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A continuous line of charge lies along the x axis, extend- ing from x + x to positive infinity. The line carries a uniform linear charge density A. What are the magni- tude and direction of the electric field at the origin?

A continuous line of charge lies along the x axis, extend- ing from x + x to positive infinity. The line carries a uniform linear charge density A. What are the magni- tude and direction of the electric field at the origin?

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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25. A continuous line of charge lies along the x axis, extend- ing from x = + x, to positive infinity. The line carries a uniform linear charge density Ap. What are the magni- tude and direction of the electric field at the origin? 26. A line of charge starts at x = +xg and extends to posi- tive infinity. If the linear charge density is A = Ap xa /x, determine the electric field at the origin. 27. A uniformly charged ring of radius 10.0 cm has a total charge of 75.0 µC. Find the electric field on the axis of the ring at (a) 1.00 cm, (b) 5.00 cm, (c) 30.0 cm, and (d) 100 cm from the center of the ring. 28. Show that the maximum field strength Emax along the axis of a uniformly charged ring occurs at x = a/V2 (sce Fig. 23.17) and has the value Q/(6v3me,a*). 29. A uniformly charged disk of radius 35.0 cm carries a charge density of 7.90 x 10- C/m?. Calculate the electric field on the axis of the disk at (a) 5.00 cm, (b) 10.0 cm, (c) 50.0 cm, and (d) 200 cm from the cen- •0 Figure P23.33 ter of the disk. 30. Example 23.9 derives the exact expression for the elec- tric field at a point on the axis of a uniformly charged disk. Consider a disk of radius R = 3.00 cm having a uniformly distributed charge of + 5.20 µC. (a) Using the result of Example 23.9, compute the electric field at a point on the axis and 3.00 mm from the center. Com- pare this answer with the field computed from the near- field approximation E = 0/2€. (b) Using the result of Example 23.9, compute the electric field at a point on the axis and 30.0 cm from the center of the disk. Com- dx Figure P23.34 carrying the same charge, which is uniformly distrib- uted through its volume. Use the result of Example 23.9 to find the field it creates at the same point. 35. A thin rod of length € and uniform charge per unit length à lies along the xaxis, as shown in Figure P23.35. (a) Show that the electric field at P, a distance y from the rod, along the perpendicular bisector has no xcom- ponent and is given by E = 2k,A sin 6/y. (b) Using your result to part (a), show that the field of a rod of in- finite length is E = 2k,A/y. (Hint: First calculate the field at Pdue to an element of length dx, which has a charge A dx. Then change variables from x to 6, using the facts that x = y tan 0 and dx = y see? 0 do, and inte- pare this result with the electric field obtained by treat- ing the disk as a +5.20-uC point charge at a distance of 30.0 ст. 31. The electric field along the axis of a uniformly charged disk of radius Rand total charge Qwas calculated in Ex- ample 23.9. Show that the electric field at distances x that are great compared with Rapproaches that of a point charge Q = onR". (Hint: First show that x/(x* + R* )!/2 = (1 + R*/x)-1/2, and use the bino- mial expansion (1 + 8)" 1 + nổ when ô« 1.) 32. A piece of Styrofoam having a mass m carries a net charge of – gand floats above the center of a very large horizontal sheet of plastic that has a uniform charge density on its surface. What is the charge per unit area on the plastic sheet? grate over 0.) WED 33. A uniformly charged insulating rod of length 14.0 cm is bent into the shape of a semicircle, as shown in Figure P23.33. The rod has a total charge of – 7.50 µC. Find the magnitude and direction of the electric field at 0, the center of the semicirele. 34. (a) Consider a uniformly charged right circular cylin- drical shell having total charge Q, radius R, and height h. Determine the electric field at a point a distance d from the right side of the cylinder, as shown in Figure P23.34. (Hint: Use the result of Example 23.8 and treat the cylinder as a collection of ring charges.) (b) Con- sider now a solid cylinder with the same dimensions and dx Figure P23.35 36. Three solid plastic cylinders all have a radius of 2.50 cm and a length of 6.00 cm. One (a) carries charge with
Transcribed Image Text:25. A continuous line of charge lies along the x axis, extend- ing from x = + x, to positive infinity. The line carries a uniform linear charge density Ap. What are the magni- tude and direction of the electric field at the origin? 26. A line of charge starts at x = +xg and extends to posi- tive infinity. If the linear charge density is A = Ap xa /x, determine the electric field at the origin. 27. A uniformly charged ring of radius 10.0 cm has a total charge of 75.0 µC. Find the electric field on the axis of the ring at (a) 1.00 cm, (b) 5.00 cm, (c) 30.0 cm, and (d) 100 cm from the center of the ring. 28. Show that the maximum field strength Emax along the axis of a uniformly charged ring occurs at x = a/V2 (sce Fig. 23.17) and has the value Q/(6v3me,a*). 29. A uniformly charged disk of radius 35.0 cm carries a charge density of 7.90 x 10- C/m?. Calculate the electric field on the axis of the disk at (a) 5.00 cm, (b) 10.0 cm, (c) 50.0 cm, and (d) 200 cm from the cen- •0 Figure P23.33 ter of the disk. 30. Example 23.9 derives the exact expression for the elec- tric field at a point on the axis of a uniformly charged disk. Consider a disk of radius R = 3.00 cm having a uniformly distributed charge of + 5.20 µC. (a) Using the result of Example 23.9, compute the electric field at a point on the axis and 3.00 mm from the center. Com- pare this answer with the field computed from the near- field approximation E = 0/2€. (b) Using the result of Example 23.9, compute the electric field at a point on the axis and 30.0 cm from the center of the disk. Com- dx Figure P23.34 carrying the same charge, which is uniformly distrib- uted through its volume. Use the result of Example 23.9 to find the field it creates at the same point. 35. A thin rod of length € and uniform charge per unit length à lies along the xaxis, as shown in Figure P23.35. (a) Show that the electric field at P, a distance y from the rod, along the perpendicular bisector has no xcom- ponent and is given by E = 2k,A sin 6/y. (b) Using your result to part (a), show that the field of a rod of in- finite length is E = 2k,A/y. (Hint: First calculate the field at Pdue to an element of length dx, which has a charge A dx. Then change variables from x to 6, using the facts that x = y tan 0 and dx = y see? 0 do, and inte- pare this result with the electric field obtained by treat- ing the disk as a +5.20-uC point charge at a distance of 30.0 ст. 31. The electric field along the axis of a uniformly charged disk of radius Rand total charge Qwas calculated in Ex- ample 23.9. Show that the electric field at distances x that are great compared with Rapproaches that of a point charge Q = onR". (Hint: First show that x/(x* + R* )!/2 = (1 + R*/x)-1/2, and use the bino- mial expansion (1 + 8)" 1 + nổ when ô« 1.) 32. A piece of Styrofoam having a mass m carries a net charge of – gand floats above the center of a very large horizontal sheet of plastic that has a uniform charge density on its surface. What is the charge per unit area on the plastic sheet? grate over 0.) WED 33. A uniformly charged insulating rod of length 14.0 cm is bent into the shape of a semicircle, as shown in Figure P23.33. The rod has a total charge of – 7.50 µC. Find the magnitude and direction of the electric field at 0, the center of the semicirele. 34. (a) Consider a uniformly charged right circular cylin- drical shell having total charge Q, radius R, and height h. Determine the electric field at a point a distance d from the right side of the cylinder, as shown in Figure P23.34. (Hint: Use the result of Example 23.8 and treat the cylinder as a collection of ring charges.) (b) Con- sider now a solid cylinder with the same dimensions and dx Figure P23.35 36. Three solid plastic cylinders all have a radius of 2.50 cm and a length of 6.00 cm. One (a) carries charge with
59. Identical thin rods of length 2a carry equal charges, +Q, uniformly distributed along their lengths. The rods lie along the x axis with thcir centers separated by a distance of b> 2a (Fig. P23.59). Show that the magni- tude of the force exerted by the left rod on the right one is given by In 62 - 4a? F
Transcribed Image Text:59. Identical thin rods of length 2a carry equal charges, +Q, uniformly distributed along their lengths. The rods lie along the x axis with thcir centers separated by a distance of b> 2a (Fig. P23.59). Show that the magni- tude of the force exerted by the left rod on the right one is given by In 62 - 4a? F
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