Electric Field. Purpose of the experiment. PES 2160 Advanced Physics Lab II

Transcription

1 Electric Field PES 2160 Advanced Physics Lab II Purpose of the experiment To see what an electric field and electric potential look like. To understand how the shape of the electric field and potential are determined by the arrangement of the charges that formed them. To understand the relationship between electric field and potential. FYI FYI Los Angeles s full name is El Pueblo de Nuestra Senora la Reina de los Angeles de Porciuncula.

2 Page2 Background The Electric Field Think about how the electric force works for a minute and you may find something a little strange. A charge over here can attract or repel a charge way over there. This is strange because this attraction or repulsion happens without the two charges ever touching each other. Common experience tells us that most forces we see in everyday life require contact between two objects. Nature has a clever way of solving this action-at-a-distance issue, the electric field. A good working definition of the electric field is the effect produced in the space surrounding an electric charge that causes forces on other electric charges. Essentially, the first electric charge generates the electric field that transmits the force through the intervening space between the charges. The second charge is then influenced, not by the first charge directly, but by the electric field produced by the first charge. If you are a little confused, here is a reasonable analogy for the electric field. Consider the charges to be like two ice skaters on a nearly friction-free ice rink. The first ice skater throws a football toward the second ice skater. As the first ice skater throws the football, he moves backward away from the second skater. The second skater then catches the football, receives the football s momentum, and moves away from the first skater. Thus, the football has caused a repulsive force to be transmitted between the two skaters without the two ever touching. (Admittedly, it is a little difficult to see how throwing a football would result in an attraction.) In this analogy, the skaters are the charges and the football is the electric field. So, what, physically speaking, is the electric field? The electric field is something very much like the football. The first charge generates many footballs, called photons, in the space surrounding it. This photon cloud that surrounds the charge is precisely the electric field. These photons, or particles of light, are exchanged with the second charge thus producing the electric or Coulomb force. Thus, the action-at-a-distance problem has disappeared because the photons do come into contact with the second charge. We say the photon mediates the electric force. We now know what the electric field is, a photon cloud. As scientists, we need to model the electric field mathematically. Experimentally, it is found that the force between two charges is given by Coulomb s Law:

3 Page3 qq F k rˆ (Coulomb s Law) 2 r At any point in space, the electric field is defined in terms of the force it would exert on a positive test charge q. F E (Definition of the electric field) q Note that since the force is a vector, the electric field will also be a vector. By combining the above two equations, we get the electric field of a single charge: kq E rˆ (Electric field caused by a single charge Q) 2 r Notice that while it requires two charges, q and Q, to have an electric force, it requires only one charge, in this case Q, to generate an electric field. Additionally, at every point where that electric field is to be specified, it must be defined in terms of both magnitude (kq/r 2 ) and direction ( r ). So, what if you have more than one charge generating an electric field? How do the combined fields of charges Q1, Q2, and so forth affect the test charge q? The wonderful thing about nature is that it always comes up with the simplest way to do something. Here, the fields simply add: E TOTAL E E (Principle of Superposition) 1 2 E3 Bear in mind that this is a vector sum, not a scalar sum. The direction is critical in adding up all the individual fields. All of this suggests a way to measure the electric field, which is the subject of this lab. If we have an electric field, we can measure its strength by placing the unit test charge in the field and measuring the force on the charge. In a roundabout way, this is essentially what the digital voltmeter does.

4 Page4 Electric Potential (Voltage) You may have heard the terms electric potential, potential difference, and voltage in connection with electricity. First of all, let me state that these three terms are synonyms and can therefore be used interchangeably. The term electric potential is somewhat unfortunate because it is easily confused for potential energy. While potential energy and electric potential are intimately related, they are not the same thing, as we shall see. Then, the next question is, what is electric potential? Electric potential is the effect produced in the space surrounding an electric charge that causes forces on other electric charges. Sound familiar? It should, it is the same definition that we used for the electric field. In fact, the electric field and electric potential are exactly the same physical thing. The electric field and potential are two different mathematical ways of describing the same physical thing, namely this photon cloud that is generated by the presence of an electric charge. The electric field is the vector description of this photon cloud and the electric potential is the scalar description of the same photon cloud. You could describe an elephant in English, French or Swahili, but the language does not matter. In the end, you are still talking about a large, gray pachyderm. Here, the mathematical language of vectors or scalars is irrelevant. We are still talking about the photon cloud. Again, as scientists, we wish to model the electric potential mathematically. The potential difference between two points in an electric field is a measure of the change of potential energy of the charge in moving from one point to the other. U V (Definition of electric potential) q Comparing this to the definition of the electric field, we see similarity. It requires two charges, Q and q, for there to be potential energy, but only one charge, Q, for there to be electric potential. This is explained by the fact that one charge sitting by itself will generate a photon cloud. But, if you want to bring in another charge to be close to the first charge, you have to do work to get them together if they are like charges, or have work done on you if they are opposite charges. This work has to go somewhere and shows up as potential energy stored with the charges. Another analogy for how electric potential describes electrical phenomenon is useful. Hikers use topographical maps to find their way around. On these maps, lines of equal elevation called contour lines are provided so hikers know how the elevation in a given area changes. So, if

