Concept explainers
(a)
The expression for the electric field at point A located at a distance l above the mid-point of the rod.
(a)
Answer to Problem 79PQ
The expression for the electric field at point A located at a distance l above the mid-point of the rod is
Explanation of Solution
Sketch the diagram showing the five charges.
The x component of the electric field is zero based on the geometry.
Write the expression for the y component of the electric field.
Here,
Write the equation for the total electric field.
Conclusion:
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute equations (III), (IV), (V), (VI) and (VII) in equation (II) to find
Thus, the expression for the electric field at point A located at a distance l above the mid-point of the rod is
(b)
The electric field at point A located at a distance l above the mid-point of the rod using the exact expression.
(b)
Answer to Problem 79PQ
The electric field at point A located at a distance l above the mid-point of the rod using the exact expression is
Explanation of Solution
Write the exact expression for the total electric field.
Here,
Conclusion:
Substitute
Thus, the electric field at point A located at a distance l above the mid-point of the rod using the exact expression is
(c)
Compare the approximate result with the exact result.
(c)
Answer to Problem 79PQ
The approximate result is
Explanation of Solution
Find the ratio of the approximate result with the exact result.
Conclusion:
Thus, the approximate result is
Want to see more full solutions like this?
Chapter 24 Solutions
Physics for Scientists and Engineers: Foundations and Connections
- Two rings of radius R-5 cm are 34 cm apart and concentric with a common vertical axis (figure below). Ring 1 carries a uniformly distributed charge Q₁ = 20 nC and ring 2 carries a uniformly distributed charge of Q₂ = -70 nC. T A Ring 2 Ring 1 1. Find the magnitude of the electric field E created by Ring 1 at point A, halfway between the two rings. E₁ = [N/C] What is the direction of the electric. 2. field E, created by Ring 1 at point A. Direction: + 3. Write the expression of the electric filed E₁ created by Ring 1 at point A. E₁ = [N/C] > [N/C] 7. If vector E₁ = [N/C] and vector Ē₂ = [N/C]. Find the net electric filed Enet at point A. Ēnet = [N/C] 8. If a charge q = 5 nC is placed at location A, what would be the force on this charge? Use the answer of question 7.arrow_forwardProblem 3: Imagine you have a very thin plastic hoop of diameter D, which is charged up to total net electrostatic charge Q. The hoop is thin in the sense that the plastic rod of which it is made has a diameter d D. In each case, give the magnitude and direction of the field. You might need to use words (rather than formulae) to express the direction unambiguously. Assume that there are no other charges anywhere!arrow_forwardA rod with a length of l lies on the x-axis as shown in the picture. It carries a total charge of Q. A) Write the integral (you do not need to solve it!) to find the electric field at the point P which lies above the line of charge on the same axis.B) If you compute the integral, you would find that e=kq/(x(x+l)) . Say the rod carries a total charge +8ncof and has a length of 8um. Find the force (as a vector) on a particle carrying a charge of -3ucwhich is located 1.5um from the end of the rod.arrow_forward
- You are helping to design a new electron microscope to investigate the structure of the HIV virus. A new device to position the electron beam consists of a charged circle of conductor. This circle is divided into two half circles separated by a thin insulator so that half of the circle can be charged positively and half can be charged negatively. The electron beam will go through the center of the circle. To complete the design your job is to calculate the electric field in the center of the circle as a function of the amount of positive charge on the half circle, the amount of negative charge on the half circle, and the radius of the circle.arrow_forwardAn infinitely long sheet of charge of width L lies in the xy-plane between x = -L/2 and x =L/2. The surface charge density is n. Derive an expression for the electric field E at height z above the centerline of the sheet. Express your answer in terms of some or all of the variables €0, 7, 7, L, z, and unit vector k. Use the 'unit vector' button to denote unit vectors in your answer. E =arrow_forward. The circular arc shown below carries a charge per unit length λ=λ0cosθ,λ=λ0cosθ, where θθ is measured from the x-axis. What is the electric field at the origin?arrow_forward
- Consider two thin disks, of negligible thickness, of radius R oriented perpendicular to the x axis such that the x axis runs through the center of each disk. The disk centered at x=0 has positive charge density η, and the disk centered at x=a has negative charge density −η, where the charge density is charge per unit area. What is the magnitude E of the electric field at the point on the x axis with x coordinate a/2? Express your answer in terms of η, R, a, and the permittivity of free space ϵ0.arrow_forwardA total charge of Q is uniformly distributed along a line, which extends along the x- axis from x=0 to x=L. What is the electric field due to this line of charge at a point P, which is on the x axis at x=a. Your answer should be a symbolic expression that only depends on the variables k, Q, a, and L. What does your expression reduce to when a≫L (far-field limit)?arrow_forwardConsider a uniformly charged ring in the xy plane, centered at the origin. The ring has radius a and positive charge q distributed evenly along its circumference. What is the direction of the electric field at any point on the z axis? What is the magnitude of the electric field along the positive z axis? Use k in your answer, where k=1/4πϵ0.arrow_forward
- Sections AB and CD of a thin non-conducting ring of radius R are uniformly (with constant linear density) charged with charge + q and −q, respectively. The points ABCD form the vertices of the square. Find the electric field in the center of the ring.arrow_forwardPositive charge is distributed with a uniform linear density 1 along the positive x-axis from r to o, along the positive y-axis from r to o, and along a 90° arc of a circle of radius r, as shown below. Determine the direction and the magnitude of the net electric field due to this charge distribution at point O. Your final answer should be in terms of 2, r, and any necessary constants. + + + + + + + + + + + + + +arrow_forwardYou are working for the summer at a research laboratory. Your research director has devised a scheme for holding small charged particles at fixed positions. The scheme is shown in the figure below. An insulating cylinder of radius a and length L ≫ a is positively charged and carries a uniform volume charge density ρ. A very thin tunnel is drilled through a diameter of the cylinder and two small spheres with charge q are placed in the tunnel. These spheres are represented by the blue dots in the figure. They find equilibrium positions at a distance of r on opposite sides of the axis of the cylinder. Your research director has had great success with this scheme. Determine the specific value of r at which equilibrium exists. (Use the following as necessary: q and ρ.)arrow_forward
- Physics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage Learning