Unit 6: Electricity and Magnetism 1

Student: Ms. Elbein & Ms. Townsend Physics, ______Due Date: Unit 6: Electricity and Magnetism1 Section 6.1: Action at a Distance The Electric Field Concept As children grow, they become very accustomed to contact forces; but an action-at-a-distance force usually surprises them. Seeing two charged balloons repel from a distance or two magnets attract from a distance raises the eyebrow of a child and maybe even causes a chuckle or a "wow." Indeed, an action-at-a-distance or non-contact force is quite unusual. Football players don't run down the field and encounter collision forces from five yards apart. The rear-end collision at a stop sign is not characterized by repulsive forces that act upon the colliding cars at a spatial separation of 10 meters. And (with the exception modern WWF wrestling matches) the fist of one fighter does not act from 12 inches away to cause the forehead of a second fighter to be knocked backwards. Contact forces are quite usual and customary to us. Explaining a contact force that we all feel and experience on a daily basis is not difficult. Non-contact forces require a more difficult explanation. After all, how can one balloon reach across space and pull a second balloon towards it or push it away? The best explanation to this question involves the introduction of the concept of electric field. Action-at-a-distance forces are sometimes referred to as field forces. The concept of a field force is utilized by scientists to explain this rather unusual force phenomenon that occurs in the absence of physical contact. While all masses attract when held some distance apart, charges can either repel or attract when held some distance apart. An alternative to describing this action-at-a-distance effect is to simply suggest that there is something rather strange about the space surrounding a charged object. Any other charged object that is in that space feels the effect of the charge. A charged object creates an electric field - an alteration of the space in the region that surrounds it. Other charges in that field would feel the unusual alteration of the space. Whether a charged object enters that space or not, the electric field exists. Space is altered by the presence of a charged object. Other objects in that space experience the strange and mysterious qualities of the space. The strangeness of the space surrounding a charged object is often experienced firsthand by the use of a Van de Graaff generator. A Van de Graaff generator is a large conducting sphere that acquires a charge as electrons are scuffed off of a rotating belt as it moves past sharp elongated prongs inside the sphere. The buildup of static charge on the Van de Graaff generator is much greater than that on a balloon rubbed with animal fur or an aluminum plate charged by induction. On a dry day, the buildup of charge becomes so great that it can exert influences on charged balloons held some distance away. If you were to walk near a Van de Graaff generator and hold out your hand, you might even notice the hairs on your hand standing up. And if you were to slowly walk near a Van de Graaff generator, your eyebrows might begin to feel quite staticy. The Van de Graaff generator, like any charged object, alters the space surrounding it. Other charged objects entering the space feel the strangeness of that space. Electric forces are exerted upon those charged objects when they enter that space. The Van de Graaff generator is said to create an electric field in the space surrounding it. A Stinky Analogy With a concept such as the electric field, analogies are often appropriate and useful. While the following analogy might be a wee-bit crude, it certainly proves useful in many respects in describing the nature of an electric field. Anyone who has ever walked into a room of an infant with a soiled diaper (as in a poopy diaper) has experienced a stinky field. There is something about the space surrounding

1 Reading adapted from www.physicsclassroom.com. Student: Ms. Elbein & Ms. Townsend Physics, ______Due Date: an infant's soiled diaper that exerts a strange influence upon other people who enter that space. When that little stinker needs a diaper change, you can't help but to notice it. When you walk into a room with such a diaper present, your detectors (i.e., the nose) begin to detect the presence of a stinky field. As you move closer and closer to the infant, the stinky field becomes more and more intense. And of course the worse the diaper, the stronger the stinky field becomes. It's not difficult to imagine that a soiled diaper could exert a smelly influence some distance away that would repel any nose that gets in that area. The diaper has altered the nature of the surrounding space and when your nose gets near, you know it. The stinky diaper has created a stinky field. In the same manner, an electric charge creates an electric field - it has altered the nature of the space surrounding the charge. And if another charge gets near enough, that charge will sense that there is an effect when present in that surrounding space. And electric field is sensed by the detector charge in the same way that a nose senses the stinky field. The strength of the stinky field is dependent upon the distance from the stinky diaper and the amount of stinky in the diaper. And in an analogous manner, the strength of the electric field is dependent upon the amount of charge that creates the field and the distance from the charge.

Section 6.2: Electric Field Intensity All charged objects create an electric field that extends outward into the space that surrounds it. The charge alters that space, causing any other charged object that enters the space to be affected by this field. The strength of the electric field depends on how charged the object creating the field is and on the distance of separation from the charged object. Now, we will investigate electric field from a numerical viewpoint - the electric field strength.