5 Page5 you wanted to plan a hike on a particular mountain, the map would easily tell you the shape of the mountain by the way the contour lines are drawn. If the lines are closer together, the mountain is steeper there. If the lines are far apart, that part of the mountain will be flatter. Similar to lines of equal elevation, lines of equal voltage are called equipotential surfaces or simply equipotentials. Equipotentials are used to describe the shape of the photon cloud like the lines of equal elevation describe the shape of the mountain. Notice that equipotential lines cannot cross each other. By analogy, if lines of equal elevation crossed, the map would be worthless because you wouldn t know whether the particular spot where the lines crossed was at an elevation of 1000 feet or 2000 feet above sea level.

6 Page6 The electric field also fits into this analogy. Since the electric field is the same thing as electric potential, there should be some way to describe the shape of a mountain with something like an electric field. While hikers do not use this idea, you could specify the shape of a mountain by indicating the downhill direction at every point on the mountainside with an arrow or vector. The longer the arrow, the steeper the descent is. This is exactly what the electric field does--it points in the direction of greatest decrease (i.e. downhill) of the electric potential. If the potential does not change, this would correspond to a flat plain rather than a mountain and the electric field would be zero because there is no downhill direction. Notice that the electric field must be perpendicular to the equipotentials. If you were to follow the lines of equal elevation (equipotentials), then you would never change elevation (voltage). If you want to go the steepest direction downhill, you would have to go perpendicular to the elevation lines. Therefore, there will be a potential difference between neighboring points only if there is a component of electric field in the direction from one point to the other. Since the electric field and electric potential describe the same thing, we should be able to switch between them mathematically. In fact, we can by exploiting the following relationships. To go from an electric field description to a potential description, we use: V b E ds (Relationship between electric field and potential) a Notice that the dot product (1) turns the vector electric field on the right-hand side into a scalar potential on the left-hand side, and (2) ensures that the electric field is perpendicular to the equipotentials. Or, to reverse the relationship, we would use: E x V x E y V y E z V (Reverse relationship) z We have three equations here to turn the scalar V back into a vector. You should also notice the negative signs. They are there because the electric field points in the downhill (negative) direction or direction of greatest decrease in potential. For those of you in the Algebra based class don t worry about the extensive math used in this description! Concentrate on the analogies. However, for those of you in the calculusbased lecture pay attention to the math it could help bring all those lectures on electric field and electric potential together and make more sense.

7 Page7 Properties of Electric Field Lines So, let s tie what we know about electric fields into this week s lab. Using a power supply, we are going to charge up a conductor and trace out the resulting electric field in the surrounding space. Since we don t have all day, we are only going to sketch out a few representative lines to show us what an electric field actually looks like. These lines are called electric field lines. To help us draw these lines from the data you will collect, here is a summary of what we know about electric field lines: 1.) Electric field lines begin on positive charges (or at infinity) and end on negative charges (or at infinity). 2.) Lines are drawn symmetrically entering or leaving a charge. 3.) The number of lines entering or leaving a charge is proportional to the charge. For example, 25 lines leaving means +5 units of charge and 15 lines entering means 3 units of charge. 4.) When electric field lines are closer together, the field is stronger in that region. 5.) No two electric field lines can cross.

8 Page8 Properties of Equipotentials Similarly, we will need to be able to draw the equipotentials from the data you collect this week. Here is a summary of what we know about equipotentials: 1.) Voltage (potential) is constant on a given equipotential. This is the definition of an equipotential. 2.) Equipotentials are perpendicular to the electric field at every point. 3.) Electric field lines point in the direction of greatest decrease (downhill) in voltage. 4.) The electric field is strongest where the equipotentials are closest. Properties of Conductors We will also be interested in how electric fields and potential is affected by conductors. So here is one last list that may help you with your lab. You should prove all of these at some point in your physics career. But for now, we will just list them. 1.) Conductors have free electrons. 2.) All net charge resides at the surface of a conductor. 3.) The electric field inside a conductor is zero.

9 Page9 4.) The potential inside a conductor is constant. 5.) The electric field just outside a conductor has a magnitude of / o where is the local surface charge density. 6.) The electric field just outside a conductor is perpendicular to the surface. 7.) Charge density is highest at sharp points on the conductor s surface. 8.) The electric field is highest at sharp points on the conductor.