) = "dU(x)dxˆ x . If a potential energy is given as ! U(x) =12k x " 2( )2 where ! k = 3 [J/m2], what is the force from this potential energy on a particle placed at ! x = 2 m? Where is the force zero? Your calculations and answers in SI units:1.2. The previous technique may be applied to motion in more than one direction: In this situation, the particle can move in the x and y directions. The potential energy landscape shows a single unstable equilibrium (“mountaintop”). Dotted lines have been drawn to show where the potential energy is constant. These are called equipotential lines. If this were a topographical map, these lines would represent constant height (called contours), and mountaineers maneuvering around the mountain at a constant height would be “contouring”. The force on the particle “down the mountain” is always perpendicular to the equipotential lines. Again the force will equal the negative slope of the potential energy graph, but how do you describe a two-dimensional slope? The answer is that the slope will be a two-dimensional vector (so is the force!), which can be found using partial derivatives: ! r F (x, y) ="#U(x, y)#xˆ x +"#U(x, y)#yˆ y . This equation makes intuitive sense. If you want the component of the force in the x-direction, find out how much the potential energy is changing in the x-direction using the partial x-derivative. If a potential energy is given as ! U(x, y) = k 2x2+ xy + 3xy2( ) where ! k = 1 [N/m], what is the force in the x-direction from this potential energy on a particle placed at ! x = 2 m and ! y = 0 m? Your calculations and answer in SI units:1.3. When charge is placed on a conductor, the excess charge creates an electric (vector) field around it. If a test charge q is place in this electric field, the test charge feels an electric force ! r F = qr E . Another equally valid point of view is that the charged conductor creates an electric potential landscape (a scalar field): Just as the force was the slope of the potential energy graph, the electric field is the slope of the electric potential graph: ! r E (x, y) ="#V (x, y)#xˆ x +"#V (x, y)#yˆ y or ! r E ="#V#xˆ x ,"#V#yˆ y $ % & ' ( ) . In a laboratory, one approximates derivatives with finite difference measurements: ! r E (x, y) "#$V (x, y)$xˆ x +#$V (x, y)$yˆ y =#(V (x + $x, y) # V (x, y)$xˆ x +#(V (x, y + $y) # V (x, y))$yˆ y This means that you can approximate the electric field at a point in space by using a DMM to measure the negative change in the voltage in each direction divided by the distance of your measurement. These ideas are easily extended to three dimensions: ! r F (x, y,z) ="#U(x, y,z)#xˆ x +"#U(x, y,z)#yˆ y +"#U(x, y,z)#zˆ z c (mathematically identical)r E (x, y,z) ="#V (x, y,z)#xˆ x +"#V (x, y,z)#yˆ y +"#V (x, y,z)#zˆ z1.4. In this lab, we will be working with two-dimensional electric vector fields (using conducting paper), but we will keep are graphs of the electric field and equipotentials restricted to two-dimensions by “looking at the mountain from above”. This “aerial view” will make it easier to make graphs, but don’t forget the underlying


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UA PHYS 241 - Physics 241 Lab: Electric Field Mapping

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