The Force per Charge Ratio Electric field strength is a vector quantity; it has both magnitude and direction. The magnitude of the electric field strength is defined in terms of how it is measured. Let's suppose that an electric charge can be denoted by the symbol Q. This electric charge creates an electric field; since Q is the source of the electric field, we will refer to it as the source charge. The strength of the source charge's electric field could be measured by any other charge placed somewhere in its surroundings. The charge that is used to measure the electric field strength is referred to as a test charge since it is used to test the field strength. The test charge has a quantity of charge denoted by the symbol q. When placed within the electric field, the test charge will experience an electric force - either attractive or repulsive. As is usually the case, this force will be denoted by the symbol F. The magnitude of the electric field is simply defined as the force per charge on the test charge.

If the electric field strength is denoted by the symbol E, then the equation can be rewritten in symbolic form as

The standard metric units on electric field strength arise from its definition. Since electric field is defined as a force per charge, its units would be force units divided by charge units. In this case, the standard metric units are Newton/Coulomb or N/C.

In the above discussion, you will note that two charges are mentioned - the source charge and the test charge. Two charges would always be necessary to encounter a force. In the electric world, it takes two to attract or repel. The equation for electric field strength (E) has one of the two charge quantities listed in it. Since there are two charges involved, a student will have to be ultimately careful to use the correct charge quantity when computing the electric field strength. The symbol q in the equation is the quantity of charge on the test charge (not the source Student: Ms. Elbein & Ms. Townsend Physics, ______Due Date: charge). Recall that the electric field strength is defined in terms of how it is measured or tested; thus, the test charge finds its way into the equation. Electric field is the force per quantity of charge on the test charge.

Another Electric Field Strength Formula The above discussion pertained to defining electric field strength in terms of how it is measured. Now we will investigate a new equation that defines electric field strength in terms of the variables that affect the electric field strength. To do so, we will have to revisit the Coulomb's law equation. Coulomb's law states that the electric force between two charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between their centers. When applied to our two charges - the source charge (Q) and the test charge (q) - the formula for electric force can be written as

If the expression for electric force as given by Coulomb's law is substituted for force in the above E =F/q equation, a new equation can be derived as shown below.

Note that the derivation above shows that the test charge q was canceled from both numerator and denominator of the equation. The new formula for electric field strength (shown inside the box) expresses the field strength in terms of the two variables that affect it. The electric field strength is dependent upon the quantity of charge on the source charge (Q) and the distance of separation (d) from the source charge. An Inverse Square Law Like all formulas in physics, the formulas for electric field strength can be used to algebraically solve physics word problems. And like all formulas, these electric field strength formulas can also be used to guide our thinking about how an alteration of one variable might (or might not) affect another variable. One feature of this electric field strength formula is that it illustrates an inverse square relationship between electric field strength and distance. The strength of an electric field as created by source charge Q is inversely related to square of the distance from the source. This is known as an inverse square law. Electric field strength is location dependent, and its magnitude decreases as the distance from a location to the source increases. And by whatever factor the distance is changed, the electric field strength will change inversely by the square of that factor. So if separation distance increases by a factor of 2, the electric field strength decreases by a factor of 4 (2^2). If the separation distance increases by a factor of 3, the electric field strength decreases by a factor of 9 (3^2). If the separation distance increases by a factor of 4, the electric field strength decreases by a factor of 16 (4^2). And finally, if separation distance decreases by a factor of 2, the electric field strength increases by a factor of 4 (2^2). Student: Ms. Elbein & Ms. Townsend Physics, ______Due Date: The Stinky Field Analogy Revisited Earlier, a somewhat crude yet instructive analogy was presented - the stinky field analogy. The analogy compares the concept of an electric field surrounding a source charge to the stinky field that surrounds an infant's stinky diaper. Just as every stinky diaper creates a stinky field, every electric charge creates an electric field. And if you want to know the strength of the stinky field, you simply use a stinky detector - a nose that (as far as I have experienced) always responds in a repulsive manner to the stinky source. In the same way, if you want to know the strength of an electric field, you simply use a charge detector - a test charge that will respond in an attractive or repulsive manner to the source charge. And of course the strength of the field is proportional to the effect upon the detector. A more sensitive detector (a better nose or a more charged test charge) will sense the effect more intensely. Yet the field strength is defined as the effect (or force) per sensitivity of the detector; so the field strength of a stinky diaper or of an electric charge is not dependent upon the sensitivity of the detector. If you measure the diaper's stinky field, it only makes sense that it would not be affected by how stinky you are. A person measuring the strength of a diaper's stinky field can create their own field, the strength of which is dependent upon how stinky they are. But that person's field is not to be confused with the diaper's stinky field. The diaper's stinky field depends on how stinky the diaper is. In the same way, the strength of a source charge's electric field is dependent upon how charged up the source charge is. Furthermore, just as with the stinky field, our electric field equation shows that as you get closer and closer to the source of the field, the effect becomes greater and greater and the electric field strength increases. The stinky field analogy proves useful in conveying both the concept of an electric field and the mathematics of an electric field. Conceptually, it illustrates how the source of a field can affect the surrounding space and exert influences upon sensitive detectors in that space. And mathematically, it illustrates how the strength of the field is dependent upon the source and the distance from the source and independent of any characteristic having to do with the detector.

The Direction of the Electric Field Vector As mentioned earlier, electric field strength is a vector quantity. Unlike a scalar quantity, a vector quantity is not fully described unless there is a direction associated with it. The magnitude of the electric field vector is calculated as the force per charge on any given test charge located within the electric field. The force on the test charge could be directed either towards the source charge or directly away from it. The precise direction of the force is dependent upon whether the test charge and the source charge have the same type of charge (in which repulsion occurs) or the opposite type of charge (in which attraction occurs). To resolve the dilemma of whether the electric field vector is directed towards or away from the source charge, a convention has been established. The worldwide convention that is used by scientists is to define the direction of the electric field vector as the direction that a positive test charge is pushed or pulled when in the presence of the electric field. By using the convention of a positive test charge, everyone can agree upon the direction of E. Given this convention of a positive test charge, several generalities can be made about the direction of the electric field vector. A positive source charge would create an electric field that would exert a repulsive effect upon a positive test charge. Thus, the electric field vector would always be directed away from positively charged objects. On the other hand, a positive test charge would be attracted to a negative source charge. Therefore, electric field vectors are always directed towards negatively charged objects. Student: Ms. Elbein & Ms. Townsend Physics, ______Due Date: Section 6.3: Electric Field Lines Earlier, the vector nature of the electric field strength was discussed. The magnitude or strength of an electric field in the space surrounding a source charge is related directly to the quantity of charge on the source charge and inversely to the distance from the source charge. The direction of the electric field is always directed in the direction that a positive test charge would be pushed or pulled if placed in the space surrounding the source charge. Since electric field is a vector quantity, it can be represented by a vector arrow. For any given location, the arrows point in the direction of the electric field and their length is proportional to the strength of the electric field at that location. Such vector arrows are shown in the diagram below. Note that the lengths of the arrows are longer when closer to the source charge and shorter when further from the source charge.

A more useful means of visually representing the vector nature of an electric field is through the use of electric field lines of force. Rather than draw countless vector arrows in the space surrounding a source charge, it is perhaps more useful to draw a pattern of several lines that extend between infinity and the source charge. These pattern of lines, sometimes referred to as electric field lines, point in the direction that a positive test charge would accelerate if placed upon the line. As such, the lines are directed away from positively charged source charges and toward negatively charged source charges. To communicate information about the direction of the field, each line must include an arrowhead that points in the appropriate direction. An electric field line pattern could include an infinite number of lines. Because drawing such large quantities of lines tends to decrease the readability of the patterns, the number of lines is usually limited. The presence of a few lines around a charge is typically sufficient to convey the nature of the electric field in the space surrounding the lines.

Rules for Drawing Electric Field Patterns There are a variety of conventions and rules to drawing such patterns of electric field lines. The conventions are simply established in order that electric field line patterns communicate the greatest amount of information about the nature of the electric field surrounding a charged object. One common convention is to surround more charged objects by more lines. Objects with greater charge create stronger electric fields. By surrounding a highly charged object with more lines, one can communicate the strength of an electric field in the space surrounding a charged object by the line density. This convention is depicted in the diagram below. Student: Ms. Elbein & Ms. Townsend Physics, ______Due Date:

Not only does the density of lines surrounding any given object reveal information about the quantity of charge on the source charge, the density of lines at a specific location in space reveals information about the strength of the field at that location. Consider the object shown at the right. Two different circular cross-sections are drawn at different distances from the source charge. These cross-sections represent regions of space closer to and further from the source charge. The field lines are closer together in the regions of space closest to the charge; and they are spread further apart in the regions of space furthest from the charge. Based on the convention concerning line density, one would reason that the electric field is greatest at locations closest to the surface of the charge and least at locations further from the surface of the charge. Line density in an electric field line pattern reveals information about the strength or magnitude of an electric field. A second rule for drawing electric field lines involves drawing the lines of force perpendicular to the surfaces of objects at the locations where the lines connect to object's surfaces. At the surface of both symmetrically shaped and irregularly shaped objects, there is never a component of electric force that is directed parallel to the surface. The electric force, and thus the electric field, is always directed perpendicular to the surface of an object. If there were ever any component of force parallel to the surface, then any excess charge residing upon the surface of a source charge would begin to accelerate. This would lead to the occurrence of an electric current within the object; this is never observed in static electricity. Once a line of force leaves the surface of an object, it will often alter its direction. This occurs when drawing electric field lines for configurations of two or more charges as discussed in the section below. A final rule for drawing electric field lines involves the intersection of lines. Electric field lines should never cross. This is particularly important (and tempting to break) when drawing electric field lines for situations involving a configuration of charges (as in the section below). If electric field lines were ever allowed to cross each other at a given location, then you might be able to imagine the results. Electric field lines reveal information about the direction (and the strength) of an electric field within a region of space. If the lines cross each other at a given location, then there must be two distinctly different values of electric field with their own individual direction at that given location. This could never be the case. Every single location in space has its own electric field strength and direction associated with it. Consequently, the lines representing the field cannot cross each other at any given location in space.

Electric Field Lines for Configurations of Two or More Charges In the examples above, we've seen electric field lines for the space surrounding single point charges. But what if a region of space contains more than one point charge? How can the electric field in the space surrounding a configuration of two or more charges be described by electric field lines? Student: Ms. Elbein & Ms. Townsend Physics, ______Due Date: The magnitude and direction of the electric field at each location is simply the vector sum of the electric field vectors for each individual charge. Ultimately, the electric field lines surrounding the configuration of our two charges would begin to emerge. For the limited number of points selected in this location, the beginnings of the electric field line pattern can be seen. This is depicted in the diagram below. Note that for each location, the electric field vectors point tangent to the direction of the electric field lines at any given point.

The construction of electric field lines in this manner is a tedious and cumbersome task. The use of a field plotting computer software program or a lab procedure produces similar results in less time (and with more phun). Whatever the method used to determine the electric field line patterns for a configuration of charges, the general idea is that the pattern is the resultant of the patterns for the individual charges within the configuration. The electric field line patterns for other charge configurations are shown in the diagrams below.

In each of the above diagrams, the individual source charges in the configuration possess the same amount of charge. Having an identical quantity of charge, each source charge has an equal ability to alter the space surrounding it. Subsequently, the pattern is symmetrical in nature and the number of lines emanating from a source charge or extending towards a source charge is the same. This reinforces a principle discussed earlier that stated that the density of lines surrounding any given source charge is proportional to the quantity of charge on that source charge. If the quantity of charge on a source charge is not identical, the pattern will take on an asymmetric nature, as one of the source charges will have a greater ability to alter the electrical nature of the surrounding space. This is depicted in the electric field line patterns below. Student: Ms. Elbein & Ms. Townsend Physics, ______Due Date:

Electric Field Lines as an Invisible Reality It has been emphasized in these readings that the concept of an electric field arose as scientists attempted to explain the action-at-a-distance that occurs between charged objects. The concept of the electric field was first introduced by 19th century physicist Michael Faraday. It was Faraday's perception that the pattern of lines characterizing the electric field represents an invisible reality. Rather than thinking in terms of one charge affecting another charge, Faraday used the concept of a field to propose that a charged object (or a massive object in the case of a gravitational field) affects the space that surrounds it. As another object enters that space, it becomes affected by the field established in that space. Viewed in this manner, a charge is seen to interact with an electric field as opposed to with another charge. To Faraday, the secret to understanding action-at-a-distance is to understand the power of charge-field-charge. A charged object sends its electric field into space, reaching from the "puller to the pullee." Each charge or configuration of charges creates an intricate web of influence in the space surrounding it. While the lines are invisible, the effect is ever so real. So as you practice the exercise of constructing electric field lines around charges or configuration of charges, you are doing more than simply drawing curvy lines. Rather, you are describing the electrified web of space that will draw and repel other charges that enter it.