Concept explainers
An electric dipole is located along the y axis as shown in Figure P24.48. The magnitude of its electric dipole moment is defined as p = 2aq. (a) At a point P, which is far from the dipole (r ≫ a), show that the electric potential is
(b) Calculate the radial component Er and the perpendicular component Eθ of the associated electric field. Note that Eθ = −(1/r)(∂V/∂θ). Do these results seem reasonable for (c) θ = 90° and 0°? (d) For r = 0? (e) For the dipole arrangement shown in Figure P24.48, express V in terms of Cartesian coordinates using r = (x2 + y2)1/2 and
(f) Using these results and again taking r ≫ a, calculate the field components Ex and Ey.
Figure P24.48
(a)
To show: The electric potential at a point P is
Answer to Problem 48CP
The charge on the insulating sphere is
Explanation of Solution
Given info: The magnitude of the electric dipole moment is
The expression to calculate the total electric potential of the dipole is,
Since,
Substitute
Substitute
Conclusion:
Therefore, the electric potential at a point P is
(b)
The radial component
Answer to Problem 48CP
The radial component is
Explanation of Solution
Given info: The magnitude of the electric dipole moment is
Write the expression of radial component
Substitute
Write the expression to calculate the and perpendicular component
Substitute
Thus, the radial component is
Conclusion:
Therefore, radial component is
(c)
Whether the values of
Answer to Problem 48CP
These values of radial component and perpendicular component of electric field seem reasonable.
Explanation of Solution
Given info: The magnitude of the electric dipole moment is
The radial component of electric filed at point P is,
Substitute
Substitute
The perpendicular component of electric field at point P is,
Substitute
Substitute
Thus, these values of radial component and perpendicular component of electric field seem reasonable.
Conclusion:
Therefore, these values of radial component and perpendicular component of electric field seem reasonable.
(d)
Whether the values of
Answer to Problem 48CP
The value of radial component and perpendicular component of electric field at
Explanation of Solution
Given info: The magnitude of the electric dipole moment is
The magnitude of the electric field between the charges of the dipole is not infinite.
The radial component of electric filed at point P is,
The perpendicular component of electric field at point P is,
The values of
Conclusion:
Therefore, the value of radial component and perpendicular component of electric field at
(e)
The electric potential of dipole in terms of Cartesian coordinates.
Answer to Problem 48CP
The electric potential of dipole in terms of Cartesian coordinates is
Explanation of Solution
Given info: The magnitude of the electric dipole moment is
The electric potential at a point P is,
If
Substitute
Conclusion:
Therefore, the electric potential of dipole in terms of Cartesian coordinates is
(e)
The field component
Answer to Problem 48CP
The field component
Explanation of Solution
Given info: The magnitude of the electric dipole moment is
The x component of the electric field is,
Substitute
The y component of the electric field is,
Substitute
Conclusion:
Therefore, the field component
Want to see more full solutions like this?
Chapter 24 Solutions
Physics for Scientists and Engineers with Modern Physics
Additional Science Textbook Solutions
Lecture- Tutorials for Introductory Astronomy
College Physics
Essential University Physics: Volume 1 (3rd Edition)
College Physics (10th Edition)
Physics for Scientists and Engineers with Modern Physics
- Figure P24.16 shows three charged particles arranged in the xy plane at the coordinates shown, with qA = qB = 3.30 nC and qC = 4.70 nC. What is the electric field due to these particles at the origin? FIGURE P24.16arrow_forwardFind an expression for the magnitude of the electric field at point A mid-way between the two rings of radius R shown in Figure P24.30. The ring on the left has a uniform charge q1 and the ring on the right has a uniform charge q2. The rings are separated by distance d. Assume the positive x axis points to the right, through the center of the rings. FIGURE P24.30 Problems 30 and 31.arrow_forwardFigure P24.17 shows a dipole. If the positive particle has a charge of 35.7 mC and the particles are 2.56 mm apart, what is the electric field at point A located 2.00 mm above the dipoles midpoint?arrow_forward
- In Figure P24.49, a charged particle of mass m = 4.00 g and charge q = 0.250 C is suspended in static equilibrium at the end of an insulating thread that hangs from a very long, charged, thin rod. The thread is 12.0 cm long and makes an angle of 35.0 with the vertical. Determine the linear charge density of the rod. FIGURE P24.49arrow_forwardWhy is the following situation impossible? A solid copper sphere of radius 15.0 cm is in electrostatic equilibrium and carries a charge of 40.0 nC. Figure P24.30 shows the magnitude of the electric field as a function of radial position r measured from the center of the sphere. Figure P24.30arrow_forwardA Figure P23.65 shows two identical conducting spheres, each with charge q, suspended from light strings of length L. If the equilibrium angle the strings make with the vertical is , what is the mass m of the spheres? Figure P23.65arrow_forward
- A uniform electric field given by E=(2.655.35j)105N/C permeates a region of space in which a small negatively charged sphere of mass 1.30 g is suspended by a light cord (Fig. P24.53). The sphere is found to be in equilibrium when the string makes an angle = 23.0. a. What is the charge on the sphere? b. What is the magnitude of the tension in the cord? FIGURE P24.53arrow_forwardA positively charged disk of radius R = 0.0366 m and total charge 56.8 C lies in the xz plane, centered on the y axis (Fig. P24.35). Also centered on the y axis is a charged ring with the same radius as the disk and a total charge of 34.1 C. The ring is a distance d = 0.0050 m above the disk. Determine the electric field at the point P on the y axis, where P is y = 0.0100 m above the origin. FIGURE P24.35 Problems 35 and 36.arrow_forwardA conducting rod carrying a total charge of +9.00 C is bent into a semicircle of radius R = 33.0 cm, with its center of curvature at the origin (Fig.P24.75). The charge density along the rod is given by = 0 sin , where is measured clockwise from the +x axis. What is the magnitude of the electric force on a 1.00-C charged particle placed at the origin?arrow_forward
- Eight small conducting spheres with identical charge q = 2.00 C are placed at the corners of a cube of side d = 0.500 m (Fig. P23.75). What is the total force on the sphere at the origin (sphere A) due to the other seven spheres? Figure P23.75arrow_forwardA circular ring of charge with radius b has total charge q uniformly distributed around it. What is the magnitude of the electric field at the center of the ring? (a) 0 (b) keq/b2 (c) keq2/b2 (d) keq2/b (e) none of those answersarrow_forwardA charged rod is curved so that it is part of a circle of radius R (Fig. P24.32). The excess positive charge Q is uniformly distributed on the rod. Find an expression for the electric field at point A in the plane of the curved rod in terms of the parameters given in the figure.arrow_forward
- Physics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPhysics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage LearningPhysics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
- Principles of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning