PSS 27.2 The Electric Field of a Continuous Distribution of Charge


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1 Chapter 27 Solutions PSS 27.2 The Electric Field of a Continuous Distribution of Charge Description: Knight ProblemSolving Strategy 27.2 The Electric Field of a Continuous Distribution of Charge is illustrated. (vector applet) Learning Goal: To practice ProblemSolving Strategy 27.2 for continuous charge distribution problems. A straight wire of length has a positive charge distributed along its length. Find the magnitude of the electric field due to the wire at a point located a distance extending from the wire. from one end of the wire along the line PROBLEMSOLVING STRATEGY 27.2 The electric field of a continuous distribution of charge MODEL: Model the distribution as a simple shape, such as a line of charge or a disk of charge. Assume the charge is uniformly distributed. VISUALIZE: For the pictorial representation: 1. Draw a picture and establish a coordinate system. 2. Identify the point P at which you want to calculate the electric field. 3. Divide the total charge into small pieces of charge using shapes for which you already know how to determine. This is often, but not always, a division into point charges. 4. Draw the electric field vector at P for one or two small pieces of charge. This will help you identify distances and angles that need to be calculated. 5. Look for symmetries in the charge distribution that simplify the field. You may conclude that some components of are zero.
2 SOLVE: The mathematical representation is. Use superposition to form an algebraic expression for each of the three components of at point P. (Note that one or more components may be zero.) Let the coordinates of the point remain as variables. Replace the small charge with an equivalent expression involving a charge density and a coordinate, such as, that describes the shape of charge. This is the critical step in making the transition from a sum to an integral because you need a coordinate to serve as the integration variable. Express all angles and distances in terms of the coordinates. Let the sum become an integral. The integration will be over the coordinate variable that is related to. The integration limits for this variable will depend on your choice of coordinate system. Carry out the integration, and simplify the result as much as possible. ASSESS: Check that your result is consistent with any limits for which you know what the field should be. Model No information is given on the cross section of the wire. For simplicity, assume that its diameter is much smaller than the wire's length, and model the wire as a line of charge. Also, assume that the total charge is uniformly distributed along the wire. Note that the point at which you want to calculate the electric field is close to one of the ends of the wire, so you cannot make use of any symmetry to simplify the problem. Instead, you will need to divide the total charge distributed along the wire into many small segments and model each segment as a pointlike charge. Visualize Part A The diagram below is an incomplete pictorial representation of the situation described in this problem: It shows the positively charged wire, a suitable coordinate system with the origin at the left end of the wire, and the point P at which you need to calculate the electric field. The total charge distributed along the wire has been divided into small segments, each of length and charge. For clarity, only the ith segment at point is shown in the diagram. Complete this pictorial representation by drawing the electric field at P due to segment i. Draw the vector starting at P. View Each small segment will contribute to the net electric field at P. By modeling each segment as a point charge and using the principle of superposition, you can make a quick estimate of the direction of the net field at P. To do that, it is helpful to draw the electric field due to one or two more charge
3 net field at P. To do that, it is helpful to draw the electric field due to one or two more charge segments. Part B With reference to the coordinate system used in the previous part, which component, if any, of the electric field due to the total charge is zero at P? both the x and y component the x component the y component neither the x nor the y component Since the point at which you want to calculate lies on a straight line extending from the wire, must be directed along the same line. Therefore, by choosing the x axis to be parallel to the wire, as we did in Part A, we simplified the problem: Now the y component of the field at P is zero and the magnitude of Solve is simply given by the magnitude of its x component. Part C Find C.1, the magnitude of the electric field at point P due to the total charge. How to approach the problem As explained in the strategy above, the mathematical expression for the net field is. Because of the nature of the problem, which deals with a continuous distribution of charge rather than a pointlike distribution, the sum must be treated as an integral. To set up the summation and find what to integrate, follow the steps listed in the strategy. After dividing the total charge into small segments, and modeling each segment as a point charge, as we did in Part A, construct a mathematical expression for, the electric field due to the ith segment. Treat, the x coordinate of the ith segment, as a variable. Express the charge on the i th segment in terms of. Then, let become an integral and apply your knowledge of calculus to evaluate the integral. C.2 Find an expression for the electric field due to segment i Find the magnitude of the electric field at point P due to a segment of charge located at on the x axis.
4 on the x axis. C.2.1 Find the distance between point P and a segment of charge at Assume the origin of the x axis is at the left end of the wire and point P is located at a distance from the other end of the wire. is the length of the wire. What is the distance between point P and a segment of charge located at Express your answer in terms of,, and. on the x axis? Express your answer in terms of some or all of the variables,,, and, and the constants and. C.3 Find an expression for the charge of a small segment Since you will integrate with respect to a coordinate, it is important to express the charge on a short segment of wire,, in terms of its length. Find a mathematical expression for assuming the charge is uniformly distributed along the wire. Recall that the total charge on the wire is and the length of the wire is. C.3.1 Find the linear charge density Assuming the wire, whose length is, has a total charge uniformly distributed along its length, what is the linear charge density of the wire? Express your answer in terms of and.
5 Now use the linear charge density to express the charge on a small segment of wire in terms of its length as. Express your answer in terms of and some or all of the variables,, and. C.4 How to make the transition from a sum to an integral At this point in the problem, you should have found an expression for the magnitude of the electric field due to a small charge segment of length located at along the x axis. Make sure that your expression for contains only the following quantities, besides the electrostatic constant :,,,, and. If this is not the case, go back and make the necessary substitutions (the previous hints may help you do that). Assuming each charge segment has the same length, you can construct an algebraic expression for the magnitude of the net field using the principle of superposition, that is,, where the sum contains as many terms as the number of charge segments. If you assume that there is an infinite number of segments in the wire, will become infinitesimally small and can be replaced with. At this point, you can replace the sum with an integral and drop the subscript i from the variable, which will then become the integration variable. Note that the remaining quantities,, and are constants: They are known quantities given in the problem statement. C.5 Find the limits of integration When you change the sum over charge segments into an integral, the variable of integration will be the coordinate. What should the limits of integration be? 0 to
6 0 to to to 0 to C.6 How to simplify the integral You can simplify the integral by changing the variable of integration from to. Don't forget to adjust the limits of integration appropriately when you make this change. The following formula will also be useful:. Express your answer in terms of some or all of the variables,, and, and the constants and. Assess Part D Imagine that distance is much greater than the length of the wire. Intuitively, what should the magnitude of the electric field at point P be in this case? D.1 Envisioning the limiting case In the limiting case, the length of the wire is not relevant and the wire appears to be a point charge in the distance. What is the magnitude of the electric field due to a point charge? Express your answer in terms of some or all of the variables and, and the constants and. The variable should not appear in your answer.
7 A short wire far away from point P creates a field very similar to the one created by a point charge. Not surprisingly, in the limiting case, the mathematical expression you derived for reduces to that of the electric field due to a point charge. To see this, note that one way of writing the answer you obtained in the Solve step is: When, becomes negligible compared to and. Thus, the denominator. reduces to, as we expected. Visualizing Electric Fields Description: Select the correct drawing of electric field lines for several situations and answer questions about why other choices are incorrect. Then, these ideas are demonstrated with an applet. Learning Goal: To understand the nature of electric fields and how to draw field lines. Electric field lines are a tool used to visualize electric fields. A field line is drawn beginning at a positive charge and ending at a negative charge. Field lines may also appear from the edge of a picture or disappear at the edge of the picture. Such lines are said to begin or end at infinity. The field lines are directed so that the electric field at any point is tangent to the field line at that point. The figure shows two different ways to visualize an electric field. On the left, vectors are drawn at various points to show the direction and magnitude of the electric field. On the right, electric field lines depict the same situation. Notice that, as stated above, the electric field lines are drawn such that their tangents point in the same
8 direction as the electric field vectors on the left. Because of the nature of electric fields, field lines never cross. Also, the vectors shrink as you move away from the charge, and the electric field lines spread out as you move away from the charge. The spacing between electric field lines indicates the strength of the electric field, just as the length of vectors indicates the strength of the electric field. The greater the spacing between field lines, the weaker the electric field. Although the advantage of field lines over field vectors may not be apparent in the case of a single charge, electric field lines present a much less cluttered and more intuitive picture of more complicated charge arrangements. Part A Which of the following figures correctly depicts the field lines from an infinite uniformly negatively charged sheet? Note that the sheet is being viewed edgeon in all pictures. A.1 Description of the field Recall that the field around an infinite charged sheet is always perpendicular to the sheet and that the field strength does not change, regardless of distance from the sheet. A B C D Part B In the diagram from part A
9 figure B? (Pick only those statements that apply to figure B.) Check all that apply. Field lines cannot cross each other. The field lines should be parallel because of the sheet's symmetry., what is wrong with The field lines should spread apart as they leave the sheet to indicate the weakening of the field with distance. The field lines should always end on negative charges or at infinity. Part C Which of the following figures
10 electric field lines for an electric dipole? A B C D shows the correct This applet shows two charges. You can alter the charge on each independently or alter the distance between them. You should try to get a feeling for how altering the charges or the distance affects the field lines. Part D In the diagram from part C
11 figure D? (Pick only those statements that apply to figure D.) Check all that apply. Field lines cannot cross each other., what is wrong with The field lines should turn sharply as you move from one charge to the other. The field lines should be smooth curves. The field lines should always end on negative charges or at infinity. In even relatively simple setups as in the figure, electric field lines are quite helpful for understanding the field qualitatively (understanding the general direction in which a certain charge will move from a specific position, identifying locations where the field is roughly zero or where the field points a specific direction, etc.). A good figure with electric field lines can help you to organize your thoughts as well
12 as check your calculations to see whether they make sense. Part E In the figure, the electric field lines are shown for a system of two point charges, and. Which of the following could represent the magnitudes and signs of and? In the following, take to be a positive quantity.,,,,, Very far from the two charges, the system looks like a single charge with value. At large enough distances, the field lines will be indistinguishable from the field lines due to a single point charge. Charged Ring
13 Description: Find the electric field from a uniformly charged ring (qualitative and quantitative parts) at points along its axis. Then use this to find the frequency of small oscillations of a oppositely charged object placed on the axis. Consider a uniformly charged ring in the xy plane, centered at the origin. The ring has radius positive charge distributed evenly along its circumference. and Part A What is the direction of the electric field at any point on the z axis? A.1 How to approach the problem Approach 1 In what direction is the field due to a point on the ring? Add to this the field from a point on the opposite side of the ring. In what direction is the net field? What if you did this for every pair of points on opposite sides of the ring? Approach 2 Consider a general electric field at a point on the z axis, i.e., one that has a z component as well as a component in the xy plane. Now imagine that you make a copy of the ring and rotate this copy about its axis. As a result of the rotation, the component of the electric field in the xy plane will rotate also. Now you ask a friend to look at both rings. Your friend wouldn't be able to tell them apart, because the ring that is rotated looks just like the one that isn't. However, they have the component of the electric field in the xy plane pointing in different directions! This apparent contradiction can be resolved if this component of the field has a particular value. What is this value? Does a similar argument hold for the z component of the field? parallel to the x axis parallel to the y axis parallel to the z axis
14 parallel to the z axis in a circle parallel to the xy plane Part B What is the magnitude of the electric field along the positive z axis? B.1 Formula for the electric field You can always use Coulomb's law,, to find the electric field (the Coulomb force per unit charge) due to a point charge. Given the force, the electric field say at due to is. In the situation below, you should use Coulomb's law to find the contribution to the electric field at the point from a piece of charge on the ring at a distance away. Then, you can integrate over the ring to find the value of. Consider an infinitesimal piece of the ring with charge. Use Coulomb's law to write the magnitude of the infinitesimal at a point on the positive z axis due to the charge shown in the figure.
15 Use in your answer, where. You may also use some or all of the variables,, and. B.2 Simplifying with symmetry By symmetry, the net field must point along the z axis, away from the ring, because the horizontal component of each contribution of magnitude component of a similar contribution of magnitude is exactly canceled by the horizontal from the other side of the ring. Therefore, all we care about is the z component of each such contribution. What is the component of the electric field caused by the charge on an infinitesimally small portion of the ring direction? in the z Express your answer in terms of, the infinitesimally small contribution to the electric field;, the coordinate of the point on the z axis; and, the radius of the ring.
16 B.3 Integrating around the ring If you combine your results from the first two hints, you will have an expression for, the vertical component of the field due to the infinitesimal charge. The total field is. If you are not comfortable integrating over the ring, change to a spatial variable. Since the total charge is distributed evenly about the ring, convince yourself that. Use in your answer, where. Notice that this expression is valid for both positive and negative charges as well as for points located on the positive and negative z axis. If the charge is positive, the electric field should point outward. For points on the positive z axis, the field points in the positive z direction, which is outward from the origin. For points on the negative z axis, the field points in the negative z direction, which is also outward from the origin. If the charge is negative, the electric field should point toward the origin. For points on the positive z axis, the negative sign from the charge causes the electric field to point in the negative z direction, which points toward the origin. For points on the negative z axis, the negative sign from the z coordinate and the negative sign from the charge cancel, and the field points in the positive z direction, which also points toward the origin. Therefore, even though we obtained the above result for postive and, the algebraic expression is valid for any signs of the parameters. As a check, it is good to see that if is much greater than the magnitude of is approximately, independent of the size of the ring: The field due to the ring is almost the same as that due to a point charge at the origin.
17 Part C Imagine a small metal ball of mass and negative charge. The ball is released from rest at the point and constrained to move along the z axis, with no damping. If, what will be the ball's subsequent trajectory? repelled from the origin attracted toward the origin and coming to rest oscillating along the z axis between and circling around the z axis at Part D The ball will oscillate along the z axis between and in simple harmonic motion. What will be the angular frequency of these oscillations? Use the approximation to simplify your calculation; that is, assume that. D.1 Simple harmonic motion Recall the nature of simple harmonic motion of an object attached to a spring. Newton's second law for the system states that, leading to oscillation at a frequency of (here, the prime on the symbol representing the spring constant is to distinguish it from ). The solution to this differential equation is a sinusoidal function of time with angular frequency. Write an analogous equation for the ball near the charged ring in order to find the term. D.2 Find the force on the charge What is, the z component of the force on the ball on the ball at the point? Use the
18 approximation. D.2.1 A formula for the force on a charge in an electric field The formula for the force on a charge in an electric field is Therfore, in particular,.. You have already found in Part B. Use that expression in the equation above to find an expression for the z component of the force on the ball at the point. Don't forget to use the approximation given. Express your answer in terms of,,,, and. Compare this to the generic force equation for an oscillator to find the oscillation frequency. Express your answer in terms of given charges, dimensions, and constants. Electric Field Vector Drawing Description: Simple conceptual question about determining the electric field vector in a region of space from the change in the path of a charged particle. (vector applet) Each of the four parts of this problem depicts a motion diagram for a charged particle moving through a region of uniform electric field. For each part, draw a vector representing the direction of the electric field. Part A A.1 Relationship between electric field and electric force The relationship between the electric force that acts on a particle and the electric field at the location of the particle is
19 location of the particle is. This formula indicates that the force and the electric field point in the same direction for a positively charged particle, and in opposite directions for a negatively charged particle. A.2 Determining the direction of the electric field The acceleration of the particle can be determined from the change in its velocity. By Newton s 2nd law, the force acting on the particle is parallel to its acceleration. Finally, since this is a positively charged particle, the electric field is parallel to the force. Putting this all together results in an electric field that is parallel to the particle s acceleration. Draw a vector representing the direction of the electric field. The orientation of the vector will be graded. The location and length of the vector will not be graded. View The motion diagram shows that the particle's acceleration points to the right. Because the particle has positive charge, the electric field should point to the right. Part B B.1 Relationship between electric field and electric force The relationship between the electric force that acts on a particle and the electric field at the location of the particle is. This formula indicates that the force and the electric field point in the same direction for a positively charged particle, and in opposite directions for a negatively charged particle. B.2 Determining the direction of the electric field The acceleration of the particle can be determined from the change in its velocity. By Newton s 2nd law, the force acting on the particle is parallel to its acceleration. Finally, since this is a negatively charged particle, the electric field is directed opposite to the force. Putting this all together results in an electric field that is directed opposite to the particle s acceleration. Draw a vector representing the direction of the electric field. The orientation of the vector will be graded. The location and length of the vector will not be graded.
20 View The motion diagram shows that the particle's acceleration points to the right. Because the particle has negative charge, the electric field should point to the left. Part C C.1 Relationship between electric field and electric force The relationship between the electric force that acts on a particle and the electric field at the location of the particle is. This formula indicates that the force and the electric field point in the same direction for a positively charged particle, and in opposite directions for a negatively charged particle. C.2 Determining the direction of the electric field The acceleration of the particle can be determined from the change in its velocity. By Newton s 2nd law, the force acting on the particle is parallel to its acceleration. Finally, since this is a positively charged particle, the electric field is parallel to the force. Putting this all together results in an electric field that is parallel to the particle s acceleration. Because the electric field is uniform, you can find the direction of the particle's acceleration by subtracting any two consecutive velocity vectors graphically. If and are any two consecutive velocities, you can subtract from by placing at the tip of. is the vector that starts at the tail of and ends at the tip of. To find the direction of the particle's acceleration graphically, use two unlabeled vectors to represent and. Pick any two vectors and that would make your subtraction easier; you can verify your result by subtracting any other pair of consecutive vectors. Draw a vector representing the direction of the electric field. The orientation of the vector will be graded. The location and length of the vector will not be graded. View Part D
21 D.1 Relationship between electric field and electric force The relationship between the electric force that acts on a particle and the electric field at the location of the particle is. This formula indicates that the force and the electric field point in the same direction for a positively charged particle, and in opposite directions for a negatively charged particle. D.2 Determining the direction of the electric field The acceleration of the particle can be determined from the change in the illustrated velocity vectors. By Newton s 2nd law, the force acting on the particle is parallel to its acceleration. Finally, since this is a negatively charged particle, the electric field is directed opposite to the electric force. Putting this all together results in an electric field that is directed opposite to the particle s acceleration. Because the electric field is uniform, you can find the direction of the particle's acceleration by subtracting any two consecutive velocity vectors graphically. If and are any two consecutive velocities, you can subtract from by placing at the tip of. is the vector that starts at the tail of and ends at the tip of. To find the direction of the particle's acceleration graphically, use two unlabeled vectors to represent and. Pick any two vectors and that would make your subtraction easier; you can verify your result by subtracting any other pair of consecutive vectors. Draw a vector representing the direction of the electric field. The orientation of the vector will be graded. The location and length of the vector will not be graded. View What About Finite Sheets? Description: The electric field near two oppositely charged finite sheets is calculated and compared to the result for infinite sheets. Then, these ideas are demonstrated with an applet. Frequently in physics, one makes simplifying approximations. A common one in electricity is the notion of infinite charged sheets. This approximation is useful when a problem deals with points whose distance from a finite charged sheet is small compared to the size of the sheet. In this problem, you will look at the electric field from two finite sheets and compare it to the results for infinite sheets to get a better idea of when this approximation is valid. Consider two thin disks, of negligible thickness, of radius that the x axis runs through the center of each disk. oriented perpendicular to the x axis such
22 The disk centered at has positive charge density, and the disk centered at has negative charge density, where the charge density is charge per unit area. Part A What is the magnitude of the electric field at the point on the x axis with x coordinate? A.1 How to approach the problem When calculating the electric field from more than one charge or a continuous charge distribution, one makes use of the superposition principle, which states that the electric field from multiple charges equals the vector sum of the fields from each individual charge. To find the electric field at a point, add the field due to each disk at that point. Be careful of whether the field magnitudes should add or subtract. The easiest way to be sure is to draw a figure with the two disks and arrows for the electric field direction on each side of each disk. If the arrows for the two disks point the same way, then the magnitudes add. If they point in opposite directions, the magnitudes subtract. A.2 The magnitude of the electric field due to a single disk The magnitude is of the electric field along the x axis for a charged disk centered at, where is the radius of the disk, is the charge density on the disk, is the permittivity of free space, and is the x coordinate. Be careful in determining the direction in which the electric field
23 space, and is the x coordinate. Be careful in determining the direction in which the electric field due to each disk points. A.3 Determine the general form of the electric field between the disks Which of the following equations represents the magnitude disks? A.3.1 Determine whether the magnitudes should add or subtract of the electric field between the two Which of the following statements properly describes the directions of the electric fields due to each disk in the region between the two disks? The electric fields due to both disks point to the left. The electric fields due to both disks point to the right. The electric field due to the positive disk points to the left, while the electric field due to the negative disk points to the right. The electric field due to the positive disk points to the right, while the electric field due to the negative disk points to the left. Since the two fields point in the same direction, their magnitudes add.
24 Express your answer in terms of,,, and the permittivity of free space. Notice that as approaches, this expression approaches, the result for two infinite sheets. Also, note that the minimum value of the electric field, which corresponds in this case to the greatest deviation from the result for two infinite sheets, occurs halfway between the disks (i.e., at ). Part B For what value of the ratio of plate radius to separation between the plates does the electric field at the point on the x axis differ by 1 percent from the result for infinite sheets? B.1 Percent difference Recall that the percent difference between two numbers, in this case the electric field from Part A and the electric field due to two infinite sheets, is given by Express your answer to two significant figures.. As mentioned above, this is the point on the x axis where the deviation from the result for two infinite sheets is greatest. A common component of electrical circuits called a capacitor is usually made from two thin charged sheets that are separated by a small distance. In such a capacitor, the ratio is far greater than 50. Based on your result, you can see that the infinite sheet approximation is quite good for a capacitor. This applet shows the electric field lines from a pair of finite plates (viewed edgeon). You can adjust the surface charge density. You can also move the test charge around and increase or decrease its charge to see what sort of force it would experience. Notice that the deviation from uniform electric field only becomes noticeable near the edges of the capacitor plates. Operation of an Inkjet Printer
25 Description: Find the charge that an ink droplet must be given to be deflected by a uniform electric field between two parallel plates. In an inkjet printer, letters and images are created by squirting drops of ink horizontally at a sheet of paper from a rapidly moving nozzle. The pattern on the paper is controlled by an electrostatic valve that determines at each nozzle position whether ink is squirted onto the paper or not. The ink drops have a mass each and leave the nozzle and travel horizontally toward the paper at velocity The drops pass through a charging unit that gives each drop a positive charge by causing it to lose some electrons. The drops then pass between parallel deflecting plates of length 1.50, where there is a uniform vertical electric field with magnitude Part A If a drop is to be deflected a distance by the time it reaches the end of the deflection plate, what magnitude of charge the ink drop is 1000 A.1 How to approach the problem must be given to the drop? Assume that the density of, and ignore the effects of gravity. First, find the amount of time spent by the drop between the plates and use the result to find the acceleration needed for the drop to have a vertical deflection of Next, using Newton's 2nd law, find the force needed for this vertical deflection. Finally, calculate the charge by setting the electric force equal to this required force.
26 A.2 Calculate the time spent between the plates How much time does it take for the ink drop to travel horizontally from the start to the end of the deflection plates? A.2.1 Relating horizontal distance and velocity The ink drop has no acceleration in the horizontal direction. Therefore, the distance traveled by the drop is equal to the product of the horizontal velocity and the time traveled. Use this relation to calculate the time the ink drop spends between the plates. Express your answer in seconds. A.3 Calculate the vertical acceleration Calculate the acceleration needed for the ink drop to be deflected vertically by during its trip between the deflection plates. A.3.1 How to calculate the vertical acceleration Since the ink drop initially has zero velocity in the vertical direction, the vertical deflection is related to the vertical acceleration by. Once you know the amount of time the ink droplet spends between the plates, you can use this equation and solve for the vertical acceleration. Express your answer in meters per second squared. A.4 Calculate the force that must be acting on the drop using Newton's law Calculate the force needed to deflect the ink drop vertically by a distance of A.4.1 How to approach the problem
27 Recall Newton's 2nd law:. Once you know the droplet's mass and its vertical acceleration, you can compute the force needed to deflect the ink droplet vertically. Express your answer in newtons. Now find the charge the ink droplet must have so that it will experience an electric force of this magnitude when in the presence of an electric field of magnitude A.5 Relating the electric force and the electric field By definition, the magnitude of the electric field is the absolute value of the ratio of the magnitude of the force acting on a charged particle in the field to the magnitude of the charge of the particle:. Express your answer numerically in coulombs. Here is something to think about: Is it reasonable to ignore the effect of gravity on the droplet in our calculations? For an average inkjet printer, the magnitude of the acceleration due to the electric field will be over ten times larger than the magnitude of the acceleration due to gravity. However, gravity will still cause a small deflection of the droplet and hence should not be ignored if the accuracy of the placement of the ink droplet is particularly important.
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